\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{color} \AtBeginDocument{{\noindent\small 2007 Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems. {\em Electronic Journal of Differential Equations}, Conference 18 (2010), pp. 67--105.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{67} \title[\hfilneg EJDE-2010/Conf/18/\hfil Spectral analysis of the p-Laplacian] {Variational methods and linearization tools towards the spectral analysis of the p-Laplacian, especially for the Fredholm alternative} \author[P. Tak\'a\v{c}\hfil EJDE/Conf/18 \hfilneg] {Peter Tak\'a\v{c}} \address{Peter Tak\'a\v{c} \newline Institut f\"ur Mathematik, Universit\"at Rostock, D-18055 Rostock, Germany} \email{peter.takac@uni-rostock.de} \dedicatory{These notes transcribe the plenary lecture at the conference\\ and are written for advanced Ph.~D.\ students} \thanks{Published July 10, 2010.} \subjclass[2000]{35J20, 49J35, 35P30, 49R50} \keywords{Nonlinear eigenvalue problem; Fredholm alternative; \hfill\break\indent degenerate or singular quasilinear Dirichlet problem; $p$-Laplacian; global minimizer; \hfill\break\indent minimax principle} \begin{abstract} We look for weak solutions $u\in W_0^{1,p}(\Omega)$ of the degenerate quasi\-linear Dirichlet boundary value problem \begin{equation*} \tag*{\rm (P)} - \Delta_p u = \lambda |u|^{p-2} u + f(x) \quad \text{in } \Omega \,;\qquad u = 0 \quad \text{on } \partial\Omega \,. \end{equation*} It is assumed that $1 \lambda_1$, even in space dimension one ($N=1$). We will restrict ourselves to the case $\lambda = \lambda_1$, the {\it Fredholm alternative for the $p$-Laplacian\/} at the first eigenvalue. Even if the functional $\mathcal{J}_{\lambda_1}$ is no longer coercive on $W_0^{1,p}(\Omega)$, for $p>2$ we will show that it is bounded from below and does possess a global minimizer. For $1 0$ small enough). A crucial ingredient in our proofs are rather {\it precise asymptotic estimates\/} for possible ``large'' solutions to problem~{\rm (P)} obtained from the linearization of problem~{\rm (P)} about the eigenfunction $\varphi_1$. These will be briefly discussed. Naturally, the (linear selfadjoint) Fredholm alternative for the linearization of problem~{\rm (P)} about $\varphi_1$ (with $\lambda = \lambda_1$) appears in the proofs. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \tableofcontents \section{Introduction} \label{s:Intro} In this lecture notes we will combine variational and topological methods to establish various results on \emph{existence\/} (or \emph{nonexistence\/}), \emph{uniqueness\/}, and \emph{multiplicity\/} of solutions corresponding to the \emph{Fredholm alternative\/} for the nonlinear Dirichlet problem % \begin{equation} - \Delta_p u = \lambda |u|^{p-2} u + f(x) \quad\text{in } \Omega \,;\qquad u = 0 \quad\text{on } \partial\Omega \,, \label{e:BVP.l} \end{equation} % with the $p$-Laplacian $\Delta_p$ defined by $\Delta_p u{\stackrel{{\mathrm {def}}}{=}} {\mathop{\rm {div}}} ( |\nabla u|^{p-2} \nabla u )$ for $p\in (1,\infty)$. Here, $\lambda\in {\mathbb{R}}$ is the spectral (control) parameter taking values near the first (smallest) eigenvalue $\lambda_1$ of $-\Delta_p$, % \begin{equation} \lambda_1 = \inf \Big\{ \int_\Omega |\nabla u|^p \,\mathrm{d}x : u\in W_0^{1,p}(\Omega) \text{ with } \int_\Omega |u|^p \,\mathrm{d}x = 1 \Big\} \,, \label{def.lam_1} \end{equation} % and $f\in L^\infty(\Omega)$ is a given function, $f\not\equiv 0$ in~$\Omega$. Using a new variational method introduced in { Tak\'a\v{c}} \cite{Takac-2, Takac-5} we will be able to show a number of results on the solvability of problem \eqref{e:BVP.l} under various conditions on $\lambda$ and~$f$. We assume that $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ ($N\geq 1$) with sufficiently smooth boundary $\partial\Omega$; $C^2$ boundary will do. As far as conditions on $f$ are concerned, we will find it convenient to work with the orthogonal sum $L^2(\Omega) = {\mathop{\rm {lin}}}\{\varphi_1\} \oplus L^2(\Omega)^\top$, where % \begin{equation} L^2(\Omega)^\top {\stackrel{{\mathrm {def}}}{=}} \left\{ f\in L^2(\Omega) : \langle f,\varphi_1 \rangle = 0 \right\} , \label{e:L^2^top} \end{equation} % \begin{math} \langle f, \varphi_1 \rangle {\stackrel{{\mathrm {def}}}{=}} \int_\Omega f\varphi_1 \,\mathrm{d}x , \end{math} % so that $f$ splits as $f = f^\parallel\cdot \varphi_1 + f^\top$, where % \begin{equation} f^\parallel {\stackrel{{\mathrm {def}}}{=}} \|\varphi_1\|_{ L^2(\Omega) }^{-2} \langle f, \varphi_1 \rangle \quad\text{and }\; \langle f^\top, \varphi_1 \rangle = 0 \,. \label{ortho:f} \end{equation} % Here, $\varphi_1$ denotes the eigenfunction associated with $\lambda_1$ which is a simple eigenvalue of $-\Delta_p$, by Anane \cite[Th\'eor\`eme~1, p.~727]{Anane-1} or Lindqvist \cite[Theorem 1.3, p.~157]{Lindqvist}. We normalize $\varphi_1$ by $\varphi_1 > 0$ in $\Omega$ and $\|\varphi_1\|_{ L^p(\Omega) } = 1$. We have $\varphi_1\in L^\infty(\Omega)$ by Anane \cite[Th\'eor\`eme A.1, p.~96]{Anane-2}. More details about $\lambda_1$ and $\varphi_1$ will be presented in Section~\ref{s:lambda_1}. We formulate our solvability conditions on $f$ in terms of $f^\parallel$ and $f^\top$ where the ratio between $| f^\parallel |$ and $\| f^\top\|_{ L^\infty(\Omega) }$ plays a decisive role. For $\lambda$ near $\lambda_1$, say, $|\lambda - \lambda_1|\leq \delta$, this ratio, combined with the signs of $\lambda - \lambda_1$ and $p-2$, determines the \emph{existence\/} (or \emph{nonexistence}) and \emph{multiplicity\/} of weak solutions to problem \eqref{e:BVP.l} in $W_0^{1,p}(\Omega)$. It is not surprising that we look for solutions to \eqref{e:BVP.l} coming also in the form $u = u^\parallel\cdot \varphi_1 + u^\top$ defined in \eqref{ortho:f}. We will establish strong relations between $f^\parallel$ and $u^\parallel$, and $f^\top$ and $u^\top$, as well, for any $1 < p < \infty$, in analogy with the case $p=2$ when these two relations are decoupled by the spectral decomposition of the (linear) Laplace operator $\Delta$ in $L^2(\Omega)$. In accordance with \eqref{e:L^2^top} we write % \begin{gather*} L^\infty(\Omega)^\top {\stackrel{{\mathrm {def}}}{=}} \big\{ u\in L^\infty(\Omega) : \langle u,\varphi_1 \rangle = 0 \big\} \,, \\ W_0^{1,p}(\Omega)^\top {\stackrel{{\mathrm {def}}}{=}} \big\{ u\in W_0^{1,p}(\Omega) : \langle u,\varphi_1 \rangle = 0 \big\} \,, \end{gather*} % and take advantage of the orthogonal sums % \begin{equation*} L^\infty (\Omega) = {\mathop{\rm {lin}}} \{ \varphi_1\} \oplus L^\infty (\Omega)^\top \,,\quad W_0^{1,p}(\Omega) = {\mathop{\rm {lin}}} \{ \varphi_1\} \oplus W_0^{1,p}(\Omega)^\top \,. \end{equation*} % The \emph{main idea\/} in our variational approach is to look for the unknowns $u^\parallel$ and $u^\top$ separately; first for $u^\top\in W_0^{1,p}(\Omega)^\top$ and then for $u^\parallel\in {\mathbb{R}}$ (depending on $u^\top$). More precisely, we look for {\it critical points\/} of the energy functional % \begin{equation} \mathcal{J}_{\lambda}(u) = \frac{1}{p} \int_\Omega |\nabla u|^p \,\mathrm{d}x - \frac{\lambda}{p} \int_\Omega |u|^p \,\mathrm{d}x - \int_\Omega f(x) u \,\mathrm{d}x \,,\quad u\in W_0^{1,p}(\Omega) \,, \label{def.jl} \end{equation} % corresponding to problem \eqref{e:BVP.l} as follows. First, we fix $u^\parallel = \tau\in {\mathbb{R}}$ arbitrary and minimize the restricted energy functional $u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$ over the orthogonal complement $W_0^{1,p}(\Omega)^\top$, thus obtaining a global minimizer $u_\tau^\top \in W_0^{1,p}(\Omega)^\top$, \[ j_{\lambda}(\tau){\stackrel{{\mathrm {def}}}{=}} \mathcal{J}_{\lambda}( \tau\varphi_1 + u_\tau^\top ) \leq \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) \quad\text{for all } u^\top\in W_0^{1,p}(\Omega)^\top \,. \] This is possible provided the functional $u^\top \mapsto \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top )$ is coercive on $W_0^{1,p}(\Omega)^\top$ which, indeed, is the case if $\lambda < \Lambda_{\infty}$. The number % \begin{equation} \Lambda_{\infty}{\stackrel{{\mathrm {def}}}{=}} \inf \big\{ \int_\Omega |\nabla u|^p \,\mathrm{d}x : u\in W_0^{1,p}(\Omega)^\top \text{ with } \int_\Omega |u|^p \,\mathrm{d}x = 1 \big\} \label{def.Lam_infty} \end{equation} % satisfies $\Lambda_{\infty} > \lambda_1$, by the simplicity of eigenvalue $\lambda_1$. Using the uniform convexity of $W_0^{1,p}(\Omega)$, we will be able to show that $j_{\lambda}: {\mathbb{R}}\to {\mathbb{R}}$ is a continuous function. Now we vary $\tau\in {\mathbb{R}}$ and look for local extrema (minima or maxima) of the function $j_{\lambda}(\tau)$ in order to determine possible critical points of the functional $\mathcal{J}_{\lambda}: W_0^{1,p}(\Omega)$ $\to {\mathbb{R}}$. Clearly, if $\tau_0$ is a \emph{local minimizer\/} for $j_{\lambda}$, then also $u_0 = \tau_0\varphi_1 + u_{\tau_0}^\top$ is a local minimizer for $\mathcal{J}_{\lambda}$ and hence a critical point of $\mathcal{J}_{\lambda}$. A more complicated situation occurs if $j_{\lambda}$ has a \emph{local maximum\/} at $\tau_0\in {\mathbb{R}}$. In general, we even do not know if $j_{\lambda}$ is differentiable at $\tau_0$. However, we are still able to show that problem \eqref{e:BVP.l} possesses a pair of \emph{sub-\/} and \emph{super\-solutions\/}, \[ \underline{u} = \tau_0\varphi_1 + \underline{u}^\top \quad\text{and}\quad \overline{u} = \tau_0\varphi_1 + \overline{u}^\top , \] respectively, such that % \begin{gather} \left. \begin{gathered} - \Delta_p \underline{u} = \lambda |\underline{u}|^{p-2} \underline{u} + f(x) + \underline{\zeta}\cdot \varphi_1 \quad\text{in } \Omega ; \\ \underline{u}= 0 \quad\text{on } \partial\Omega , \end{gathered} \right\} \label{e:BVP.l_sub} \\ \left. \begin{gathered} - \Delta_p \overline{u} = \lambda |\overline{u}|^{p-2} \overline{u} + f(x) + \overline{\zeta}\cdot \varphi_1 \quad\text{in } \Omega ; \\ \overline{u}= 0 \quad\text{on } \partial\Omega , \end{gathered} \right\} \label{e:BVP.l_sup} \end{gather} % for some $\underline{\zeta} , \overline{\zeta}\in {\mathbb{R}}$ satisfying $\underline{\zeta}\leq 0\leq \overline{\zeta}$. Fortunately, another method (\cite[Theorem 8.2, p.~448]{DeCoster} or \cite[Lemma 2.4, p.~191]{DrabHolub}), which combines the existence of such a pair of sub- and super\-solutions with the topological (Leray\--Schauder) degree, renders the existence of a (weak) \emph{solution\/} $u = \tau\varphi_1 + u^\top$ of problem \eqref{e:BVP.l} with $\tau$ ``close enough'' to $\tau_0$, relative to the magnitude of $|\tau_0|$. In this way we are able to distinguish the solution $u$ from those other local minima or maxima of $j_{\lambda}$ whose absolute value is of a different order of magnitude than $|\tau_0|$ (meaning either much smaller or much larger) or whose sign is opposite to ${\mathop{\rm {sgn}}}\tau_0$. If the sub- and super\-solutions coincide, then $u_0 = \underline{u} = \overline{u}$ is a critical point of $\mathcal{J}_{\lambda}$ and, moreover, it is an easy exercise to show that $j_{\lambda}$ is differentiable at $\tau_0$ with vanishing derivative. Employing this procedure we are able to obtain multiple solutions of the form $u = \tau\varphi_1 + u^\top$ to problem \eqref{e:BVP.l} which can be distinguished \emph{either\/} by the order of magnitude of $|\tau|$ \emph{or\/} by the sign of parameter $\tau\in {\mathbb{R}}$. The variational method sketched above is somewhat different from { Rabinowitz}' ``Saddle Point Theorem'' \cite[Theorem 4.6, p.~24]{Rabin} applied to the functional $\mathcal{J}_{\lambda}$. We actually work with a ``maximin'' expression, % \begin{equation} \max_{a < \tau < b}\ j_{\lambda}(\tau) = \max_{a < \tau < b}\ \min_{ u^\top\in W_0^{1,p}(\Omega)^\top } \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) \label{e:maximin:a2$ and $\lambda_1 < \lambda\leq \lambda_1 + \delta$ is of special interest, for it features at least \emph{three\/} (pairwise distinct) solutions of problem \eqref{e:BVP.l}: two critical points of the functional $\mathcal{J}_{\lambda}$, which are probably not local minimizers, and a local minimizer ``between'' them. Occasionally, variational proofs provide generalizations of earlier results. For these reasons we feel that the method is worth of being further explored in order to derive existence and multiplicity results for \eqref{e:BVP.l} and related problems, with $f(x)$ replaced by a more general function $f(x,u)$, see suggestions in Section~\ref{s:Discuss}. We will see that problem~\eqref{e:BVP.l} possesses multiple solutions in some cases provided $| \langle f, \varphi_1 \rangle | > 0$ is small enough relative to $\| f\|_{ L^\infty(\Omega) }$: at least two solutions if $\lambda = \lambda_1$, and at least three solutions if $|\lambda - \lambda_1| > 0$ is small enough. We start from the basic case $\langle f, \varphi_1 \rangle = 0$ and $\lambda = \lambda_1$. Our variational method is stable enough for the energy functional $\mathcal{J}_{\lambda}$, so that we can slightly relax the orthogonality condition $\langle f, \varphi_1 \rangle = 0$ (relative to the size of $\| f\|_{ L^\infty(\Omega) }$) and/or the condition $\lambda = \lambda_1$. Multiple solutions to problem~\eqref{e:BVP.l} with $N=1$, any $0 < \lambda < \lambda_1$, and a suitable function $f$ have been constructed earlier in \cite{FHTT} (for $10}), and its impact on the uniqueness of a solution to problem~\eqref{e:BVP.l}. In Section~\ref{s:Minimum} we present a variational method which is based on minimization with constraint. Elementary analysis in Section~\ref{s:Elem_anal} suggests how to obtain critical points for $\mathcal{J}_{\lambda}$. In Section~\ref{s:Topological} we apply the Leray\--Schauder degree to a suitable fixed point mapping in order to establish the existence of a critical point for $\mathcal{J}_{\lambda}$ (which probably is not a local minimizer). Multiple (and ``large'') critical points of $\mathcal{J}_{\lambda}$ are discussed in Section~\ref{s:Asympt}. A collection of main results that can be obtained by the variational and topological methods presented in these notes is given in Section~\ref{s:Main}. Finally, Section~\ref{s:Discuss} is devoted to a brief discussion of the general problem % \begin{equation} - \Delta_p u = \lambda |u|^{p-2} u + f(x,u(x)) \quad\text{ in } \Omega \,;\qquad u = 0 \quad \text{ on } \partial\Omega \,, \label{e:BVP.f_u} \end{equation} % when the function $f(x,u)$ is allowed to depend also on the state variable $u\in {\mathbb{R}}$. The lecture notes conclude with two appendices: Appendix~\ref{s:Appendix_A} with some auxiliary functional\--analytic results from { Tak\'a\v{c}} \cite{Takac-2} and Appendix~\ref{s:Appendix_B} which contains some (mostly highly nontrivial) results from { Dr\'abek} et al.\ \cite[Theorem 4.1, pp.\ 445--446]{DGTU} and { Tak\'a\v{c}} \cite{Takac-2, Takac-3}. \section{The Riesz representation theorem in $L^p(\Omega)$} \label{s:Riesz_Proof} A given (equivalent) norm $\|\cdot\|_X$ on a Banach space $X$ (and, hence, also $X$ itself endowed with this norm) is called {\it uniformly convex\/} if for every ${\varepsilon} > 0$ there exists some $\delta\equiv \delta({\varepsilon}) > 0$ such that, for all $u,v\in X$, one has \[ \| u\|_X ,\ \| v\|_X\leq 1 \quad\text{and}\quad \big\|\frac{u+v}{2}\big\|_X > 1 - \delta \quad\Longrightarrow\quad \| u-v\|_X < {\varepsilon} \,. \] The uniform convexity of the standard norm on $L^p(\Omega)$ for $1 < p < \infty$, \begin{equation} \| u\|_{ L^p(\Omega) }{\stackrel{{\mathrm {def}}}{=}} \Big( \int_\Omega |u(x)|^{p} \,\mathrm{d}x \Big)^{1/p} ,\quad u\in L^{p}(\Omega) , \label{norm:L^p} \end{equation} % is a direct consequence of {\it Clarkson's inequalities\/} (see e.g.\ { Adams} and { Four\-nier} \cite[Theorem 2.38, p.~44]{AdamsFournier} for a proof): \begin{lemma}[Clarkson's inequalities] \label{lem-Clarkson} Let\/ $1 < p < \infty$ and\/ $p'= p/(p-1)$. For\/ $u,v\in L^p(\Omega)$ (real or complex-valued, and vector-valued functions from $[ L^p(\Omega) ]^N$, as well) we have % \begin{itemize} % \item[{\rm (a)}] $\;$ if\/ $2\leq p < \infty$ then % \begin{align} \big\|\frac{u+v}{2}\big\|_{ L^p(\Omega) }^p + \big\|\frac{u-v}{2}\big\|_{ L^p(\Omega) }^p & \leq \frac12 \left( \| u\|_{ L^p(\Omega) }^p + \| v\|_{ L^p(\Omega) }^p \right) \,, \label{ineq-Clarks_p>2} \\ \big\|\frac{u+v}{2}\big\|_{ L^p(\Omega) }^{p'} + \big\|\frac{u-v}{2}\big\|_{ L^p(\Omega) }^{p'} & \geq \Big[ \frac12 \left( \| u\|_{ L^p(\Omega) }^p + \| v\|_{ L^p(\Omega) }^p \right) \Big]^{p'-1} \,; \nonumber \end{align} % \item[{\rm (b)}] $\;$ if\/ $1 < p\leq 2$ then % \begin{align} \big\|\frac{u+v}{2}\big\|_{ L^p(\Omega) }^{p'} + \big\|\frac{u-v}{2}\big\|_{ L^p(\Omega) }^{p'} & \leq \Big[ \frac12 \left( \| u\|_{ L^p(\Omega) }^p + \| v\|_{ L^p(\Omega) }^p \right) \Big]^{p'-1} \,, \label{ineq-Clarks_p<2} \\ \big\|\frac{u+v}{2}\big\|_{ L^p(\Omega) }^p + \big\|\frac{u-v}{2}\big\|_{ L^p(\Omega) }^p & \geq \frac12 \left( \| u\|_{ L^p(\Omega) }^p + \| v\|_{ L^p(\Omega) }^p \right) \,. \nonumber \end{align} % \end{itemize} % \end{lemma} Notice that if $p = p'= 2$, the inequalities above reduce to the parallelogram law in the Hilbert space $L^2(\Omega)$. Clarkson's inequalities \eqref{ineq-Clarks_p>2} (if $2\leq p < \infty$) and \eqref{ineq-Clarks_p<2} (if $1 < p\leq 2$) can be used to give a simple {\it variational proof\/} of the Riesz representation theorem for continuous linear functionals on $L^p(\Omega)$; see e.g.\ the monograph by { Adams} and { Four\-nier} \cite[Theorem 2.44, p.~47]{AdamsFournier}. We are now ready to give a similar proof. Our approach is even more ``variational'' than the one presented in \cite[Proof of Theorem 2.44, pp.\ 46--47]{AdamsFournier}. We apply a standard minimization procedure to the functional % \begin{equation} \mathcal{E}(u)\equiv \mathcal{E}(u;\ell) = \frac{1}{p} \int_\Omega |u|^p \,\mathrm{d}x - {\mathop{\Re\mathfrak{e}}\,}\ell(u) \,,\quad u\in L^p(\Omega) \,, \label{def.E} \end{equation} % where $\ell: L^p(\Omega)\to {\mathbb{R}}$ (or ${\mathbb{C}}$) is a continuous linear functional defined on the real (or complex, respectively) Lebesgue space $L^p(\Omega)$, $1 < p < \infty$, which is given. Hence, we will show that this functional has a global minimizer $\tilde{u}$ in $L^p(\Omega)$; see Lemma~\ref{lem-Riesz_conv} below. Moreover, $\mathcal{E}$ being strictly convex, by Clarkson's inequalities, we can conclude that it possesses a unique critical point. This point will provide us with the desired representation function from the dual space $L^{p'}(\Omega)$, $p'= p/(p-1)$. In other words, we will apply Clarkson's inequalities (Lemma~\ref{lem-Clarkson}) to show \smallskip \noindent\textbf{The Riesz representation theorem.} \begingroup\it Given a continuous linear functional\/ $\ell$ on $L^p(\Omega)$, there exists a unique function $f\in L^{p'}(\Omega)$ such that\/ % \begin{equation} \ell(u) = \int_{\Omega} uf\,\mathrm{d}x \quad\text{holds for all\/ } u\in L^p(\Omega) \,. \label{eq:Riesz} \end{equation} % \endgroup Clearly, the standard complexification procedure for the real Lebesgue space $L_{{\mathbb{R}}}^p(\Omega)$ (over the field ${\mathbb{R}}$) yields the complex Lebesgue space $L_{{\mathbb{C}}}^p(\Omega)$ (over the field ${\mathbb{C}}$), so that % \begin{math} L_{{\mathbb{C}}}^p(\Omega) = L_{{\mathbb{R}}}^p(\Omega)\oplus \mathrm{i} L_{{\mathbb{R}}}^p(\Omega) \end{math} % is a direct sum over the field ${\mathbb{R}}$ ($\mathrm{i}^2 = -1$). As a consequence, both, the real and imaginary parts of the continuous linear functional $\ell$ are also continuous linear functionals on the direct sum $L_{{\mathbb{R}}}^p(\Omega)\oplus \mathrm{i} L_{{\mathbb{R}}}^p(\Omega)$. Hence, it suffices to verify the Riesz representation theorem for the real Lebesgue space $L^p(\Omega) = L_{{\mathbb{R}}}^p(\Omega)$ and $\ell: L_{{\mathbb{R}}}^p(\Omega)\to {\mathbb{R}}$. To this end, observe that if $f\in L^{p'}(\Omega)$ satisfies \eqref{eq:Riesz} then $v = |f|^{p'-2} f$ is a critical point of the functional % \begin{equation*} \mathcal{E}(u) = \frac{1}{p} \int_\Omega |u|^p \,\mathrm{d}x - \ell(u) \,,\quad u\in L^p(\Omega) \,. %\label{def.E_real} \end{equation*} % Vice versa, if $v\in L^p(\Omega)$ is a critical point of $\mathcal{E}$ then $f = |v|^{p-2} v$ satisfies \eqref{eq:Riesz}. Notice that the Fr\'echet derivative $\mathcal{E}'(v): L^p(\Omega)\to {\mathbb{R}}$ of the functional $\mathcal{E}$ at a given point $v\in L^p(\Omega)$ is given by the formula % \begin{equation*} \mathcal{E}'(v) u = \int_\Omega u\, |v|^{p-2} v\,\mathrm{d}x - \ell(u) \,,\quad u\in L^p(\Omega) \,. \end{equation*} % It remains to establish the following lemma which is valid in both, the real and complex versions of the Lebesgue space $L^p(\Omega)$ (over the field ${\mathbb{R}}$ or ${\mathbb{C}}$), that is, also for the functional $\mathcal{E}$ defined in \eqref{def.E}. \begin{lemma}\label{lem-Riesz_conv} The functional\/ $\mathcal{E}: L^p(\Omega)\to {\mathbb{R}}$ defined in \eqref{def.E} is continuous, strictly convex, and coercive on $L^p(\Omega)$, $1 < p < \infty$. Moreover, every minimizing sequence for $\mathcal{E}$ converges \emph{strongly\/} in $L^p(\Omega)$ to the (unique) global minimizer\/ $\tilde{u}$ of\/ $\mathcal{E}$. % \end{lemma} \begin{proof} The continuity and coercivity of $\mathcal{E}$ are obvious. As mentioned above, $\mathcal{E}$ is strictly convex, by Clarkson's inequalities \eqref{ineq-Clarks_p>2} (if $2\leq p < \infty$) and \eqref{ineq-Clarks_p<2} (if $1 < p\leq 2$). Set $\mu{\stackrel{{\mathrm {def}}}{=}} \inf_{ u\in L^p(\Omega) } \mathcal{E}(u)$ and consider any minimizing sequence $\{ u_n\}_{n=1}^\infty \subset L^p(\Omega)$ for $\mathcal{E}$, i.e., $\mathcal{E}(u_n)\to \mu$ as $n\to \infty$. We wish to show that $u_n\to \tilde{u}$ strongly in $L^p(\Omega)$ as $n\to \infty$. Since the space $L^p(\Omega)$ is complete, this is equivalent to $\{ u_n\}_{n=1}^\infty$ being a Cauchy sequence, i.e., % \begin{equation} \lim_{m,n\to \infty} \| u_n - u_m\|_{ L^p(\Omega) } \to 0 \quad\text{as } m,n\to \infty \,. \label{e:u_n-Cauchy} \end{equation} % The last claim follows from the facts that % \begin{eqnarray} & \mathcal{E}\left( \frac{1}{2}(u_n + u_m) \right) \geq \mu \quad\text{and}\quad \ell\left( \frac{1}{2}(u_n + u_m) \right) = \frac{1}{2} \left( \ell(u_n) + \ell(u_m) \right) \label{e:(u_n+u_m)/2} \end{eqnarray} % combined with Clarkson's inequalities \eqref{ineq-Clarks_p>2} and \eqref{ineq-Clarks_p<2} which force \eqref{e:u_n-Cauchy}, thanks to $\mathcal{E}(u_n)\to \mu$ as $n\to \infty$. To see this, for $p\geq 2$ one applies inequality \eqref{ineq-Clarks_p>2} directly to the functional $\mathcal{E}$ as follows: % \begin{align*} \mu + \frac{1}{p}\, \big\|\frac{u_n - u_m}{2}\big\|_{ L^p(\Omega) }^p & \leq \mathcal{E} \big(\frac{u_n + u_m}{2}\big) + \frac{1}{p}\, \big\|\frac{u_n - u_m}{2}\big\|_{ L^p(\Omega) }^p \\ & \leq \frac12 \Big( \mathcal{E}(u_n) + \mathcal{E}(u_m) \Big) \,. \end{align*} % Using $\mathcal{E}(u_n)\to \mu$ as $n\to \infty$ we arrive at \eqref{e:u_n-Cauchy}. For $12} (if $2\leq p < \infty$) and \eqref{ineq-Clarks_p<2} (if $1 < p\leq 2$) to obtain \eqref{e:u_n-Cauchy}. \end{proof} \section{Notation and basic hypotheses} \label{s:Notations} All Banach and Hilbert spaces used in these lecture notes are real. We work with the standard inner product in $L^2(\Omega)$ defined by $\langle u,v \rangle{\stackrel{{\mathrm {def}}}{=}} \int_\Omega uv \,\mathrm{d}x$ for $u,v\in L^2(\Omega)$. The orthogonal complement in $L^2(\Omega)$ of a set $\mathcal{M}\subset L^2(\Omega)$ is denoted by $\mathcal{M}^{\perp, L^2}$, \[ \mathcal{M}^{\perp, L^2} {\stackrel{{\mathrm {def}}}{=}} \{ u\in L^2(\Omega): \langle u,v \rangle = 0 \text{ for all } v\in \mathcal{M} \} . \] The inner product $\langle \,\cdot\, ,\,\cdot\, \rangle$ in $L^2(\Omega)$ induces a duality between the Lebesgue spaces $L^p(\Omega)$ and $L^{p'}(\Omega)$, where $1\leq p < \infty$ and $1 < p'\leq \infty$ with $\frac{1}{p} + \frac{1}{p'} = 1$, and between the Sobolev space $W_0^{1,p}(\Omega)$ and its (strong) dual space $W^{-1,p'}(\Omega)$, as well. We keep the same notation also for the duality between the Cartesian products $[ L^p(\Omega) ]^N$ and $[ L^{p'}(\Omega) ]^N$. The closure, interior, and boundary of a set $S\subset {\mathbb{R}}^N$ are denoted by $\overline{S}$, ${\mathop{\rm {int}}}(S)$, and $\partial S$, respectively, and the characteristic function of $S$ by $\chi_S: {\mathbb{R}}^N\to \{ 0,1\}$. We write $|S|_N{\stackrel{{\mathrm {def}}}{=}} \int_{{\mathbb{R}}^N} \chi_S(x) \,\mathrm{d}x$ if $S$ is also Lebesgue measurable. We always assume the following hypothesis: \begin{enumerate} \renewcommand{\labelenumi}{(H\arabic{enumi})} % \item[{\bf (H1)}] \makeatletter \def\@currentlabel{H1}\label{hyp:Omega} \makeatother % If $N\geq 2$ then $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ whose boundary $\partial\Omega$ is a compact manifold of class $C^{1,\alpha}$ for some $\alpha\in (0,1)$, and $\Omega$ satisfies also the interior sphere condition at every point of $\partial\Omega$. If $N=1$ then $\Omega$ is a bounded open interval in ${\mathbb{R}}^1$. % \end{enumerate} % For $N\geq 2$, \eqref{hyp:Omega} is satisfied if $\Omega\subset {\mathbb{R}}^N$ is a bounded domain with $C^2$ boundary. \section{The energy functional} \label{s:Energy} Recall that the energy functional defined in \eqref{def.jl}, % \begin{equation} \mathcal{J}_{\lambda}(u)\equiv \mathcal{J}_{\lambda}(u;f) = \frac{1}{p} \int_\Omega |\nabla u|^p \,\mathrm{d}x - \frac{\lambda}{p} \int_\Omega |u|^p \,\mathrm{d}x - \int_\Omega f(x) u \,\mathrm{d}x \nonumber \tag{\color{green}\ref{def.jl}\color{black}} \end{equation} % for $u\in W_0^{1,p}(\Omega)$, corresponds to problem \eqref{e:BVP.l}. This problem is equivalent to $\mathcal{J}_{\lambda}'(u) = 0$ in $W^{-1,p'}(\Omega)$, where $\mathcal{J}_{\lambda}'(u)$ denotes the (first) Fr\'echet derivative of the functional $\mathcal{J}_{\lambda}$ on $W_0^{1,p}(\Omega)$. If $\lambda < \lambda_1$, the functional $\mathcal{J}_{\lambda}$ is coercive on $W_0^{1,p}(\Omega)$ which means that $\| u\|_{ W_0^{1,p}(\Omega) }$ $\to \infty$ forces $\mathcal{J}_{\lambda}(u) \to +\infty$. Thus, the Sobolev space $W_0^{1,p}(\Omega)$ being reflexive, there exists a weakly convergent (minimizing) sequence $u_n\rightharpoonup u_0$ in $W_0^{1,p}(\Omega)$ as $n\to \infty$, such that \[ \mathcal{J}_{\lambda}(u_n) \to \mathcal{J}_{\lambda}(u_0) = \inf_{ u\in W_0^{1,p}(\Omega) } \mathcal{J}_{\lambda}(u) > -\infty \quad\text{as } n\to \infty \,. \] Notice that this weak convergence implies only % \[ \liminf_{n\to \infty} \int_\Omega |\nabla u_n|^p \,\mathrm{d}x \geq \int_\Omega |\nabla u_0|^p \,\mathrm{d}x \,. \] % Since the Sobolev embedding $W_0^{1,p}(\Omega) \hookrightarrow L^p(\Omega)$ is compact, by the Rellich\--Kondrachov embedding theorem, we have also $u_n\to u_0$ strongly in $L^p(\Omega)$ as $n\to \infty$. We combine these convergence results with the definition of the functional $\mathcal{J}_{\lambda}$ to conclude that we must have also % \[ \int_\Omega |\nabla u_n|^p \,\mathrm{d}x \,\longrightarrow\, \int_\Omega |\nabla u_0|^p \,\mathrm{d}x \quad\text{as } n\to \infty \,. \] % But this and the weak convergence $u_n\rightharpoonup u_0$ in $W_0^{1,p}(\Omega)$ force the strong convergence \[ \| u_n - u\|_{ W_0^{1,p}(\Omega) } = \|\nabla (u_n - u)\|_{ L^p(\Omega) } \to 0 \quad\text{in $W_0^{1,p}(\Omega)$ as $n\to \infty$, } \] owing to the fact that $L^p(\Omega)$ (and similarly $[ L^p(\Omega) ]^N$) is a {\it uniformly convex\/} Banach space, by Clarkson's inequalities (Lemma~\ref{lem-Clarkson}). \begin{lemma}\label{lem-strong_conv} Let\/ $\lambda\in \mathbb{R}$ be arbitrary. If\/ $u_n\rightharpoonup u_0$ weakly in $W_0^{1,p}(\Omega)$ and\/ $\mathcal{J}_{\lambda}(u_n) \to \mathcal{J}_{\lambda}(u_0)$ as\/ $n\to \infty$, then also $u_n\to u_0$ strongly in $W_0^{1,p}(\Omega)$. % \end{lemma} We will use such convergence reasoning often throughout the entire lecture notes. % In what follows we show a few applications of the reasoning from our proof of Lemma \ref{lem-strong_conv}. \section{The first eigenvalue of $-\Delta_p$} \label{s:lambda_1} Consider the Rayleigh quotient \eqref{def.lam_1} for the first (smallest) eigenvalue $\lambda_1$ of $-\Delta_p$, % \begin{equation} \lambda_1 = \inf \Big\{ \frac{ \int_\Omega |\nabla u|^p \,\mathrm{d}x }% { \int_\Omega |u|^p \,\mathrm{d}x } : 0\neq u\in W_0^{1,p}(\Omega) \Big\} \,. \nonumber \tag{\color{green}\ref{def.lam_1}\color{black}} \end{equation} % The embedding $W_0^{1,p}(\Omega) \hookrightarrow L^p(\Omega)$ being compact, there exists a function $\varphi_1\in W_0^{1,p}(\Omega)$ such that % \[ \int_\Omega |\varphi_1|^p \,\mathrm{d}x = 1 \quad\text{and}\quad \int_\Omega |\nabla\varphi_1|^p \,\mathrm{d}x = \lambda_1 \,. \] % Hence, $0 < \lambda_1 < \infty$. Next, we use the polar decomposition $u = u^{+} - u^{-}$ of $u\in L^p(\Omega)$, $u^{+}{\stackrel{{\mathrm {def}}}{=}} \max\{ u,\, 0\}$ and $u^{-}{\stackrel{{\mathrm {def}}}{=}} \max\{ -u,\, 0\}$, with $\nabla u = \nabla u^{+} - \nabla u^{-}$ a.e.\ in $\Omega$ for $u\in W_0^{1,p}(\Omega)$; see, e.g., { Gilbarg} and { Trudinger} \cite[Theorem 7.8, p.~153]{GilbargTrud}. We observe that also the positive and negative parts $\varphi_1^{\pm}$ of $\varphi_1$ must satisfy % \begin{eqnarray*} & \int_\Omega |\nabla\varphi_1^{\pm}|^p \,\mathrm{d}x = \lambda_1\int_\Omega (\varphi_1^{\pm})^p \,\mathrm{d}x \,. \end{eqnarray*} % We will see in the next section, {\S}\ref{ss:Convex:lamda>0}, Theorem~\ref{thm-Uni-EV}, that $\varphi_1\in W_0^{1,p}(\Omega)$ is determined {\it uniquely\/} by the conditions % \begin{itemize} % \item[($\mathrm{i}_{\varphi_1}$)] $\varphi_1\geq 0$ almost everywhere in $\Omega$; % \item[($\mathrm{ii}_{\varphi_1}$)] % $\int_\Omega \varphi_1^p \,\mathrm{d}x = 1$ and $\int_\Omega |\nabla\varphi_1|^p \,\mathrm{d}x = \lambda_1 $. % \end{itemize} % This uniqueness result is due to { Anane} \cite[Th\'eor\`eme~1, p.~727]{Anane-1} and { Lindqvist} \cite[Theorem 1.3, p.~157]{Lindqvist}. Hence, either $\varphi_1^{+}\equiv 0$ or else $\varphi_1^{-}\equiv 0$ in $\Omega$, and so we may assume $\varphi_1\geq 0$ a.e.\ in $\Omega$. It follows that $\lambda_1$ is a {\it simple\/} eigenvalue of the positive Dirichlet $p$-Laplacian $-\Delta_p$ with a nonnegative eigenfunction $\varphi_1$ satisfying % \begin{equation} - \Delta_p \varphi_1 = \lambda_1 |\varphi_1|^{p-2} \varphi_1 \quad\text{in } \Omega ;\qquad \varphi_1 = 0 \quad\text{on } \partial\Omega . \label{e:varphi.l_1} \end{equation} % Using this equation, { Anane} \cite[Th\'eor\`eme A.1, p.~96]{Anane-2} has derived also $\varphi_1\in L^\infty(\Omega)$. \section{Convexity and uniqueness} \label{s:Convexity} Convexity of a function (or functional) is a standard tool for proving the {\it uniqueness\/} of a critical point for that function (or functional), provided the convexity is strict in some sense; cf.\ {\S}\ref{ss:Convex:lamda<0} below. We will see in {\S}\ref{ss:Convex:lamda>0} that if the strictness in the convexity is lost in some sense (partially), so may be the uniqueness of critical points (only partially, as well). The following simple example explains the key tool for this method. \begin{example}\label{exam-Convexity}\rm Let\/ $X$ be a Banach space, $x_0\in X$ a point, and\/ $v\in X\setminus \{ 0\}$ a direction. Consider the straight line $L = \{ x = x_0 + tv\in X: t\in {\mathbb{R}}\}$ in $X$. Given a functional\/ $\Phi: X\to {\mathbb{R}}$ on $X$, consider its restriction $\Phi\vert_L$ to the line $L$ or, equivalently, consider the function $\phi(t) = \Phi(x_0 + tv)$ of the variable $t\in {\mathbb{R}}$. If $\phi$ happens to be convex and differentiable on ${\mathbb{R}}$ then the set of all critical points of $\phi$ coincides with a closed interval in ${\mathbb{R}}$ (which may be empty or unbounded). This interval coincides, in turn, with the set of all global minimizers for $\phi$. If $\phi$ is also strictly convex then it may possess at most one critical point in ${\mathbb{R}}$; this point is the global minimizer for $\phi$. Consequently, if $\Phi$ is convex and G{\^a}teaux\--differentiable on $X$, it suffices to investigate the strict convexity of $\Phi$ on every line $L$ in $X$ in order to determine the set of all critical points for $\Phi$ in $X$. This set is convex and consists precisely of all global minimizers for~$\Phi$; it may be empty or unbounded in~$X$. \end{example} Now let us treat the energy functional defined in \eqref{def.jl}, % \begin{equation} \mathcal{J}_{\lambda}(u)\equiv \mathcal{J}_{\lambda}(u;f) = \frac{1}{p} \int_\Omega |\nabla u|^p \,\mathrm{d}x - \frac{\lambda}{p} \int_\Omega |u|^p \,\mathrm{d}x - \int_\Omega f(x) u \,\mathrm{d}x \nonumber \tag{\color{green}\ref{def.jl}\color{black}} \end{equation} % for $u\in W_0^{1,p}(\Omega)$. Recall that problem \eqref{e:BVP.l} is equivalent to $\mathcal{J}_{\lambda}'(u) = 0$ in $W^{-1,p'}(\Omega)$, where $\mathcal{J}_{\lambda}'(u)$ denotes the (first) Fr\'echet derivative of the functional $\mathcal{J}_{\lambda}$ on $W_0^{1,p}(\Omega)$. From Section~\ref{s:Energy} we know that $\mathcal{J}_{\lambda}$ is coercive on $W_0^{1,p}(\Omega)$ provided $\lambda < \lambda_1$. Hence, if $\lambda < \lambda_1$ then $\mathcal{J}_{\lambda}$ has a global minimizer in $W_0^{1,p}(\Omega)$. We investigate its uniqueness (or multiplicity) in the next two paragraphs, for $\lambda\leq 0$ and $0 < \lambda\leq \lambda_1$, respectively. From the end of Section~\ref{s:Minimum} we recall $j_{\lambda_1}(\tau) = - c\cdot |\tau|^{2-p} + o(|\tau|^{2-p})$ as $\tau\to \pm\infty$, where $c = c(f^\top) > 0$ is a constant depending on $f^\top$ ($f^\top\not\equiv 0$ in $\Omega$). The function $j_{\lambda_1}$ being continuous, it attains a global minimum at $\tau_0\in {\mathbb{R}}$ if $p>2$. It follows that, even if $\mathcal{J}_{\lambda_1}$ is not coercive on $W_0^{1,p}(\Omega)$, for $p>2$ it possesses a global minimizer $u_0 = \tau_0\varphi_1 + u_0^\top$ with some $u_0^\top\in W_0^{1,p}(\Omega)^\top$. \subsection{Convexity for $\lambda\leq 0$ and any $f$} \label{ss:Convex:lamda<0} Let $\lambda\leq 0$ and $f\in W^{-1,p'}(\Omega)$. Then both nonlinear terms on the right\--hand side in \eqref{def.jl} are nonnegative and convex, the first one even strictly convex on $W_0^{1,p}(\Omega)$. The uniqueness of the global minimizer for $\mathcal{J}_{\lambda}$ in $W_0^{1,p}(\Omega)$ now follows from Example~\ref{exam-Convexity} above. Moreover, this global minimizer is the unique critical point for $\mathcal{J}_{\lambda}$. Thus, we have proved the following theorem. \begin{theorem}\label{thm-Convexity} Let\/ $1 < p < \infty$, $-\infty < \lambda\leq 0$, and\/ $f\in W^{-1,p'}(\Omega)$. Then the functional\/ $\mathcal{J}_{\lambda}$ defined in \eqref{def.jl} has a unique critical point in $W_0^{1,p}(\Omega)$. This critical point is its (unique) global minimizer. Moreover, $\mathcal{J}_{\lambda}$ is strictly convex on $W_0^{1,p}(\Omega)$. % \end{theorem} \subsection{Convexity for $0 < \lambda\leq \lambda_1$ and $f\geq 0$} \label{ss:Convex:lamda>0} In this paragraph we restrict ourselves to the special case $0\leq f\in L^\infty(\Omega)$ and investigate the functional $\mathcal{J}_{\lambda}(u)$ for $u\in W_0^{1,p}(\Omega)$, $u > 0$ a.e.\ in $\Omega$. We follow the approach used in { Fleckinger} et al.\ \cite[Sect.~6 (Appendix)]{FHTT}; cf.\ also { Girg} and { Tak\'a\v{c}} \cite[{\S}4.1]{GirgTakac}, { Tak\'a\v{c}} \cite[Sect.~3]{Takac-4}, or { Tak\'a\v{c}}, { Tello}, and { Ulm} \cite[Sect.~2]{TakacUT} for some generalizations. \begin{lemma}\label{lem-Posit} Let\/ $0\leq f\in L^\infty(\Omega)$ and\/ $-\infty < \lambda\leq \lambda_1$. If\/ $\lambda = \lambda_1$, assume also $f\not\equiv 0$ in $\Omega$. Then every critical point for\/ $\mathcal{J}_{\lambda}$ is nonnegative. \end{lemma} \begin{proof} Consider any critical point $u\in W_0^{1,p}(\Omega)$ for $\mathcal{J}_{\lambda}$. On the contrary, assume $u^-\not\equiv 0$ in $\Omega$. Then we have % \begin{align*} 0 & = \langle \mathcal{J}_{\lambda}'(u) ,\, u^-\rangle \\ {} & = \int_\Omega |\nabla u|^{p-2} (\nabla u\cdot \nabla u^-) \,\mathrm{d}x - \lambda \int_\Omega |u|^{p-2} u\, u^- \,\mathrm{d}x - \int_\Omega f(x)\, u^- \,\mathrm{d}x \\ {} & = - \int_\Omega |\nabla u^-|^p \,\mathrm{d}x + \lambda \int_\Omega |u^-|^p \,\mathrm{d}x - \int_\Omega f(x)\, u^- \,\mathrm{d}x \\ {} & \leq (\lambda - \lambda_1) \int_\Omega |u^-|^p \,\mathrm{d}x - \int_\Omega f(x)\, u^- \,\mathrm{d}x \leq 0 \,. \end{align*} % Since $u^-\not\equiv 0$ and $f\geq 0$ in $\Omega$, these inequalities force $\lambda = \lambda_1$, $u^-$ is a minimizer for $\lambda_1$ in eq.~\eqref{def.lam_1}, and $u^-(x) = 0$ almost everywhere in the set $\{ x\in \Omega: f(x) > 0 \}$. On the other hand, from the strong maximum principle of { Tolksdorf} \cite[Prop. 3.2.2, p.~801]{Tolksdorf-1} or { V\'azquez} \cite[Theorem~5, p.~200]{Vazquez} we deduce $u^- > 0$ a.e.\ in $\Omega$. This forces $f = 0$ a.e.\ in $\Omega$, a contradiction. The lemma is proved. \end{proof} We say that a functional $\mathcal{K}$ is \emph{ray\--strictly convex\/} if it is convex on a convex set $\mathcal{C}$ (say, $\mathcal{C}\subset L_\mathrm{loc}^1(\Omega)$), $\mathcal{K}: \mathcal{C}\to \mathbb{R}\cup \{ +\infty\}$, and if $v_1, v_2\in \mathcal{C}$ obey the equality \[ \mathcal{K} ( (1 - \theta) v_1 + \theta v_2 ) = (1 - \theta) \mathcal{K}(v_1) + \theta \mathcal{K}(v_2) < \infty \] for some $\theta\in (0,1)$, then $v_1$ and $v_2$ are linearly dependent. We now define the functional \[ \mathcal{K}(v){\stackrel{{\mathrm {def}}}{=}} \int_{\Omega} |\nabla (v^{1/p})|^p \,\mathrm{d}x = p^{-p} \int_{\Omega} v\, |\nabla (\log v)|^p \,\mathrm{d}x \] for all functions $v: \Omega\to (0,\infty)$ such that $v^{1/p}\in W_0^{1,p}(\Omega)$. More generally, we may replace $\boldsymbol{v}\mapsto |\boldsymbol{v}|^p: {\mathbb{R}}^N\to {\mathbb{R}}_+$ by any continuous, strictly convex function $K: {\mathbb{R}}^N\to {\mathbb{R}}_+$ and define (up to the constant factor $p^{-p}$) \[ \mathcal{K}(v) = \int_{\Omega} v\, K(\nabla (\log v)) \,\mathrm{d}x \in \mathbb{R}\cup \{ +\infty\} \] for all functions $v: \Omega\to (0,\infty)$ such that $v^{1/p}\in W_0^{1,p}(\Omega)$, cf.\ { Girg} and { Tak\'a\v{c}} \cite[{\S}4.1]{GirgTakac}, { Tak\'a\v{c}} \cite[Sect.~3]{Takac-4}, or { Tak\'a\v{c}}, { Tello}, and { Ulm} \cite[Sect.~2]{TakacUT}. (Notice that $\nabla (v^{1/p}) = p^{-1} v^{1/p}\, \nabla (\log v)$.) \begin{lemma}\label{lem-RayConv} The functional\/ $\mathcal{K}$ defined above is ray\--strictly convex. \end{lemma} \begin{proof} For $\theta\in (0,1)$, $u_1 > 0$, $u_2 > 0$ and $\xi_1,\xi_2\in {\mathbb{R}}^N$, we compute %\begin{multline*} \begin{align*} & ( (1-\theta) u_1 + \theta u_2 ) K \left( \frac{(1-\theta)\xi_1 + \theta\xi_2}{(1-\theta) u_1 + \theta u_2} \right) \\ & {} = ( (1 - \theta) u_1 + \theta u_2 ) K \left( \frac{(1 - \theta) u_1}{(1-\theta) u_1 + \theta u_2}\, \frac{\xi_1}{u_1} + \frac{\theta u_2}{(1-\theta) u_1 + \theta u_2}\, \frac{\xi_2}{u_2} \right) \\ & {} \leq (1-\theta) u_1\, K \left( \frac{\xi_1}{u_1} \right) + \theta u_2\, K \left( \frac{\xi_2}{u_2} \right) , \end{align*} %\end{multline*} where equality holds if and only if $\xi_1 / u_1 = \xi_2 / u_2$. Putting $u_i = v_i(x)$ and $\xi_i = \nabla v_i(x)$ for a.e.\ $x\in \Omega$ and $i=1,2$, and integrating the last inequality over $\Omega$, we obtain % \begin{equation} \mathcal{K} ( (1-\theta) v_1 + \theta v_2 ) \leq (1-\theta) {\mathcal{K}}(v_1) + \theta {\mathcal{K}}(v_2) , \label{eqkconv} \end{equation} % where equality holds if and only if $\nabla v_1 / v_1 = \nabla v_2 / v_2$ a.e.\ in $\Omega$. The latter equality is equivalent to $v_1$ and $v_2$ being linearly dependent. \end{proof} %%%%%%%%%%%%%%%%% \par\vskip 10pt We combine Lemmas \ref{lem-Posit} and~\ref{lem-RayConv} to obtain the following theorem: \begin{theorem}\label{thm-ExUniPos} Let\/ $0\leq f\in L^\infty(\Omega)$ with $f\not\equiv 0$ in $\Omega$. If\/ $\lambda\in (0,\lambda_1)$ then the functional\/ $\mathcal{J}_{\lambda}$ possesses a unique critical point $u\in W_0^{1,p}(\Omega)$. This critical point is the (unique) minimizer for\/ $\mathcal{J}_{\lambda}$ and satisfies $u>0$ a.e.\ in $\Omega$. If, in addition, the boundary $\partial\Omega$ of $\Omega$ is of class $C^{1,\alpha}$ for some $\alpha\in (0,1)$, then $u$ satisfies also the \emph{Hopf maximum principle\/} % \begin{equation} u > 0 \quad\text{in } \Omega \quad\text{and}\quad \frac{\partial u}{\partial\nu} < 0 \quad\text{on } \partial\Omega . \label{Hopf.mp} \end{equation} % \end{theorem} \begin{proof} By our assumption $0 < \lambda < \lambda_1$, the functional $\mathcal{J}_{\lambda}$ is coercive on $W_0^{1,p}(\Omega)$. Thus, it has a minimizer (which is a critical point). From Lemma~\ref{lem-Posit} we know that every critical point $u\in W_0^{1,p}(\Omega)$ for $\mathcal{J}_{\lambda}$ must be nonnegative throughout $\Omega$. Moreover, the strong maximum principle of { Tolksdorf} \cite[Prop. 3.2.2, p.~801]{Tolksdorf-1} or { V\'azquez} \cite[Theorem~5, p.~200]{Vazquez} guarantees $u > 0$ a.e.\ in $\Omega$. For such $u$, let us now consider the functional $u\mapsto \mathcal{J}_\lambda (u^{1/p})$ which is strictly convex, by Lemma~\ref{lem-RayConv} combined with our hypotheses on $f$. Therefore, the critical point of this new functional and, consequently, also the critical point of $\mathcal{J}_\lambda$ are unique. Both are minimizers of the corresponding functionals. Finally, assume that $\partial\Omega$ is of class $C^{1,\alpha}$. Then we have $u\in C^{1,\beta} (\overline{\Omega})$ for some $\beta\in (0,\alpha)$, by a regularity result which is due to { DiBenedetto} \cite[Theorem~2, p.~829]{DiBene-1} { and Tolksdorf} \cite[Theorem~1, p.~127]{Tolksdorf-2} (interior regularity, shown independently), and to { Lieberman} \cite[Theorem~1, p.~1203]{Lieberman} (regularity near the boundary). More precisely, in order to apply their regularity result, one needs to invoke the boundedness of $u\in L^\infty(\Omega)$ due to { Anane} \cite[Th\'eor\`eme A.1, p.~96]{Anane-2}. Thus, the Hopf maximum principle \cite[Prop.\ 3.2.1 and 3.2.2, p.~801]{Tolksdorf-1} or \cite[Theorem~5, p.~200]{Vazquez} can be applied to obtain~\eqref{Hopf.mp} as desired. \end{proof} Applying similar arguments as in the proof above, one can verify the following complementary result for $\lambda = \lambda_1$: \begin{theorem}\label{thm-NonExist} Let\/ $0\leq f\in L^\infty(\Omega)$ with $f\not\equiv 0$ in $\Omega$. Then the functional\/ $\mathcal{J}_{\lambda_1}$ possesses \emph{no\/} critical point $u\in W_0^{1,p}(\Omega)$. Furthermore, $\mathcal{J}_{\lambda_1}$ is unbounded from below with % \begin{eqnarray} \label{e:J=-infty} & \mathcal{J}_{\lambda_1}(t\varphi_1) = - t\int_{\Omega} f\varphi_1 \,\mathrm{d}x \;\longrightarrow\; -\infty \quad\text{as }\ t\to +\infty \,. \end{eqnarray} % \end{theorem} \begin{proof} On the contrary, assume that $\mathcal{J}_{\lambda_1}$ has a critical point $u_0\in W_0^{1,p}(\Omega)$. In analogy with the proof of Theorem~\ref{thm-ExUniPos} above, one shows that $u_0\in C^{1,\beta} (\overline{\Omega})$ for some $\beta\in (0,\alpha)$, together with the Hopf maximum principle \eqref{Hopf.mp} for $u_0$. The functional $u\mapsto \mathcal{J}_\lambda (u^{1/p})$ being strictly convex on the cone of all functions $u\in C^1(\overline{\Omega})$ satisfying \eqref{Hopf.mp}, we conclude that $u_0$ is a global minimizer for $\mathcal{J}_{\lambda_1}$ over that cone. However, this conclusion contradicts \eqref{e:J=-infty}. The theorem is proved. \end{proof} \begin{remark}\label{rem-NonUniq} \begingroup\rm If\/ $p\neq 2$, $f\in L^\infty(\Omega)$ changes sign, and\/ $0 < \lambda < \lambda_1$, then the functional\/ $\mathcal{J}_{\lambda}$ may possess two or more distinct critical points in $W_0^{1,p}(\Omega)$. Examples of a suitable function $f$, such that $\mathcal{J}_{\lambda}$ exhibits multiple critical points, have been constructed in { Fleckinger} et al.\ \cite[Example~2, p.~148]{FHTT} for $1 0$ a.e.\ in $\Omega$ and\/ $\int_\Omega \varphi_1^p \,\mathrm{d}x = 1$. If, in addition, the boundary $\partial\Omega$ of\/ $\Omega$ is of class $C^{1,\alpha}$ for some $\alpha\in (0,1)$, then $\varphi_1$ satisfies also the Hopf maximum principle~\eqref{Hopf.mp}. \end{theorem} This theorem is due to { Anane} \cite[Th\'eor\`eme~1, p.~727]{Anane-1} and { Lindqvist} \cite[Theorem 1.3, p.~157]{Lindqvist}. \begin{proof} Let $u$ be any (nontrivial) minimizer for $\lambda_1$. First, suppose that $u$ changes sign in $\Omega$. Then we have % \begin{align*} \lambda_1 & {} = \frac{\int_\Omega (u^+)^p}{\int_\Omega |u|^p}\cdot \frac{\int_\Omega \left| \nabla u^+\right|^p \,\mathrm{d}x}% {\int_\Omega (u^+)^p \,\mathrm{d}x} + \frac{\int_\Omega (u^-)^p}{\int_\Omega |u|^p}\cdot \frac{\int_\Omega \left| \nabla u^-\right|^p \,\mathrm{d}x}% {\int_\Omega (u^-)^p \,\,\mathrm{d}x} \\ & {} \geq \lambda_1 \left( \frac{\int_\Omega (u^+)^p}{\int_\Omega |u|^p} + \frac{\int_\Omega (u^-)^p}{\int_\Omega |u|^p} \right) = \lambda_1 \,. \end{align*} % Consequently, both $u^+$ and $u^-$ are (nontrivial) minimizers for $\lambda_1$. Hence, we must have $u^+ > 0$ and $u^- > 0$ a.e.\ in $\Omega$, by the strong maximum principle \cite[Prop. 3.2.2, p.~801]{Tolksdorf-1} or \cite[Theorem~5, p.~200]{Vazquez}. But this is impossible. We conclude that either $u > 0$ a.e.\ in $\Omega$ or else $u < 0$ a.e.\ in $\Omega$. Second, recall the proof of Theorem~\ref{thm-ExUniPos} above and take there $f\equiv 0$ in $\Omega$. Replacing $u$ by $-u$ if necessary, we may assume that $u > 0$ a.e.\ in $\Omega$. For such $u$, the functional $u\mapsto \mathcal{J}_\lambda (u^{1/p})$ is ray-strictly convex, by Lemma~\ref{lem-RayConv}. Therefore, the critical point of this new functional and, consequently, also the critical point of $\mathcal{J}_\lambda$ are unique up to a constant multiple. Finally, the Hopf maximum principle~\eqref{Hopf.mp} for $\varphi_1$ is proved in the same way as in Theorem~\ref{thm-ExUniPos}. \end{proof} \section{Minimization with constraint} \label{s:Minimum} Recall the orthogonal decomposition % \begin{math} W_0^{1,p}(\Omega) = {\mathop{\rm {lin}}} \{ \varphi_1\} \oplus W_0^{1,p}(\Omega)^\top \end{math} % induced by the inner product in $L^2(\Omega)$. For $u\in W_0^{1,p}(\Omega)$ we write $u = \tau\varphi_1 + u^\top$ with $\tau\in \mathbb{R}$ and $\int_{\Omega} u^\top\varphi_1 \,\mathrm{d}x = 0$. In analogy with the Rayleigh quotient in \eqref{def.lam_1}, we have introduced another quotient in \eqref{def.Lam_infty}, % \begin{equation} \Lambda_{\infty}{\stackrel{{\mathrm {def}}}{=}} \inf \Big\{ \frac{ \int_\Omega |\nabla u^\top|^p \,\mathrm{d}x }% { \int_\Omega |u^\top|^p \,\mathrm{d}x } : 0\neq u\in W_0^{1,p}(\Omega)^\top \Big\} . \nonumber \tag{\color{green}\ref{def.Lam_infty}\color{black}} \end{equation} % Clearly, $\lambda_1 < \Lambda_{\infty} < \infty$, because $\lambda_1$ is a simple eigenvalue of $-\Delta_p$. (If $\Omega$ has reflection symmetry, one can show $\Lambda_{\infty} = \lambda_2$, the second eigenvalue of $-\Delta_p$.) Let $\lambda < \Lambda_{\infty}$ and consider the functional % \begin{math} \mathcal{J}_{\lambda}(u) = \mathcal{J}_{\lambda}(\tau\varphi_1 + u^\top) \end{math} % with $\tau\in {\mathbb{R}}$ being {\it fixed\/} (but arbitrary) and $u^\top\in W_0^{1,p}(\Omega)^\top$ {\it variable}. Then the restricted functional $u^\top \mapsto \mathcal{J}_{\lambda}(\tau\varphi_1 + u^\top)$ is coercive on $W_0^{1,p}(\Omega)^\top$ and thus possesses a global minimizer $u_\tau^\top\in W_0^{1,p}(\Omega)^\top$. Such a global minimizer satisfies the boundary value problem % \begin{equation} \begin{gathered} - \Delta_p( \tau\varphi_1 + u^\top ) = \lambda\, | \tau\varphi_1 + u^\top |^{p-2}( \tau\varphi_1 + u^\top ) + f(x) + \zeta\cdot \varphi_1(x) \quad\text{in } \Omega ; \\ u^\top = 0 \quad\text{on } \partial\Omega ; \\ \langle u^\top, \varphi_1 \rangle = 0 , \end{gathered} \label{tau:BVP.lam} \end{equation} % where $\zeta\in {\mathbb{R}}$ is a Lagrange multiplier (which is unknown). In particular, when investigating the solvability of this problem, without loss of generality we may assume that $f = f^\top\in L^{\infty}(\Omega)^\top$, by simply substituting $\zeta$ for $f^\parallel + \zeta$. Notice that, for $u\equiv \tau \varphi_1 + u^\top$ and $f\equiv \zeta\varphi_1 + f^\top$, with $\tau, \zeta\in {\mathbb{R}}$, $u^\top\in W_0^{1,p}(\Omega)^\top$, and $f^\top\in L^\infty(\Omega)^\top$, we have % \begin{equation} \mathcal{J}_{\lambda}(u;f) = \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ; f^\top ) - \tau\zeta\, \|\varphi_1\|_{ L^2(\Omega) }^2 . \label{J_lam_f:ortho} \end{equation} % It is now clear that it suffices to determine the properties of $\mathcal{J}_{\lambda}$ in the special case $f = f^\top\in L^\infty(\Omega)^\top$. By arguments used in the proof of Lemma \ref{lem-strong_conv} above, we conclude that {\it any\/} minimizing sequence for the restricted functional $u^\top \mapsto \mathcal{J}_{\lambda}(\tau\varphi_1 + u^\top)$ on $W_0^{1,p}(\Omega)^\top$ contains a strongly convergent subsequence $u_{\tau,n}^\top\to u_{\tau}^\top$ in $W_0^{1,p}(\Omega)$ as $n\to \infty$, which converges to a global minimizer $u_{\tau}^\top\in W_0^{1,p}(\Omega)^\top$. Taking advantage of this technique, it is not difficult to see that % \begin{equation} j_{\lambda}(\tau) {\stackrel{{\mathrm {def}}}{=}} \inf \Big\{ \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) : u^\top\in W_0^{1,p}(\Omega)^\top \Big\} = \mathcal{J}_{\lambda}( \tau\varphi_1 + u_{\tau}^\top ) \label{def.j_lam,tau} \end{equation} % is a {\it continuous\/} function of $\tau\in {\mathbb{R}}$; a complete proof is in { Tak\'a\v{c}} \cite[Lemma 7.2, p.~222]{Takac-2}. We notice that eq.~\eqref{tau:BVP.lam} yields % \begin{math} \mathcal{J}_{\lambda}'( \tau\varphi_1 + u^\top ) = \zeta\cdot \varphi_1 \end{math} % in $W^{-1,p'}(\Omega)$, which entails % \begin{align*} \tau\zeta\, \|\varphi_1\|_{ L^2(\Omega) }^2 & = \zeta\, \langle \varphi_1, \tau\varphi_1 + u^\top \rangle = \langle \mathcal{J}_{\lambda}'( \tau\varphi_1 + u^\top ) , \tau\varphi_1 + u^\top \rangle \\ & = p\cdot \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) + (p-1)\, \langle f, \tau\varphi_1 + u^\top \rangle \,. \end{align*} % Thus, if $f = f^\top\in L^\infty(\Omega)^\top$ and if $u_\tau^\top\in W_0^{1,p}(\Omega)^\top$ is any global minimizer for the restricted energy functional $u^\top \mapsto \mathcal{J}_{\lambda}(\tau\varphi_1 + u^\top)$ on $W_0^{1,p}(\Omega)^\top$, then the Lagrange multiplier $\zeta = \zeta_\tau\in {\mathbb{R}}$ from the boundary value problem \eqref{tau:BVP.lam} satisfies % \begin{align*} \tau\zeta_\tau\, \|\varphi_1\|_{ L^2(\Omega) }^2 & = p\cdot \mathcal{J}_{\lambda}( \tau\varphi_1 + u_\tau^\top ) + (p-1)\, \langle f^\top, u_\tau^\top \rangle \\ & = p\cdot j_{\lambda}(\tau) + (p-1)\, \langle f^\top, u_\tau^\top \rangle \,. \end{align*} % Unfortunately, we do {\it not\/} know whether % \begin{eqnarray} & \tau\mapsto \langle f^\top, u_\tau^\top \rangle = \int_\Omega f^\top u_\tau^\top \,\mathrm{d}x : {\mathbb{R}}\to {\mathbb{R}} \label{e:tau.zeta_tau} \end{eqnarray} % is a continuous function of $\tau$ or whether this function is independent from the choice of the global minimizer $u_\tau^\top$ for the restricted energy functional. Consequently, also $\tau\mapsto \tau\zeta_\tau$ might not be continuous. Fortunately, and this fact is very important for us, we will be able to determine the {\it asymptotic behavior\/} of both, the global minimizer $u_\tau^\top$ and the Lagrange multiplier $\zeta_\tau$ rather precisely, depending on $\tau$, as $\tau\to \pm\infty$. Applying these asymptotic estimates to eq.~\eqref{e:tau.zeta_tau} we will easily derive the asymptotic behavior of the (continuous) function $j_{\lambda}(\tau)$ as $\tau\to \pm\infty$. It will depend on whether $1 0$ is a constant depending on $f^\top$. (Of course, the symbol ``$o(\,\cdot\,)$'' means $o(t) / t\to 0$ as either $t\to \pm\infty$ or $t\to 0$ in ${\mathbb{R}}$, $t\neq 0$.) The proof of this result relies on formula \eqref{z_n.asympt} from Proposition~\ref{prop-v_n-t=0} in the Appendix, {\S}\ref{ss:Unif_Bound}. For $p>2$ it follows that the function $j_{\lambda_1}$ possesses a global minimizer $\tau_0\in {\mathbb{R}}$, even though $j_{\lambda_1}$ is not coercive. For $1 0$. In order \emph{to deduce\/} the existence of a solution to $\mathcal{J}_{\lambda}'(u) = 0$ \emph{from\/} the existence of a pair of (strict) sub- and super\-solutions, $\underline{u}$ and $\overline{u}$, we will apply a topological method (Leray\--Schauder degree theory). Needless to say, we will lose all information about the ``geometry'' of the functional $\mathcal{J}_{\lambda}$ near such a critical point. \section{Existence by a topological degree} \label{s:Topological} Let us recall that the first (smallest) eigenvalue $\lambda_1$ of the positive Dirichlet $p$-Lapla\-cian $-\Delta_p$ is simple with the associated eigenfunction $\varphi_1$ normalized by $\varphi_1 > 0$ in $\Omega$ and $\|\varphi_1\|_{ L^p(\Omega) } = 1$, by { Anane} \cite[Th\'eor\`eme~1, p.~727]{Anane-1} or { Lindqvist} \cite[Theorem 1.3, p.~157]{Lindqvist}. We have $\varphi_1\in L^\infty(\Omega)$ by { Anane} \cite[Th\'eor\`eme A.1, p.~96]{Anane-2}. Consequently, recalling hypothesis~\eqref{hyp:Omega}, we get even $\varphi_1\in C^{1,\beta} (\overline{\Omega})$ for some $\beta\in (0,\alpha)$, by a regularity result due to { DiBenedetto} \cite[Theorem~2, p.~829]{DiBene-1} and { Tolksdorf} \cite[Theorem~1, p.~127]{Tolksdorf-2} (interior regularity), and to { Lieberman} \cite[Theorem~1, p.~1203]{Lieberman} (regularity near the boundary). The constant $\beta$ depends solely on $\alpha$, $N$, and~$p$. We keep the meaning of the constants $\alpha$ and $\beta$ throughout the entire lecture notes. Finally, the Hopf maximum principle (see { Tolksdorf} \cite[Prop.\ 3.2.1 and 3.2.2, p.~801]{Tolksdorf-1} or { V\'azquez} \cite[Theorem~5, p.~200]{Vazquez}) renders % \begin{equation} \varphi_1 > 0 \quad\text{in } \Omega \quad\text{and}\quad \frac{\partial\varphi_1}{\partial\nu} < 0 \quad\text{on } \partial\Omega \,. \label{Hopf.varphi_1} \end{equation} % As usual, ${\partial}/{\partial\nu}$ denotes the outer normal derivative on $\partial\Omega$. We set \[ U{\stackrel{{\mathrm {def}}}{=}} \{ x\in \Omega: \nabla\varphi_1(x) \not= \mathbf{0} \} , \quad\text{hence }\; \Omega\setminus U = \{ x\in \Omega: \nabla\varphi_1(x) = \mathbf{0} \} , \] and observe that $\Omega\setminus U$ is a compact subset of $\Omega$, by~\eqref{Hopf.varphi_1}. As we have already mentioned at the end of Section~\ref{s:Minimum}, if $f = f^\top\in L^\infty(\Omega)$, $f^\top\not\equiv 0$ in $\Omega$, then for $11$ and any $\lambda < \Lambda_{\infty}$; see { Tak\'a\v{c}} \cite{Takac-5}. So fix $1 0$ is small enough, a ``saddle point theorem'' from { Rabinowitz} \cite[Theorem 4.6, p.~24]{Rabin} guarantees the existence of a critical point $u_0\in W_0^{1,p}(\Omega)$ for $\mathcal{J}_{\lambda_1}$. If $\zeta = 0$, the validity of the Palais\--Smale condition for $\mathcal{J}_{\lambda_1}$ is still an open question and, as indicated in { Dr\'abek} and { Tak\'a\v{c}} \cite{DrabTakac}, it might not be satisfied at all; see \cite[Theorem 4.1, pp.\ 47--48]{DrabTakac} for difficulties and hints to ramifications. This means that we will not be able to apply the saddle point theorem \cite[Theorem 4.6, p.~24]{Rabin} to treat the most natural case $\zeta = 0$. Taking $|\zeta| > 0$ small enough one can apply this theorem to construct only a pair of strict sub- and super\-solutions, $\underline{u}$ and $\overline{u}$, as in the previous section (Section~\ref{s:Elem_anal}). Notice that the construction in the previous section does not require $f^\top$ to be continuous in a an open neighborhood of the (compact) set $\Omega\setminus U$ defined above. Besides, it gives % \begin{equation} \langle \underline{u}, \varphi_1\rangle = \langle \overline{u}, \varphi_1\rangle = \tau_0\, \|\varphi_1\|_{ L^2(\Omega) }^2 \quad\text{with some } \tau_0\in {\mathbb{R}} . \label{eq:sub=super:tau_0} \end{equation} % However, it uses the asymptotic behavior $j_{\lambda_1}(\tau)\to -\infty$ as $\tau\to \pm\infty$ which is not easy to establish (except for the case when $f^\top$ is continuous in an open neighborhood of $\Omega\setminus U$). We will therefore use a topological (hence, nonvariational) method to handle this situation. It has been introduced in { De Coster} and { Henrard} \cite[Theorem 8.2, p.~448]{DeCoster} for semilinear elliptic boundary value problems. Given a pair of {\it unordered\/} sub- and super\-solutions, a fixed point mapping is constructed first and then its Leray\--Schauder degree is computed. We follow the presentation given in { Tak\'a\v{c}} \cite[{\S}4.5, pp.\ 727--733]{Takac-5}. \begin{theorem}\label{thm-large-sol} Let\/ $1 < p < \infty$, $\lambda = \lambda_1$, and\/ $f = \zeta\varphi_1 + f^\top$ with some $\zeta\in {\mathbb{R}}$ and\/ $f^\top\in L^\infty(\Omega)^\top$, $f^\top\not\equiv 0$ in~$\Omega$. Assume that\/ $\underline{u}, \overline{u}\in W_0^{1,p}(\Omega)$ is a pair of strict sub- and super\-solutions of the problem $\mathcal{J}_{\lambda_1}'(u) = 0$ as described in\/ {\rm Lemma~\ref{lem-Weak-Crit}}. (In particular, $\underline{u}$ and\/ $\overline{u}$ are unordered, by eq.\ \eqref{eq:sub=super:tau_0}.) Then the Dirichlet problem $\mathcal{J}_{\lambda_1}'(u) = 0$ possesses a weak solution $u\in W_0^{1,p}(\Omega)$. % \end{theorem} We remark that $\underline{u}, \overline{u}\in C^{1,\beta} (\overline{\Omega})$ for some $\beta\in (0,\alpha)$, by the regularity result mentioned above \cite{Anane-2, DiBene-1, Lieberman, Tolksdorf-2}. We use this fact repeatedly in an essential way. We will prove Theorem~\ref{thm-large-sol} using the topological (Leray\--Schauder) degree. % In the proof of the next lemma we obtain another pair of sub- and super\-solutions, ordered by ``$\leq$'', which provides lower and upper bounds for the unordered pair. Recalling $\varphi_1\in C^{1,\beta} (\overline{\Omega})$ and the Hopf maximum principle \eqref{Hopf.varphi_1}, we introduce the space $X$ of all functions $\phi\in C(\overline{\Omega})$ such that % \begin{equation} \label{norm:X} \|\phi\|_X {\stackrel{{\mathrm {def}}}{=}} \sup_{\Omega} \left( {|\phi|} / {\varphi_1} \right) < \infty \,. \end{equation} % Then $X$ endowed with the norm $\|\cdot\|_X$ is a Banach space. Notice that the embeddings % \begin{math} C_0^1(\overline{\Omega}) \hookrightarrow X \hookrightarrow C(\overline{\Omega}) \end{math} % are continuous, where % \begin{equation*} C_0^1(\overline{\Omega}) {\stackrel{{\mathrm {def}}}{=}} \{ \phi\in C^1(\overline{\Omega}): \phi = 0 \text{ on } \partial\Omega \} \end{equation*} % is a closed linear subspace of $C^1(\overline{\Omega})$. We denote ${\mathbb{R}}_+ = [0,\infty)$. Given any $R>0$, let us define the function $\gamma_{R}: {\mathbb{R}}_+\to [0,1]$ by % \begin{equation} \label{def:gamma_R} \gamma_{R}(\xi) {\stackrel{{\mathrm {def}}}{=}} \begin{cases} 1 & \text{if } 0\leq \xi\leq R \,; \\ 2 - ({\xi} / R) & \text{if } R < \xi\leq 2R \,; \\ 0 & \text{if } \xi > 2R \,. \end{cases} \end{equation} % Notice that $\gamma_{R}$ is a monotone decreasing, Lipschitz\--continuous function. Next, for $u\in X$ we define $G_{R}(u): \Omega\to {\mathbb{R}}$ by % \begin{equation} \label{def:G_R} [ G_{R}(u) ](x) {\stackrel{{\mathrm {def}}}{=}} \begin{cases} \lambda_1 |u|^{p-2} u + \gamma_{R}\left( \frac{|u(x)|}{\varphi_1(x)} \right) f(x) & \text{if } \frac{|u(x)|}{\varphi_1(x)} \leq 2R \,; \\ - \lambda_1\, [ 2R\, \varphi_1(x) ]^{p-1} &\text{if } \frac{u(x)}{\varphi_1(x)} < -2R \,;\\ \lambda_1\, [ 2R\, \varphi_1(x) ]^{p-1} & \text{if } \frac{u(x)}{\varphi_1(x)} > 2R \,, \end{cases} \end{equation} % at every $x\in \Omega$. As $\gamma_{R}(2{\varepsilon}) = 0$, $f\in L^\infty(\Omega)$, and $u / \varphi_1 \in L^\infty(\Omega)$, it is easy to see that the mapping \[ G_{R}: u\mapsto G_{R}(u): X\to L^\infty(\Omega) \] is continuous. \begin{lemma}\label{lem-large-sol} Assume that\/ $f$ and the sub- and super\-solutions $\underline{u}$ and\/ $\overline{u}$ are exactly as in\/ {\rm Lemma~\ref{lem-Weak-Crit}}. Let\/ $\{\eta_n\}_{n=1}^\infty \subset (0,1)$ and\/ $\{ R_n\}_{n=1}^\infty \subset (0,\infty)$ satisfy\/ $\eta_n\to 0$ and\/ $R_n\to \infty$ as $n\to \infty$. Finally, for each\/ $n=1,2,\dots$, let\/ $u_n\in W_0^{1,p}(\Omega)$ have the following properties: % \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} % \item[(i)] $u_n\in C_0^1(\overline{\Omega})$; % \item[(ii)] $u_n$ is a weak solution of the boundary value problem % \begin{equation} - \Delta_p u_n = [ G_{R_n}(u_n) ](x) \quad\text{in } \Omega \,;\quad u_n = 0 \quad\text{on } \partial\Omega \,; \label{e.sol:BVP.l+mu_n} \end{equation} % \item[(iii)] there exist points $x_n', x_n''\in \Omega$ such that\/ % \begin{gather} u_n(x_n') \leq \underline{u}(x_n') + \eta_n R_n\, \varphi_1(x_n') \,, \label{u_n:sub-unord} \\ u_n(x_n'') \geq \overline{u}(x_n'') - \eta_n R_n\, \varphi_1(x_n'') \,. \label{u_n:sup-unord} \end{gather} % \end{enumerate} % Then, given any $0 < {\varepsilon}\leq 1$, there exists an integer\/ $n_{{\varepsilon}}\geq 1$ such that inequalities % \begin{equation} \label{e:w_n/phi_1} \| u_n\|_{ C^1(\overline{\Omega}) } + \sup_{\Omega} \left( {|u_n|} / {\varphi_1} \right) \leq {\varepsilon} R_n \end{equation} % hold for every\/ $n\geq n_{{\varepsilon}}$. In particular, for every\/ $n\geq n_{{\varepsilon}}$, $u_n$ is a weak solution of problem~\eqref{e:BVP.l_1} as well, i.e., $\mathcal{J}_{\lambda_1}'(u_n) = 0$. % \end{lemma} In fact, we wish to prove Theorem~\ref{thm-large-sol} by computing the Leray\--Schauder degree in subsets of % \begin{gather*} \mathcal{U}_n' = \{ u\in X: u(x) > \underline{u}(x) + \eta_n\, \varphi_1(x)\} \,, \\ \mathcal{U}_n''= \{ u\in X: u(x) < \overline{u}(x) - \eta_n\, \varphi_1(x)\} \,. \end{gather*} % Here, we have dropped the factor $R_n$ ($\geq 1$ for all $n$ sufficiently large) from the product $\eta_n R_n$ in \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord}; the (weaker) inequalities above will do. Lemma~\ref{lem-large-sol} is needed to take care of the complement (in $X$) of the union of these two sets, $X\setminus (\mathcal{U}_n'\cup \mathcal{U}_n'')$. Clearly, the ``pointwise boundedness'' conditions \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord} are very generous as $\eta_n$ and $R_n$ are not related. \begin{proof}[Proof of Lemma~\ref{lem-large-sol}] The ``normalized'' sequence $v_n{\stackrel{{\mathrm {def}}}{=}} R_n^{-1} u_n$ ($n=1,2,\dots$) satisfies % \begin{equation} - \Delta_p v_n = R_n^{-(p-1)}\, [ G_{R_n}(R_n v_n) ](x) \quad\text{in } \Omega ;\quad v_n = 0 \quad\text{on } \partial\Omega . \label{e:BVP.th_n_eps} \end{equation} % Since $\partial\Omega$ is assumed to be of class $C^{1,\alpha}$, for some $0 < \alpha < 1$, we conclude that $v_n\in C^{1,\beta} (\overline{\Omega})$, for some $\beta\in (0,\alpha)$, and the sequence $\{ v_n\}_{n=1}^\infty$ is bounded in $C^{1,\beta} (\overline{\Omega})$, by the regularity result mentioned above \cite{Anane-2, DiBene-1, Lieberman, Tolksdorf-2}. Now, for any fixed $\beta'\in (0,\beta)$, we can apply Arzel\`a\--Ascoli's theorem in $C^{1,\beta'} (\overline{\Omega})$ to the sequence $\{ v_n\}_{n=1}^\infty$ to obtain a convergent subsequence $v_n\to v^*$ in $C^{1,\beta'} (\overline{\Omega})$ as $n\to \infty$. Letting $n\to \infty$ in the weak formulation of problem~\eqref{e:BVP.th_n_eps}, and recalling our notation from \eqref{def:gamma_R}, we arrive at % \begin{equation} - \Delta_p v^* = [ H(v^*) ](x) \quad\text{in } \Omega \,;\quad v^* = 0 \quad\text{on } \partial\Omega \,, \label{e.sol:BVP.l} \end{equation} % where $H(v): \Omega\to {\mathbb{R}}$ is defined by % \begin{equation*} %\label{def:G_eps} [ H(v) ](x) {\stackrel{{\mathrm {def}}}{=}} \begin{cases} \lambda_1 |v|^{p-2} v &\text{if }\frac{|v(x)|}{\varphi_1(x)} \leq 2 \,; \\ - \lambda_1\, [ 2\, \varphi_1(x) ]^{p-1} &\text{if }\frac{v(x)}{\varphi_1(x)} < -2 \,; \\ \lambda_1\, [ 2\, \varphi_1(x) ]^{p-1} &\text{if } \frac{v(x)}{\varphi_1(x)} > 2 \,, \end{cases} \end{equation*} % at every $x\in \Omega$, for $v\in X$. As both functions $v_\pm = \pm 2\varphi_1$ are solutions of problem \eqref{e.sol:BVP.l} and $H(v_-) \leq H(v^*) \leq H(v_+)$ in $\Omega$, we conclude that $v^*$ itself must satisfy $v_-\leq v^*\leq v_+$ in $\Omega$, by the weak comparison principle (see e.g.\ \cite[Lemma 3.1, p.~800]{Tolksdorf-1}). Consequently, eq.\ \eqref{e.sol:BVP.l} reads % \begin{equation*} - \Delta_p v^* = \lambda_1 |v^*|^{p-2} v^* \quad\text{in } \Omega ;\quad v^* = 0 \quad\text{on } \partial\Omega . %\label{e:w.l_1} \end{equation*} % Eigenvalue $\lambda_1$ of $-\Delta_p$ being simple, this equation forces $v^* = \kappa\varphi_1$ in~$\Omega$ for some $\kappa\in {\mathbb{R}}$ with $|\kappa|\leq 2$. Next, we observe that \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord}, respectively, are equivalent to % \begin{gather} v_n(x_n') \leq R_n^{-1}\, \underline{u}(x_n') + \eta_n\varphi_1(x_n') , \label{v_n:sub-unord} \\ v_n(x_n'') \geq R_n^{-1}\, \overline{u}(x_n'') - \eta_n\varphi_1(x_n'') . \label{v_n:sup-unord} \end{gather} % From $v_n\to v^* = \kappa\varphi_1$ in $C^{1,\beta'} (\overline{\Omega})$ as $n\to \infty$ we deduce that % \begin{math} {v_n} / {\varphi_1} \to {v^*} / {\varphi_1} = \kappa \end{math} % in $L^\infty(\Omega)$. We claim that $\kappa = 0$ which implies the conclusion of our lemma immediately. So, on the contrary, suppose that $\kappa\neq 0$. If $\kappa > 0$ then there exists an integer $n_0\geq 1$ such that % \begin{math} {v_n} / {\varphi_1} \geq \frac12\kappa \end{math} % in $\Omega$ for all $n\geq n_0$. Combining this inequality with \eqref{v_n:sub-unord} we arrive at % \begin{equation*} R_n^{-1}\, \frac{ \underline{u}(x_n') }{ \varphi_1(x_n') } + \eta_n \geq \frac{ v_n(x_n') }{ \varphi_1(x_n') } \geq \frac{\kappa}{2} > 0 \quad\text{for all } n\geq n_0 . \end{equation*} % But this contradicts $R_n\to \infty$ and $\eta_n\to 0$ as $n\to \infty$. Similarly, if $\kappa < 0$ then % \begin{math} {v_n} / {\varphi_1} \leq \frac12\kappa \end{math} % in $\Omega$ for all $n\geq n_0$. Combining this inequality with \eqref{v_n:sup-unord} we get % \begin{equation*} R_n^{-1}\, \frac{ \overline{u}(x_n'') }{ \varphi_1(x_n'') } - \eta_n \leq \frac{ v_n(x_n'') }{ \varphi_1(x_n'') } \leq \frac{\kappa}{2} < 0 \quad\text{for all } n\geq n_0 . \end{equation*} % Again, this contradicts $R_n\to \infty$ and $\eta_n\to 0$. The lemma follows from $R_n^{-1} u_n = v_n\to \kappa\varphi_1 = 0$ in $C_0^1(\overline{\Omega})$. \end{proof} Let us denote by $X_+$ the positive cone in $X$, that is, \[ X_+{\stackrel{{\mathrm {def}}}{=}} \{ \phi\in X: \phi\geq 0 \text{ in } \Omega \} \,, \] and by ${\stackrel{\circ}{X}}_+$ its (topological) interior, \[ {\stackrel{\circ}{X}}_+ = \{ \phi\in X: \phi\geq \kappa\varphi_1 \text{ in } \Omega \quad\text{for some }\; \kappa\in (0,\infty) \} \,. \] Given any $a,b\in X$, we write $a\ll b$ (or, equivalently, $b\gg a$) if and only if $b-a\in {\stackrel{\circ}{X}}_+$. We denote \[ [a,b]{\stackrel{{\mathrm {def}}}{=}} \{ \phi\in X: a\leq \phi\leq b \text{ in } \Omega \} \text{ and } [[a,b]]{\stackrel{{\mathrm {def}}}{=}} \{ \phi\in X: a\ll \phi\ll b \} . \] Notice that $[[a,b]]$ is the (topological) interior of $[a,b]$ in~$X$. \begin{proof}[ Proof of Theorem~\ref{thm-large-sol}] With regard to Lemma~\ref{lem-large-sol} above, it suffices to construct a weak solution $u_n\in W_0^{1,p}(\Omega)$ of problem~\eqref{e.sol:BVP.l+mu_n} with properties {\rm (i)}, {\rm (ii)}, and {\rm (iii)} of Lemma~\ref{lem-large-sol}, for each $n=1,2,\dots$. The sequences $\{\eta_n\}_{n=1}^\infty \subset (0,1)$ and $\{ R_n\}_{n=1}^\infty \subset (0,\infty)$ may be chosen arbitrarily with $\eta_n\to 0$ and $R_n\to \infty$ as $n\to \infty$. Given $0 < r < \infty$, we denote by \[ \mathcal{B}_r = \{ u\in X: \| u\|_X < r\} \] the open ball in $X$ with radius $r$ centered at the origin, and by \[ \overline{\mathcal{B}}_r = \{ u\in X: \| u\|_X\leq r\} \] its closure in~$X$. Set \[ R_0 = \max\left\{ \|\underline{u}\|_X ,\, \|\overline{u}\|_X \right\} + 1 \,. \] In particular, recalling \eqref{def:gamma_R}, for $R\geq R_0$ we have $\gamma_{R}( |\underline{u}| / {\varphi_1} ) = 1$ and $\gamma_{R}( |\overline{u} | / {\varphi_1} ) = 1$ throughout $\Omega$. Hence, from now on, we may assume $R_n\geq R_0$ for all $n\geq 1$. Let us fix any $n\geq 1$ and recall $0 < \eta_n < 1\leq R_n < \infty$. Now we follow { Dr\'abek} and { Holubov\'a} \cite{DrabHolub}, proof of Lemma 2.4, pp.\ 191--192. First, fix any number $\varrho > 3 R_n$. We define the (fixed point) mapping % \begin{equation*} \mathcal{T}: \overline{\mathcal{B}}_{\varrho} \to X: u\mapsto \mathcal{T} u{\stackrel{{\mathrm {def}}}{=}} \tilde{u} \,, \end{equation*} % where $\tilde{u}\in C_0^1(\overline{\Omega})$ is the unique weak solution of % \begin{equation} - \Delta_p \tilde{u} = [ G_{R_n}(u) ](x) \quad\text{in } \Omega \,;\quad \tilde{u} = 0 \quad\text{on } \partial\Omega \,. \label{fix.sol:BVP.l+mu_n} \end{equation} % We claim that $\mathcal{T}$ is compact, i.e., $\mathcal{T}$ is continuous and its image is contained in a compact set. Indeed, it is easy to see that % \begin{math} \mathcal{T}: \overline{\mathcal{B}}_{\varrho} \subset X\to W_0^{1,p}(\Omega) \end{math} % is continuous with the image $\mathcal{T} (\overline{\mathcal{B}}_{\varrho})$ being a bounded set in $C^{1,\beta} (\overline{\Omega})$, by regularity \cite{Anane-2, DiBene-1, Lieberman, Tolksdorf-2}. Consequently, for any fixed $\beta'\in (0,\beta)$, % \begin{equation*} \mathcal{T}: \overline{\mathcal{B}}_{\varrho} \subset X\to C^{1,\beta'} (\overline{\Omega}) \cap W_0^{1,p}(\Omega) \end{equation*} % is continuous with $\mathcal{T} (\overline{\mathcal{B}}_{\varrho})$ having compact closure in each of the spaces % \begin{equation*} C^{1,\beta'} (\overline{\Omega}) \cap W_0^{1,p}(\Omega) \hookrightarrow C_0^1(\overline{\Omega}) \hookrightarrow X , \end{equation*} % by Arzel\`a\--Ascoli's theorem. Thus, $\mathcal{T}: \overline{\mathcal{B}}_{\varrho} \to X$ is compact. If there exists a fixed point $u_n\in \overline{\mathcal{B}}_{\varrho}$ of $\mathcal{T}$, i.e., $\mathcal{T} u_n = u_n$, such that both inequalities \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord} are satisfied, then we are done. So let us assume the contrary, that is, if $u_n\in \overline{\mathcal{B}}_{\varrho}$ satisfies $\mathcal{T} u_n = u_n$ then {\it at least one\/} of the following two inequalities must be valid: % \begin{gather} u_n(x) > \underline{u}(x) + \eta_n\, \varphi_1(x) \quad\text{for all } x\in \Omega \,, \label{u_n:sub-unord_Om} \\ u_n(x) < \overline{u}(x) - \eta_n\, \varphi_1(x) \quad\text{for all } x\in \Omega \,. \label{u_n:sup-unord_Om} \end{gather} % Notice that we have dropped the factor $R_n$ ($\geq 1$) from the product $\eta_n R_n$ in \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord}; the (weaker) inequalities above will suffice to get a contradiction. Consequently, we get $u_n\gg \underline{u}$ or $u_n\ll \overline{u}$ in~$X$. Moreover, both inequalities cannot hold simultaneously; for otherwise we would have $\underline{u}\leq \overline{u}$ throughout $\Omega$ which forces $\underline{u} = \overline{u}$ in $\Omega$, owing to eq.~\eqref{eq:sub=super:tau_0}. Hence, there exists a point $y\in \Omega$ such that $\overline{u}(y) < \underline{u}(y)$. From either inequality $u_n\gg \underline{u}$ or $u_n\ll \overline{u}$ in~$X$ we deduce easily that $u_n\in X\setminus \overline{\mathcal{S}}$ where $\overline{\mathcal{S}}$ denotes the closure in $X$ of the set % \[ \mathcal{S} = \big\{ u\in X: u(x') < \underline{u}(x') \text{ and } u(x'') > \overline{u}(x'') \text{ for some } x', x''\in \Omega \big\} \,. \] % Clearly, $\mathcal{S}$ is open in $X$ with the complement \[ X\setminus \mathcal{S} = \{ u\in X: u\geq \underline{u} \text{ in } \Omega \} \cup \{ u\in X: u\leq \overline{u} \text{ in } \Omega \} \,. \] Next, we introduce the functions $A_\pm{\stackrel{{\mathrm {def}}}{=}} \pm 3 R_n\varphi_1$. We have $A_\pm\in \mathcal{B}_{\varrho}$, together with $A_-\ll \underline{u}\ll A_+$ and $A_-\ll \overline{u} \ll A_+$. Hence, also $\underline{u}, \overline{u}\in \mathcal{B}_{\varrho}$. Observe that both $A_-$ and $\underline{u}$ ($A_+$ and $\overline{u}$) are strict subsolutions (supersolutions, respectively) of the boundary value problem \eqref{e.sol:BVP.l+mu_n}; more precisely, they satisfy % \begin{equation} \begin{gathered} - \Delta_p u - [ G_{R_n}(u) ](x) \leq - \phi_1(x) < 0\ \quad ( \geq \phi_1(x) > 0 ) \quad\text{in } \Omega \,; \\ u = 0 \quad\text{on } \partial\Omega \,, \end{gathered} \label{sub.sol:BVP.l+mu_n} \end{equation} % where $\phi_1 = c_{R_n}\cdot \min\{ \varphi_1 ,\, \varphi_1^{p-1} \}$ with some constant $c_{R_n} > 0$. This in turn implies $u\ll \mathcal{T} u$ ($u\gg \mathcal{T} u$) in $X$, by the strong comparison principle due to { Cuesta} and { Tak\'a\v{c}} \cite[Theorem~1, p.~81]{CueTakac-1} (see also \cite[Theorem 2.1, p.~725]{CueTakac-2}). We remark that the connectedness of the boundary $\partial\Omega$, assumed in \cite[Theorem~1]{CueTakac-1}, is not needed here owing to the strict inequality in eq.~\eqref{sub.sol:BVP.l+mu_n} throughout the domain $\Omega$. Hence, in the proof of \cite[Theorem~1, p.~81]{CueTakac-1}, one may apply \cite[Prop.~3, p.~82]{CueTakac-1} on any connected component of the boundary $\partial\Omega$. Finally, let $\deg [ I - \mathcal{T} ;\, \mathcal{U} ,\, 0 ]$ denote the Leray\--Schauder degree of the mapping $I - \mathcal{T}: \overline{\mathcal{U}} \to X$ relative to the origin $0\in X$, where $\mathcal{U}$ is any open set in $X$ such that $\mathcal{U} \subset \mathcal{B}_{\varrho}$ and $0\not\in (I - \mathcal{T})( \partial\mathcal{U} )$. As usual, $I$ stands for the identity mapping and $\partial\mathcal{U}$ for the boundary of $\mathcal{U}$ in $X$. We compute this degree in the sets $[[A_-,A_+]]$, $[[ A_- , \overline{u}_n ]]$, $[[ \underline{u}_n , A_+ ]]$, and $\mathcal{S} \cap [[A_-,A_+]]$, all of which are open in $X$. Clearly, the last three sets, $[[ A_- , \overline{u}_n ]]$, $[[ \underline{u}_n , A_+ ]]$, and $\mathcal{S} \cap [[A_-,A_+]]$, are pairwise disjoint and the union of their closures equals $[A_-,A_+] \subset \overline{\mathcal{B}}_{\varrho}$. % Using the excision property of the Leray\--Schauder degree, we compute % \begin{equation} \label{e:degree} \begin{split} & \deg [ I - \mathcal{T} ;\, [[A_-,A_+]] ,\, 0 ] \\ & = \deg [ I - \mathcal{T} ;\, [[ A_- , \overline{u}_n ]] ,\, 0 ] + \deg [ I - \mathcal{T} ;\, [[ \underline{u}_n , A_+ ]] ,\, 0 ] \\ & \quad + \deg [ I - \mathcal{T} ;\, \mathcal{S} \cap [[A_-,A_+]] ,\, 0 ] . \end{split} \end{equation} % Recalling the fact that $\mathcal{T}$ has no fixed point $u_n\in \overline{\mathcal{B}}_{\varrho}$ on the boundary of any of the sets $[[A_-,A_+]]$, $[[ A_- , \overline{u}_n ]]$, and $[[ \underline{u}_n , A_+ ]]$, we may apply \cite[Lemma 2.3, p.~190]{DrabHolub} to conclude that % \begin{align*} \deg [ I - \mathcal{T} ;\, [[A_-,A_+]] ,\, 0 ] & = \deg [ I - \mathcal{T} ;\, [[ A_- , \overline{u}_n ]] ,\, 0 ]\\ &= \deg [ I - \mathcal{T} ;\, [[ \underline{u}_n , A_+ ]] ,\, 0 ] = 1 . \end{align*} % Furthermore, since $\mathcal{T}$ has no fixed point in $\overline{\mathcal{S}}$, we must have also % \begin{equation*} \deg [ I - \mathcal{T} ;\, \mathcal{S} \cap [[A_-,A_+]] ,\, 0 ] = 0 . \end{equation*} % Inserting these results into eq.~\eqref{e:degree} we arrive at a contradiction, $1 = 1 + 1 + 0$. Hence, we have proved that, indeed, there exists a fixed point $u_n\in \overline{\mathcal{B}}_{\varrho}$ of $\mathcal{T}$ such that both inequalities \eqref{u_n:sub-unord} and \eqref{u_n:sup-unord} are valid. So Lemma~\ref{lem-large-sol} can be applied and Theorem~\ref{thm-large-sol} is proved. \end{proof} \section{Large critical points of $\mathcal{J}_{\lambda}$} \label{s:Asympt} In the previous two sections we have shown how to obtain a (weak) solution to the resonant problem \eqref{e:BVP.l_1} (i.e., when $\lambda = \lambda_1$), first for $p>2$ (in Section~\ref{s:Elem_anal}) and then for $12$ ($1 0$ is a constant depending on $f^\top$. The function $j_{\lambda_1}$ being continuous, it attains a global minimum (maximum, respectively) at $\tau_0\in {\mathbb{R}}$ if $p>2$ ($1 0$ is small enough then $j_{\lambda_1}$ possesses a local minimizer and a local maximizer, such that one of them stays in a bounded interval and the other tends to $+\infty$ or $-\infty$ as $\zeta\to 0$, depending on the signs of $p-2$ and~$\zeta$. Consequently, the local minimizer and maximizer are distinguished from each other by the size of their absolute values. Finally, keeping $\zeta$ constant (with $|\zeta| > 0$ small enough), we perturb $\lambda = \lambda_1$ to $\lambda$ near $\lambda_1$ (with $|\lambda - \lambda_1| > 0$ small enough) in order to obtain a second local minimizer (local maximizer, respectively) for $j_{\lambda}$ if $\lambda > \lambda_1$ ($\lambda < \lambda_1$), whose absolute value is even much larger than the absolute values of the local minimizers and maximizers constructed before for the case $\lambda = \lambda_1$. With some additional caution about the size of critical points, the (multiple) critical points of $\mathcal{J}_{\lambda}$ corresponding to the local minimizers and maximizers of $j_{\lambda}$ are now obtained by the methods presented in Sections \ref{s:Elem_anal} and~\ref{s:Topological}. We refer the interested reader to the article { Tak\'a\v{c}} \cite[Sect.\ 4 and~5]{Takac-5} for (rather complicated) technical details. An essential tool in the perturbation process just described is, of course, {\it continuous dependence\/} of the function $j_{\lambda}(\tau; \zeta\varphi_1 + f^\top)$ on {\it all\/} (real) variables $\tau$, $\lambda$, and~$\zeta$, which is proved in { Tak\'a\v{c}} \cite[Lemma 7.2, p.~222]{Takac-2}. \section{A collection of main results} \label{s:Main} In order to give an idea to the reader interested in what kinds of results can be obtained by the techniques from Sections \ref{s:Elem_anal} and~\ref{s:Topological} applied to the energy functional $\mathcal{J}_{\lambda}$ (concerning multiple critical points and their classification), below we present a collection of the main results from { Tak\'a\v{c}} \cite[Sect.~2, pp.\ 698--705]{Takac-5}. Some of them appeared before in \cite{Drabek-3, DrabGirgMan, DrabGirgTakac, DGTU, DrabHolub, FleckTakac, GirgTakac, ManTakac, PinoDrabMan, Pohozaev, Takac-2, Takac-3, Takac-4}. Recall that we always assume that the domain $\Omega\subset \mathbb{R}^N$ satisfies hypothesis~\eqref{hyp:Omega}. The first (smallest) eigenvalue $\lambda_1$ of the positive Dirichlet $p$-Lapla\-cian $-\Delta_p$ for $1 0$ in $\Omega$ and $\|\varphi_1\|_{ L^p(\Omega) } = 1$. Furthermore, we have $\varphi_1\in C^{1,\beta} (\overline{\Omega})$ for some $\beta\in (0,\alpha)$, together with the Hopf maximum principle % \begin{equation} \varphi_1 > 0 \quad\text{in } \Omega \quad\text{and}\quad \frac{\partial\varphi_1}{\partial\nu} < 0 \quad\text{on } \partial\Omega . \nonumber \tag{\color{green}\ref{Hopf.varphi_1}\color{black}} \end{equation} % There, we have also introduced the set \[ U = \{ x\in \Omega: \nabla\varphi_1(x) \not= \mathbf{0} \} . \] Often, a function $u\in L^1(\Omega)$ will be decomposed as the orthogonal sum $u = u^\parallel\cdot \varphi_1 + u^\top$ according to \eqref{ortho:f}. Given a linear subspace $\mathcal{M}$ of $L^1(\Omega)$ with $\varphi_1\in \mathcal{M}$, we write \[ \mathcal{M}^\top {\stackrel{{\mathrm {def}}}{=}} \{ u\in \mathcal{M}: \langle u,\varphi_1 \rangle = 0 \} . \] The following concept is tailored for our treatment of the functional $\mathcal{J}_{\lambda}$ defined in~\eqref{def.jl}; we recall % \begin{equation} j_{\lambda}(\tau) {\stackrel{{\mathrm {def}}}{=}} \min_{ u^\top\in W_0^{1,p}(\Omega)^\top } \mathcal{J}_{\lambda}( \tau\varphi_1 + u^\top ) \quad\text{for } \tau\in {\mathbb{R}} . \nonumber \tag{\color{green}\ref{def.j_lam,tau}\color{black}} \end{equation} % \begin{definition}\label{def-Saddle} \begingroup\rm $u_0\in W_0^{1,p}(\Omega)$ will be called a \emph{simple saddle point\/} for $\mathcal{J}_{\lambda}$ %(with respect to the orthogonal sum~\eqref{ortho:f}) if $u_0 = \tau_0\varphi_1 + u_0^\top$ is a critical point for~$\mathcal{J}_{\lambda}$, $u_0^\top$ is a global minimizer for the restricted functional $u^\top \mapsto \mathcal{J}_{\lambda}( \tau_0\varphi_1 + u^\top )$ on $W_0^{1,p}(\Omega)^\top$, and the function $j_{\lambda}: {\mathbb{R}}\to {\mathbb{R}}$ attains a local maximum at $\tau_0$. \endgroup \end{definition} A more general type of a saddle point is obtained in { Rabinowitz} \cite[Theorem 4.6, p.~24]{Rabin}. From now on we separate the cases $p>2$ and $12} Let $2 0$ (in place of\/ $\delta$ and\/ $\delta'$) are arbitrary and\/ $\lambda\leq \lambda_1 - \delta_0$, $|\zeta|\leq \delta_0'$, the same bound (depending also on $\delta_0$ and\/ $\delta_0'$) holds for any global minimizer of\/ $\mathcal{J}_\lambda$. % \end{theorem} A variational proof of this theorem relies on the methods described in Sections \ref{s:Minimum} and~\ref{s:Elem_anal}, case~{\rm (i)}. It can be found in \cite[{\S}6.1, pp.\ 738--739]{Takac-5} (proof of Theorem 2.2 in \cite{Takac-5}). For all $\lambda < \lambda_1$ and $\zeta\in {\mathbb{R}}$, this result follows from the coercivity of the energy functional $\mathcal{J}_{\lambda}$, whereas for $0 < \lambda - \lambda_1\leq \delta$ and any $\zeta\in {\mathbb{R}}$, it can be proved by a well\--known argument employing topological degree; see { Dr\'abek} \cite[Theorem 14.18, p.~189]{Drabek-1}. Finally, for $\lambda = \lambda_1$ it has been established in { Fleckinger} and { Tak\'a\v{c}} \cite[Theorem 3.3]{FleckTakac} and { Tak\'a\v{c}} \cite[Theorem 2.2]{Takac-2} if $\zeta = 0$, and in { Tak\'a\v{c}} \cite[Theorem 3.1]{Takac-3} if $|\zeta|\leq \delta'$. \begin{remark}\label{rem-Exist} \begingroup\rm The local minimizer $u_1\in W_0^{1,p}(\Omega)$ described in Theorem~\ref{thm-Exist} is the same as the one obtained in Theorems \ref{thm-Two}, \ref{thm-Subcr}, and \ref{thm-Super} below. \endgroup \end{remark} Our second theorem is a multiplicity result for the resonant value $\lambda = \lambda_1$. Although it has been obtained originally in \cite[Theorem 3.1]{Takac-3}, its present form (taken from \cite[Theorem 2.4, p.~701]{Takac-5}) is more specific about the qualitative description of solutions. \begin{theorem}\label{thm-Two} There exists a constant\/ $\delta'\equiv \delta'(K) > 0$ such that problem~\eqref{e:BVP.l} with\/ $\lambda = \lambda_1$ and\/ $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{two} (distinct) weak solutions $u_1, u_2$ specified as follows, whenever\/ $f^\top\in K$ and\/ $0 < |\zeta|\leq \delta'$: Functional\/ $\mathcal{J}_{\lambda_1}$ (which is unbounded from below) possesses a local minimizer\/ $u_1\in W_0^{1,p}(\Omega)$ and another critical point\/ $u_2\in W_0^{1,p}(\Omega)$. % \end{theorem} The proof of this theorem can be derived from that of Theorem~\ref{thm-Exist}; see \cite[{\S}6.2, pp.\ 739--741]{Takac-5} (proof of Theorem 2.4 in \cite{Takac-5}). \begin{remark}\label{rem-Two-Saddle} \begingroup\rm The critical point $u_2\in W_0^{1,p}(\Omega)$ obtained in {\rm Theorem~\ref{thm-Two}} above is constructed from a suitable pair of sub- and super\-solutions to problem~\eqref{e:BVP.l} satisfying \eqref{e:BVP.l_sub} and \eqref{e:BVP.l_sup}, respectively, with $\lambda = \lambda_1$. This method is a refinement of the topological degree arguments described in Section~\ref{s:Topological}. If the sub- and super\-solutions coincide, then $u_2 = \underline{u} = \overline{u}$ is a simple saddle point for $\mathcal{J}_{\lambda_1}$, $u_2 = \tau_2\varphi_1 + u_2^\top$, and, moreover, $j_{\lambda_1}$ is differentiable at $\tau_2$ with vanishing derivative, $j_{\lambda_1}^\prime (\tau_2) = 0$. Analogous remarks apply to \emph{all\/} critical points (other than local minimizers) obtained in our theorems below, Theorems \ref{thm-Subcr} and \ref{thm-Super} in this paragraph, and Theorems \ref{thm-Exist:p<2}, \ref{thm-Two:p<2}, \ref{thm-Subcr:p<2}, and \ref{thm-Super:p<2} in the next one ({\S}\ref{ss:Case.p<2}). \endgroup \end{remark} The following two theorems on the existence of at least \emph{three\/} solutions to the Dirichlet problem~\eqref{e:BVP.l} are the main new results for $p>2$ obtained in \cite{Takac-5}. First, we consider the subcritical case $\lambda_1 - \delta\leq \lambda < \lambda_1$. \begin{theorem}\label{thm-Subcr} There exists a constant\/ $\delta'\equiv \delta'(K) > 0$ such that, for any\/ $d\in (0,\delta')$, there is another constant\/ $\delta\equiv \delta(K,d) > 0$ such that problem~\eqref{e:BVP.l} with\/ $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{three} (pairwise distinct) weak solutions $u_1, u_2, u_3$ specified as follows, whenever\/ $\lambda_1 - \delta\leq \lambda < \lambda_1$, $f^\top\in K$, and\/ $d\leq |\zeta|\leq \delta'$: Functional\/ $\mathcal{J}_{\lambda}$ (which is bounded from below) possesses\/ \emph{two\/} local minimizers $u_1, u_2\in W_0^{1,p}(\Omega)$ of which at least one is global, and another critical point\/ $u_3\in W_0^{1,p}(\Omega)$. % \end{theorem} The proof of this theorem is derived from those of Theorems \ref{thm-Exist} and~\ref{thm-Two}; see \cite[{\S}6.3, pp.\ 742--744]{Takac-5} (proof of Theorem 2.6 in \cite{Takac-5}). Finally, we treat the supercritical case $\lambda_1 < \lambda\leq \lambda_1 + \delta$. Here we obtain the following multiplicity and uniform boundedness results for problem~\eqref{e:BVP.l}: \begin{theorem}\label{thm-Super} There exist constants $\delta\equiv \delta(K) > 0$ and\/ $\delta'\equiv \delta'(K) > 0$ such that problem~\eqref{e:BVP.l} with\/ $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{three} (pairwise distinct) weak solutions $u_1, u_2, u_3$ specified as follows, whenever\/ $\lambda_1 < \lambda\leq \lambda_1 + \delta$, $f^\top\in K$, and\/ $|\zeta|\leq \delta'$: Functional $\mathcal{J}_{\lambda}$ (which is unbounded from below) possesses a local minimizer $u_1\in W_0^{1,p}(\Omega)$ and\/ \emph{two} other critical points $u_2, u_3\in W_0^{1,p}(\Omega)$. % \end{theorem} The proof of this theorem is derived from that of Theorem~\ref{thm-Exist}; see \cite[{\S}6.4, pp.\ 745--746]{Takac-5} (proof of Theorem 2.7 in \cite{Takac-5}). \begin{remark}\label{rem-Ordered} \begingroup\rm In Theorem \ref{thm-Two}, the orthogonal projections of $u_1$ and $u_2$ onto ${\mathop{\rm lin}} \{ \varphi_1\}$ satisfy $u_1^\parallel < u_2^\parallel$ if $\zeta > 0$, and $u_1^\parallel > u_2^\parallel$ if $\zeta < 0$. In Theorem~\ref{thm-Subcr}, $u_3^\parallel$ lies between $u_1^\parallel$ and $u_2^\parallel$, and in Theorem~\ref{thm-Super}, $u_1^\parallel$ lies between $u_2^\parallel$ and $u_3^\parallel$. \endgroup \end{remark} For the question of \emph{boundedness\/} of the solution set for problem~\eqref{e:BVP.l} with $\lambda$ near $\lambda_1$, we refer to { Dr\'abek} et al.~\cite{DGTU} and { Tak\'a\v{c}} \cite[Sect.~2]{Takac-2} and \cite[Prop.\ 6.1]{Takac-3}. Although the variational methods developed in our present lecture notes are clearly not suitable for resolving this question, in the proofs of all our theorems one makes essential use of \cite[Prop.\ 6.1]{Takac-3} (stated as Proposition~\ref{prop-v_n-t=0} in the Appendix, {\S}\ref{ss:Unif_Bound}) which in turn provides the following answer in the special case $\lambda = \lambda_1$; see \cite[Theorem 3.2]{Takac-3}: \begin{theorem}\label{thm-Bound} Let\/ $f^\top$ be as in\/ {\rm Theorem~\ref{thm-Exist}} above. If\/ $\zeta = 0$ then the set of all weak solutions to problem~\eqref{e:BVP.l} with\/ $\lambda = \lambda_1$ is bounded in $C^{1,\beta} (\overline{\Omega})$. Given any $\delta > 0$, this set is bounded in $C^{1,\beta} (\overline{\Omega})$ uniformly for all\/ $|\zeta|\geq \delta$ as well. % \end{theorem} This theorem is proved in \cite[{\S}6.5, p.~747]{Takac-5} (proof of Theorem 2.9 in \cite{Takac-5}). In contrast with Theorem~\ref{thm-Exist}, if $|\zeta|$ in $f\equiv f^\top + \zeta\varphi_1$ is ``too large'' relative to the size of $\| f^\top\|_{ L^\infty(\Omega) }$, say $|\zeta|\geq \delta > 0$, then problem~\eqref{e:BVP.l} with $\lambda = \lambda_1$ has \emph{no\/} weak solution; see \cite[Corollary 2.4, p.~195]{Takac-2} or \cite[Theorem 3.1]{Takac-3}: \begin{corollary}\label{cor-Bound} Given an arbitrary function $g\in L^\infty(\Omega)$ with\/ $0\leq g\not\equiv 0$ in $\Omega$, there exists a constant $\gamma\equiv \gamma(g) > 0$ with the following property: If $f\in L^\infty(\Omega)$, $f\not\equiv 0$, is such that \[ f = f^g\cdot g + \bar f^g \quad\text{with some }\; f^g\in {\mathbb{R}} \quad\text{and }\; \bar f^g\in L^\infty(\Omega) , \] and $\|\bar f^g\|_{ L^\infty(\Omega) } \leq \gamma\, |f^g|$, then problem~\eqref{e:BVP.l} with\/ $\lambda = \lambda_1$ has no weak solution. % \end{corollary} Equivalently, given $g$ as above, notice that there is an open cone $\mathcal{C}$ in $L^\infty(\Omega)$ with vertex at the origin ($0\not\in \mathcal{C}$) such that $g\in \mathcal{C}$ and problem~\eqref{e:BVP.l} with $\lambda = \lambda_1$ has no weak solution whenever $f\in \mathcal{C}$. This result improves a nonexistence result due to \cite[Th\'eor\`eme~1]{FGTT-1} (see also \cite[Theorem 7.2, p.~154]{Takac-1}) for $0\leq f\not\equiv 0$ in~$\Omega$. As already mentioned in the Introduction, one needs a number of auxiliary results to prove these theorems. Complete proofs can be found in { Tak\'a\v{c}} \cite[Sect.~6, pp.\ 738--747]{Takac-5}. Additional results revealing more details about the structure of solutions to problem~\eqref{e:BVP.l} have been established within these proofs, e.g., the positivity or negativity of solutions with a sufficiently large norm. \subsection{The singular case $12$, needs to be redefined for $1 0$ (in place of\/ $\delta$ and\/ $\delta'$) are arbitrary and\/ $\lambda\leq \lambda_1 - \delta_0$, $|\zeta|\leq \delta_0'$, the same bound (depending also on $\delta_0$ and\/ $\delta_0'$) holds for any global minimizer of\/ $\mathcal{J}_\lambda$. % \end{theorem} A proof is given in \cite[{\S}7.1, pp.\ 747--749]{Takac-5} (proof of Theorem 2.12 in \cite{Takac-5}). It is based on the topological (Leray\--Schauder) degree as described in Section~\ref{s:Topological}. It has been originally taken from \cite[{\S}8.4, p.~229]{Takac-2} and is analogous to that of Theorem~\ref{thm-Exist}. For all $\lambda\leq \lambda_1 + \delta$, $\lambda\neq \lambda_1$, and $\zeta\in {\mathbb{R}}$, this result can be proved in the same way as for $p>2$; see \cite[Theorem 14.18, p.~189]{Drabek-1}. For $\lambda = \lambda_1$ it was established in \cite[Theorem 1.1]{DrabHolub} and \cite[Theorem 3.5]{Takac-3} if $|\zeta|\leq \delta'$ (by completely different methods), and in \cite[Theorem 2.6]{Takac-2} if $\zeta = 0$ (by the same variational method we use here). \begin{remark}\label{rem-Exist:p<2} \begingroup\rm The critical point $u_1\in W_0^{1,p}(\Omega)$ described in Theorem~\ref{thm-Exist:p<2} is the same as the one obtained in Theorems \ref{thm-Two:p<2}, \ref{thm-Subcr:p<2}, and \ref{thm-Super:p<2} below. We recall that Remark~\ref{rem-Two-Saddle} applies also to \emph{all\/} critical points (other than local minimizers) obtained in our theorems throughout this paragraph, Theorems \ref{thm-Exist:p<2}, \ref{thm-Two:p<2}, \ref{thm-Subcr:p<2}, and \ref{thm-Super:p<2}. \endgroup \end{remark} Again, our second theorem for $p<2$ is a multiplicity result for $\lambda = \lambda_1$ taken from \cite[Theorem 3.5]{Takac-3} in a more specific form obtained in \cite[Theorem 2.14, p.~704]{Takac-5}. \begin{theorem}\label{thm-Two:p<2} There exists a constant\/ $\delta'\equiv \delta'(K) > 0$ such that problem~\eqref{e:BVP.l} with\/ $\lambda = \lambda_1$ and $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{two} (distinct) weak solutions $u_1, u_2$ specified as follows, whenever\/ $f^\top\in K$ and\/ $0 < |\zeta|\leq \delta'$: Functional $\mathcal{J}_{\lambda_1}$ (which is unbounded from below) possesses a critical point\/ $u_1\in W_0^{1,p}(\Omega)$ and a local minimizer\/ $u_2\in W_0^{1,p}(\Omega)$. % \end{theorem} The proof of this theorem can be derived from that of Theorem~\ref{thm-Exist:p<2}; see \cite[{\S}7.2, pp.\ 749--750]{Takac-5} (proof of Theorem 2.14 in \cite{Takac-5}). The following theorem on the existence of at least \emph{three\/} solutions to the Dirichlet problem~\eqref{e:BVP.l} in the subcritical case $\lambda_1 - \delta\leq \lambda < \lambda_1$ is a generalization of \cite[Theorem 2.7, p.~196]{Takac-2} where it is established for $\zeta = 0$ only. \begin{theorem}\label{thm-Subcr:p<2} There exist constants $\delta\equiv \delta(K) > 0$ and\/ $\delta'\equiv \delta'(K) > 0$ such that problem~\eqref{e:BVP.l} with\/ $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{three} (pairwise distinct) weak solutions $u_1, u_2, u_3$ specified as follows, whenever\/ $\lambda_1 - \delta\leq \lambda < \lambda_1$, $f^\top\in K$, and\/ $|\zeta|\leq \delta'$: Functional $\mathcal{J}_{\lambda}$ (which is bounded from below) possesses a critical point\/ $u_1\in W_0^{1,p}(\Omega)$ and\/ \emph{two} (distinct) local minimizers\/ $u_2, u_3\in W_0^{1,p}(\Omega)$ of which at least one is global. % \end{theorem} Again, the proof of this theorem is derived from that of Theorem~\ref{thm-Exist:p<2}; see \cite[{\S}7.3, p.~751]{Takac-5} (proof of Theorem 2.15 in \cite{Takac-5}). Our last theorem on the existence of at least \emph{three\/} solutions to problem~\eqref{e:BVP.l} appeared for the first time in { Tak\'a\v{c}} \cite[Theorem 2.16, p.~705]{Takac-5}. Here we consider the supercritical case $\lambda_1 < \lambda\leq \lambda_1 + \delta$. \begin{theorem}\label{thm-Super:p<2} There exists a constant\/ $\delta'\equiv \delta'(K) > 0$ such that, for any $d\in (0,\delta')$, there is another constant\/ $\delta\equiv \delta(K,d) > 0$ such that problem~\eqref{e:BVP.l} with\/ $f\equiv f^\top + \zeta\varphi_1$ has at least\/ \emph{three} (pairwise distinct) weak solutions $u_1, u_2, u_3$ specified as follows, whenever\/ $\lambda_1 < \lambda\leq \lambda_1 + \delta$, $f^\top\in K$, and\/ $d\leq |\zeta|\leq \delta'$: Functional $\mathcal{J}_{\lambda}$ (which is unbounded from below) possesses\/ \emph{two} (distinct) critical points $u_1, u_2\in W_0^{1,p}(\Omega)$ and a local minimizer $u_3\in W_0^{1,p}(\Omega)$. % \end{theorem} The proof is derived from those of Theorems \ref{thm-Exist:p<2} and~\ref{thm-Subcr:p<2}, see \cite[{\S}7.4, pp.\ 751--754]{Takac-5} (proof of Theorem 2.16 in \cite{Takac-5}). \begin{remark}\label{rem-Ordered:p<2} \begingroup\rm In Theorem~\ref{thm-Two:p<2}, the orthogonal projections of $u_1$ and $u_2$ on ${\mathop{\rm {lin}}} \{ \varphi_1\}$ satisfy $u_1^\parallel < u_2^\parallel$ if $\zeta < 0$, and $u_1^\parallel > u_2^\parallel$ if $\zeta > 0$. In Theorem~\ref{thm-Subcr:p<2}, $u_1^\parallel$ lies between $u_2^\parallel$ and $u_3^\parallel$, and in Theorem~\ref{thm-Super:p<2}, $u_3^\parallel$ lies between $u_1^\parallel$ and $u_2^\parallel$. \endgroup \end{remark} Under the same hypotheses, we obtain the corresponding uniform boundedness result \cite[Theorem 3.6]{Takac-3}; we refer to \cite{DGTU}, \cite[Sect.~2]{Takac-2}, and \cite[Prop.\ 6.1]{Takac-3} for additional results: \begin{theorem}\label{thm-Bound:p<2} The conclusion of\/ {\rm Theorem~\ref{thm-Bound}} is valid also for\/ $12$, and \eqref{e:BVP.w:p<2} if $p<2$. (We refer to \cite{Takac-5}, Lemmas 5.2 (p.~736) and Lemmas 5.3 (p.~737), respectively.) With regard to Proposition~\ref{prop-v_n-t=0}, this means investigating sequences of large solutions, $$ u_n = t_n^{-1}\varphi_1 + u_n^\top = t_n^{-1} ( \varphi_1 + v_n^\top ) \text{ with $t_n\to 0$ and } \| v_n^\top\|_{ C^{1,\beta'} (\overline{\Omega}) } \to 0 \text{ as $n\to \infty$, } $$ to the following generalized version of problem~\eqref{t:BVP.l_1}, see {\S}\ref{ss:Unif_Bound} (Appendix): % \begin{equation} \begin{gathered} \begin{aligned} & - \Delta_p( t^{-1}\varphi_1 + u^\top ) - \lambda_1 | t^{-1}\varphi_1 + u^\top |^{p-2} ( t^{-1}\varphi_1 + u^\top )\\ & = f(x,u(x))^\top + \zeta\cdot \varphi_1(x) \quad \text{in } \Omega ; \end{aligned} \\ u^\top = 0 \quad \text{ on } \partial\Omega ; \\ \langle u^\top, \varphi_1 \rangle = 0 . \end{gathered} \label{u:t:BVP.l_1} \end{equation} % After a careful inspection of the proof of Proposition~\ref{prop-v_n-t=0} one finds out that only Theorem~\ref{thm-v_n-asympt} is needed. Although we will not provide formal proofs of our claims in the present lecture notes, all these auxiliary results can be established without major changes provided $f$ satisfies the following two conditions, cf.\ Girg and Tak\'a\v{c} \cite[{\S}2.3, hypothesis $(H_\infty')$]{GirgTakac}: \begin{enumerate} \renewcommand{\labelenumi}{(f\arabic{enumi})} % \item[({\bf f1})] \makeatletter \def\@currentlabel{f1}\label{f:Carath} \makeatother % $f: \Omega\times {\mathbb{R}}\to {\mathbb{R}}$ is a given function of Carath\'eodory type such that the function $u\mapsto f(\cdot\, u)$ maps bounded intervals in ${\mathbb{R}}$ into bounded sets in $L^\infty(\Omega)$. % \item[({\bf f2})] \makeatletter \def\@currentlabel{f2}\label{f:Growth} \makeatother % $f$ satisfies % $ f(\cdot,u) / \theta(u) {\stackrel{*}{\rightharpoonup}} f_{\pm\infty} \quad\text{weakly\--star in $L^\infty(\Omega)$ as } u\to \pm\infty$, % where $f_{\pm\infty}\not\equiv 0$ in $\Omega$, and \emph{either\/} $\theta(u)\equiv 1$ for $u\in {\mathbb{R}}$, \emph{or else\/} $\theta: {\mathbb{R}}\to {\mathbb{R}}$ is some $C^1$ function such that $\theta(0) = 0$, $\theta'(u)\neq 0$ for $u\neq 0$, $$ \lim_{|u|\to \infty} \left( \theta(u) / |u|^{p-1} \right) = 0 ,\quad \sup_{u\neq 0} \left| \theta'(u)\, u / \theta(u) \right| < \infty , $$ and % \begin{eqnarray*} \theta(\tau\varphi_1(x)) / \theta(\tau) \to \theta_{\pm\infty}(x) \quad\text{uniformly for $x\in \Omega$ as } \tau\to \pm\infty , %\label{def.th_inf} \end{eqnarray*} % with $\theta_{\pm\infty} > 0$ in $\Omega$. % \end{enumerate} \smallskip In particular, the expression $V_n = t_n^{1-p} v_n^\top$ (for $\theta\equiv 1$) in Theorem \ref{thm-v_n-asympt} becomes $$ V_n = \left[ t_n^{p-1} \theta( t_n^{-1} ) \right]^{-1} \, v_n^\top . $$ Notice that $t_n\to 0$ entails $t_n^{p-1} \theta( t_n^{-1} ) \to 0$. Of course, first, formula \eqref{z_n.asympt} has to be recalculated, and then also formula~\eqref{tau_n.asympt}. In contrast with the case $\theta\equiv 1$, two obvious nontrivial examples when $f$ satisfies conditions \eqref{f:Carath} and \eqref{f:Growth} are as follows: $$ f(x,u) = f_\infty(x)\, |u|^{q-1} + f_0(x,u) ; \quad x\in \Omega ,\ u\in {\mathbb{R}} , $$ and $$ f(x,u) = f_\infty(x)\, |u|^{q-2} u + f_0(x,u) ; \quad x\in \Omega ,\ u\in {\mathbb{R}} , $$ where $q$ is a constant, $12$, we set also $\mathbf{A}(\mathbf{0}) {\stackrel{{\mathrm {def}}}{=}} \mathbf{0}\in {\mathbb{R}}^{N\times N}$. For $\mathbf{a}\not= \mathbf{0}$, $\mathbf{A}(\mathbf{a})$ is a positive definite, symmetric matrix. The spectrum of $|\mathbf{a}|^{2-p} \mathbf{A}(\mathbf{a})$ consists of eigenvalues $1$ and $p-1$, whence % \begin{equation} \min\{ 1,\,p-1\} \leq \frac{ \langle \mathbf{A}(\mathbf{a}) \mathbf{v}, \mathbf{v} \rangle_{{\mathbb{R}}^N} }% { |\mathbf{a}|^{p-2} |\mathbf{v}|^2 } \leq \max\{ 1,\,p-1\} ,\quad \mathbf{a}, \mathbf{v}\in {\mathbb{R}}^N\setminus \{ \mathbf{0}\} . \label{A.ellipt} \end{equation} % From this point on, until the end of this paragraph, we restrict ourselves to $p\geq 2$. The case $1 0$. Then there exists a constant $c_p(\Theta) > 0$ such that the following inequalities hold true for all $\mathbf{a}, \mathbf{b}\in {\mathbb{R}}^N$: If $p > 2$ then % \begin{equation} \label{ineq.geom} \begin{aligned} c_p(\Theta)^{p-2} \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} & \leq \int_0^1 |\mathbf{a} + s\mathbf{b}|^{p-2} \,\Theta(s) \,\mathrm{d}s \\ & \leq T\cdot \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} , \end{aligned} \end{equation} % and if $1 < p < 2$ and $|\mathbf{a}| + |\mathbf{b}| > 0$ then % \begin{equation} \label{ineq.geom:p<2} \begin{aligned} T\cdot \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} & \leq \int_0^1 |\mathbf{a} + s\mathbf{b}|^{p-2} \,\Theta(s) \,\mathrm{d}s \\ & \leq c_p(\Theta)^{p-2} \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} . \end{aligned} \end{equation} % The inequalities below are a combination of \eqref{A.ellipt} with \eqref{ineq.geom} (if $p>2$) and \eqref{ineq.geom:p<2} (if $p<2$), see \cite[Lemma A.2]{Takac-2}: % If $p > 2$ then % \begin{equation} \label{A.geom} \begin{aligned} c_p(\Theta)^{p-2} \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} |\mathbf{v}|^2 & \leq \int_0^1 \langle \mathbf{A}(\mathbf{a} + s\mathbf{b}) \mathbf{v}, \mathbf{v} \rangle \,\Theta(s) \,\mathrm{d}s \\ & \leq (p-1) T\cdot \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} |\mathbf{v}|^2 \end{aligned} \end{equation} % for all $\mathbf{a}, \mathbf{b}, \mathbf{v}\in {\mathbb{R}}^N$, and if $1 < p < 2$ and $|\mathbf{a}| + |\mathbf{b}| > 0$ then % \begin{equation} \label{A.geom:p<2} \begin{aligned} (p-1) T\cdot \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} |\mathbf{v}|^2 &\leq \int_0^1 \langle \mathbf{A}(\mathbf{a} + s\mathbf{b}) \mathbf{v}, \mathbf{v} \rangle \,\Theta(s) \,\mathrm{d}s \\ & \leq c_p(\Theta)^{p-2} \Big( \max_{0\leq s\leq 1} |\mathbf{a} + s\mathbf{b}| \Big)^{p-2} |\mathbf{v}|^2 . \end{aligned} \end{equation} % Several important properties of $\mathcal{D}_{\varphi_1}$ established in { Tak\'a\v{c}} \cite{Takac-2} are listed below. The following claim is obvious (\cite[Lemma 4.1]{Takac-2}): If $12$ and $1 0$ small enough, $\|\cdot\|_{\varphi_1}$ is an equivalent norm on $W_0^{1,2}(\Omega_\delta)$. % \item[(b)] The embedding $\mathcal{D}_{\varphi_1} \hookrightarrow L^2(\Omega)$ is compact. % \end{itemize} % \end{lemma} Due to inequality~\eqref{norm:A.equiv} combined with Lemma~\ref{lem-Q-L^2}, Part~{\rm (b)} above, we can extend the domain of $\mathcal{Q}_0$ to $\mathcal{D}_{\varphi_1}\times \mathcal{D}_{\varphi_1}$. This extension of $\mathcal{Q}_0$ is unique and closed in $L^2(\Omega)$. % We denote by $\mathcal{A}_{\varphi_1}$ the Friedrichs representation of the quadratic form $2\cdot \mathcal{Q}_0$ in $L^2(\Omega)$; see \cite[Theorem VI.2.1, p.~322]{Kato}. This means that $\mathcal{A}_{\varphi_1}$ is a positive semidefinite, selfadjoint linear operator on $L^2(\Omega)$ with domain ${\mathop{\rm {dom}}}(\mathcal{A}_{\varphi_1})$ dense in $\mathcal{D}_{\varphi_1}$ and % \[ \langle \mathcal{A}_{\varphi_1} v, w\rangle = 2\cdot \mathcal{Q}_0(v,w) \quad\text{for all } v,w\in {\mathop{\rm {dom}}}(\mathcal{A}_{\varphi_1}) . \] % Notice that our definition of $\mathcal{Q}_0$ yields $\mathcal{A}_{\varphi_1} \varphi_1 = 0$. Since the embedding $\mathcal{D}_{\varphi_1} \hookrightarrow L^2(\Omega)$ is compact, by Lemma~\ref{lem-Q-L^2}, Part~{\rm (b)}, the null space of $\mathcal{A}_{\varphi_1}$ denoted by \[ \ker (\mathcal{A}_{\varphi_1}) = \{ v\in {\mathop{\rm dom}}(\mathcal{A}_{\varphi_1}): \mathcal{A}_{\varphi_1} v = 0 \} \] is finite-dimensional, by the Riesz\--Schauder theorem \cite[Theorem III.6.29, p.~187]{Kato}. Owing to hypothesis~\eqref{hyp:phi_1}, we have even $\ker (\mathcal{A}_{\varphi_1}) = {\mathop{\rm {lin}}} \{ \varphi_1\}$, by Proposition~\ref{prop-Uni-EV}. Now we switch to the case $1 0$ and\/ $t_n u_n\geq \frac12\varphi_1$ in $\Omega$ for all $n$ large enough; moreover, $t_n\to 0$ and\/ $v_n^\top\to 0$ in $C^{1,\beta'} (\overline{\Omega})$ as $n\to \infty$, with $\langle v_n^\top, \varphi_1\rangle = 0$ for\/ $n=1,2,\dots$. % \end{lemma} As a consequence of this lemma, we can rewrite problem~\eqref{e:BVP.th_n} as % \begin{equation} \begin{gathered} - \Delta_p (\varphi_1 + v_n^\top) = (\lambda_1 + \mu_n) |\varphi_1 + v_n^\top|^{p-2} (\varphi_1 + v_n^\top) + t_n^{p-1} f_n(x) \quad \text{ in } \Omega ; \\ v_n^\top = 0 \quad \text{ on } \partial\Omega ; \\ \langle v_n^\top, \varphi_1 \rangle = 0 , \end{gathered} \label{e:BVP.t_n} \end{equation} % with all $t_n > 0$, $t_n\searrow 0$ as $n\to \infty$. Indeed, if $\kappa < 0$, we take advantage of the $(p-1)$-homogeneity of problem~\eqref{e:BVP.u_n} and replace all functions $f_n$, $f$ and $u_n$ by $-f_n$, $-f$ and $-u_n$, respectively, thus switching to the case $\kappa > 0$. Hence, without loss of generality, we may assume $t_n > 0$ and $t_n u_n = \varphi_1 + v_n^\top\geq \frac12\varphi_1 > 0$ in $\Omega$ for all $n\geq 1$. A useful equivalent form of problem \eqref{e:BVP.u_n} is obtained by subtracting equation \eqref{e:varphi.l_1} from \eqref{e:BVP.t_n} and using the Taylor formula with a help from identity \eqref{F''.Matrix}, for $n=1,2,\dots$: % \begin{equation} \begin{gathered} {}- {\mathop{\rm {div}}}( \mathbf{A}_n \nabla v_n^\top ) = (p-1) (\lambda_1 + \mu_n) a_n v_n^\top + \mu_n \varphi_1^{p-1} + |t_n|^{p-2} t_n f_n(x) \quad \text{ in } \Omega ; \\ v_n^\top = 0 \quad \text{ on } \partial\Omega ; \\ \langle v_n^\top, \varphi_1 \rangle = 0 , \end{gathered} \label{eq:BVP.t_n} \end{equation} % with the abbreviations % \begin{equation} \label{def:A_n,a_n} \mathbf{A}_n {\stackrel{{\mathrm {def}}}{=}} \int_0^1 \mathbf{A} ( \nabla\varphi_1 + s\nabla v_n^\top ) \,\mathrm{d}s \quad\text{and}\quad a_n {\stackrel{{\mathrm {def}}}{=}} \int_0^1 | \varphi_1 + s v_n^\top |^{p-2} \,\mathrm{d}s . \end{equation} % Recall that the matrix $\mathbf{A}(\mathbf{a})$ is defined in \eqref{def.A=F''}. We abbreviate also % \begin{equation} \mathbf{A}_{\varphi_1} {\stackrel{{\mathrm {def}}}{=}} \mathbf{A}(\nabla\varphi_1) \quad\text{and write}\quad \mathbf{A}_{\varphi_1}^{1/2} = \sqrt{ \mathbf{A}_{\varphi_1} }\, . \label{def:A_varphi_1} \end{equation} % Equivalently, $V_n{\stackrel{{\mathrm {def}}}{=}} t_n^{1-p} v_n^\top$ $\in C^{1,\beta'} (\overline{\Omega})$ satisfies the linear boundary value problem % \begin{equation} \begin{gathered} - {\mathop{\rm {div}}} ( \mathbf{A}_n \nabla V_n ) = (p-1) (\lambda_1 + \mu_n) a_n V_n + \frac{\mu_n}{ |t_n|^{p-2} t_n }\, \varphi_1^{p-1} + f_n(x) \quad \text{ in } \Omega ; \\ V_n = 0 \quad\text{on } \partial\Omega ; \\ \langle V_n, \varphi_1 \rangle = 0 . \end{gathered} \label{eq:BVP.V_n} \end{equation} % The asymptotic behavior of $V_n$ and $\mu_n / ( |t_n|^{p-2} t_n )$ as $n\to \infty$ is determined as follows (\cite[Theorem 4.1]{DGTU}). \begin{theorem}\label{thm-v_n-asympt} Let\/ $12$ and in $W_0^{1,2}(\Omega)$ if $1 0$ holds by Proposition~\ref{prop-Uni-EV} and Remark~\ref{rem-Uni-EV}. In addition, we have % \begin{math} \int_\Omega f^\dagger\varphi_1 \,\mathrm{d}x = 0 . \end{math} % The linear degenerate Dirichlet problem \eqref{eq:V^top} above has been obtained by linearizing \eqref{e:BVP.l} with $\lambda = \lambda_1$ about $\varphi_1$, that is to say, from % \begin{equation} \begin{gathered} - {\mathop{\rm {div}}} \left( {\bf A}(\nabla\varphi_1) \nabla w \right) = \lambda_1 (p-1) \varphi_1^{p-2} w + f(x) \quad \text{in } \Omega ; \\ w = 0 \quad \text{on } \partial\Omega . \end{gathered} \label{e:BVP.w} \end{equation} % It plays a crucial role in our asymptotic formulas as $|\tau|\to \infty$. Its solution set in $\mathcal{D}_{\varphi_1}$ is described in Theorem \ref{thm-v_n-asympt}. % If $p<2$ then $\mathcal{D}_{\varphi_1}$ is not necessarily dense in $L^2(\Omega)$, and so equation~\eqref{e:BVP.w} can be satisfied only in the following weak sense, cf.\ eq.~\eqref{eq:V^top}: For all test functions $\phi\in \mathcal{D}_{\varphi_1}$, % \begin{equation} \int_U \langle {\bf A} (\nabla\varphi_1) \nabla w , \nabla\phi \rangle \,\mathrm{d}x = \lambda_1 (p-1) \int_\Omega \varphi_1^{p-2} w\phi \,\mathrm{d}x + \int_\Omega f\phi \,\mathrm{d}x \,. \label{e:BVP.w:p<2} \end{equation} % The following two special cases of formula \eqref{mu_n:2.Order} are of particular interest in the present work; both force $\int_\Omega f\varphi_1 \,\mathrm{d}x = 0$. \begin{corollary}\label{cor-v_n-asympt} In the situation of\/ {\rm Theorem~\ref{thm-v_n-asympt}} we have: If\/ $\int_\Omega f_n\varphi_1 \,\mathrm{d}x = 0$ for all\/ $n\in {\mathbb{N}}$, then % \begin{equation} \label{mu_n:2.Order:ortho} \mu_n = (p-2)\, |t_n|^{2(p-1)}\, \mathcal{Q}_0( V^\top, V^\top ) + o \big( |t_n|^{2(p-1)} \big) . \end{equation} % On the other hand, if\/ $\mu_n = 0$ for all\/ $n\in {\mathbb{N}}$, then % \begin{equation} \label{mu_n=0:2.Order} \lim_{n\to \infty}\ \frac{1}{ |t_n|^{p-2} t_n } \int_\Omega f_n\varphi_1 \,\mathrm{d}x = (p-2)\, \mathcal{Q}_0( V^\top, V^\top ) . \end{equation} % In particular, in both cases we must have $\int_\Omega f\varphi_1 \,\mathrm{d}x = 0$. % \end{corollary} \subsection{Uniform boundedness of the solution set} \label{ss:Unif_Bound} Finally, we are ready to establish the asymptotic behavior (blow\--up) of every weak solution to the resonant problem \eqref{e:BVP.l_1}, that is, % \begin{equation} - \Delta_p u = \lambda_1 |u|^{p-2} u + f(x) \quad\text{in } \Omega ;\quad u = 0 \quad\text{on } \partial\Omega , \nonumber \tag{\color{green}\ref{e:BVP.l_1}\color{black}} \end{equation} % with $f\equiv f^\top + \zeta\varphi_1$, where $f^\top\in L^\infty(\Omega)^\top$ and $\zeta\in {\mathbb{R}}$. It turns out to be convenient to apply Lemma~\ref{lem-u_n-tilde}, that is, to fix an arbitrary number $t\in {\mathbb{R}}\setminus \{ 0\}$ and consider only those weak solutions to problem~\eqref{e:BVP.l_1} that take the form $u = t^{-1}\varphi_1 + u^\top$ where $u^\top\in W_0^{1,p}(\Omega)^\top$ is an unknown function. Hence $u^\top\in C^{1,\beta} (\overline{\Omega})$ by regularity \cite{Anane-2, DiBene-1, Lieberman, Tolksdorf-2}. While determining the asymptotic behavior of $u$ as $t\to 0$, we regard $\zeta\in {\mathbb{R}}$ in eq.~\eqref{e:BVP.l_1} as a parameter depending on $t$ as well. Let $F(t)$ denote the set of all pairs $(\zeta, u^\top)\in {\mathbb{R}}\times W_0^{1,p}(\Omega)^\top$ satisfying the boundary value problem % \begin{equation} \begin{gathered} \begin{aligned} & - \Delta_p( t^{-1}\varphi_1 + u^\top ) - \lambda_1 | t^{-1}\varphi_1 + u^\top |^{p-2} ( t^{-1}\varphi_1 + u^\top )\\ & = f^\top(x) + \zeta\cdot \varphi_1(x) \quad \text{in } \Omega ; \end{aligned} \\ u^\top = 0 \quad \text{ on } \partial\Omega ; \\ \langle u^\top, \varphi_1 \rangle = 0 . \end{gathered} \label{t:BVP.l_1} \end{equation} % The asymptotic behavior of $F(t)$ as $t\to 0$ is determined next. The following equivalent form of problem~\eqref{t:BVP.l_1} will be needed with the new unknown function $v^\top{\stackrel{{\mathrm {def}}}{=}} t u^\top$: % \begin{equation} \begin{gathered} \begin{aligned} & - \Delta_p( \varphi_1 + v^\top ) - \lambda_1 | \varphi_1 + v^\top |^{p-2} ( \varphi_1 + v^\top )\\ & = |t|^{p-2} t\left( f^\top(x) + \zeta\cdot \varphi_1(x) \right) \quad \text{in } \Omega ; \end{aligned} \\ v^\top = 0 \quad \text{on } \partial\Omega ; \\ \langle v^\top, \varphi_1 \rangle = 0 . \end{gathered} \label{t:BVP.v} \end{equation} % We take a sequence of nonzero real numbers $t_n\to 0$ as $n\to \infty$ and a sequence of pairs $(\zeta_n, u_n^\top)\in F(t_n)$. We substitute $v_n^\top{\stackrel{{\mathrm {def}}}{=}} t_n u_n^\top$ and $V_n{\stackrel{{\mathrm {def}}}{=}} |t_n|^{-(p-2)} u_n^\top = |t_n|^{-(p-2)} t_n^{-1} v_n^\top$, and abbreviate $f_n{\stackrel{{\mathrm {def}}}{=}} f^\top + \zeta_n\varphi_1$. More generally, we can replace function $f^\top\in L^\infty(\Omega)^\top$, $f^\top\not\equiv 0$ in $\Omega$, by a bounded sequence $\{ f_n^\top\}_{n=1}^\infty$ of functions from $L^\infty(\Omega)^\top$ satisfying the following hypothesis (cf.\ {\rm ({\bf S2})} in {\S}\ref{ss:Approx}): % \begin{enumerate} \renewcommand{\labelenumi}{(S\arabic{enumi}$^\top$)} % \item[({\bf S2}$^\top$)] $f_n^\top{\stackrel{*}{\rightharpoonup}} f^\top$ in $L^\infty(\Omega)$ (weakly\--star) as $n\to \infty$. We require $f^\top \not\in \mathcal{D}_{\varphi_1}^{\perp, L^2}$. % \end{enumerate} % The following \emph{a~priori\/} asymptotic formula was obtained in { Tak\'a\v{c}} \cite[Prop.\ 6.1]{Takac-3}: \begin{proposition}\label{prop-v_n-t=0} Assume\/ {\rm ({\bf S2}$^\top$)}. If $t_n\neq 0$ and $t_n\to 0$ as $n\to \infty$, then $\zeta_n\to 0$, all conclusions of\/ {\rm Theorem \ref{thm-v_n-asympt}} and\/ {\rm Corollary \ref{cor-v_n-asympt}} remain valid with $f = f^\top$, and moreover % \begin{equation} \lim_{n\to \infty} \,\frac{ \zeta_n }{ |t_n|^{p-2} t_n } = (p-2) \|\varphi_1\|_{ L^2(\Omega) }^{-2} \cdot \mathcal{Q}_0(w,w) \neq 0 . \label{z_n.asympt} \end{equation} % \end{proposition} Here, for $p>2$, $w\in \mathcal{D}_{\varphi_1}$ is the unique weak solution of problem~\eqref{e:BVP.w} with $f = f^\top$ satisfying $\langle w, \varphi_1 \rangle = 0$. For $p<2$, equation~\eqref{e:BVP.w:p<2} replaces~\eqref{e:BVP.w}. Finally, Proposition~\ref{prop-Uni-EV} and Remark~\ref{rem-Uni-EV} guarantee $\mathcal{Q}_0(w,w) > 0$. \subsection*{Acknowledgments} % This work was supported in part by the German Academic Exchange Service (DAAD, Germany) within the exchange programs ``PROCOPE'' and ``Acciones Integradas'' with France and Spain. %\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.} \makeatother \begin{thebibliography}{00} \bibitem{AdamsFournier} R.~A. Adams and J.~J.~F. Fournier, \emph{``Sobolev Spaces''}, 2nd Ed., Academic Press, New York\--Oxford, 2003. \bibitem{Anane-1} A. Anane, {\it Simplicit\'e et isolation de la premi\`ere valeur propre du $p$-laplacien avec poids}, Comptes Rendus Acad. Sc. Paris, S\'erie I, {\bf 305} (1987), 725--728. \bibitem{Anane-2} A. Anane, \emph{``Etude des valeurs propres et de la r\'esonance pour l'op\'erateur $p$-Laplacien''}, Th\`ese de doctorat, Universit\'e Libre de Bruxelles, 1988, Brussels. \bibitem{CueTakac-1} M. Cuesta and P. Tak\'a\v{c}, {\it A strong comparison principle for the Dirichlet $p$-Laplacian}, in G.~Caristi and E.~Mitidieri; eds., \emph{``Proceedings of the Conference on Reaction-Diffusion Equations''}, 1995, Trieste, Italy. Lecture Notes in Pure and Applied Math., Vol.~{\bf 194}, pp.~79--87. Marcel Dekker, New York and Basel, 1998. \bibitem{CueTakac-2} M. Cuesta and P. Tak\'a\v{c}, {\it A strong comparison principle for positive solutions of degenerate elliptic equations}, Diff. and Int. Equations, {\bf 13}(4--6) (2000), 721--746. \bibitem{DeCoster} C. De Coster and M. Henrard, {\it Existence and localization of solution for second order elliptic BVP in presence of lower and upper solutions without any order}, J. Differential Equations, {\bf 145} (1998), 420--452. \bibitem{DiazSaa} J.~I. D{\'\i}az and J.~E. Saa, {\it Existence et unicit\'e de solutions positives pour certaines \'equations elliptiques quasilin\'eaires}, Comptes Rendus Acad. Sc. Paris, S\'erie I, {\bf 305} (1987), 521--524. \bibitem{DiBene-1} E. DiBenedetto, {\it $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations}, Nonlinear Anal., {\bf 7}(8) (1983), 827--850. \bibitem{Drabek-1} P. Dr\'abek, \emph{``Solvability and Bifurcations of Nonlinear Equations''}, Pitman Research Notes in Mathematics Series, Vol.~{\bf 264}. Longman Scientific {\&} Technical, U.K., 1992. \bibitem{Drabek-3} P. Dr\'abek, {\it Geometry of the energy functional and the Fredholm alternative for the $p$-Laplacian in higher dimensions}, Electr. J. Diff. Equations, Conference {\bf 08} (2002), 103--120. \bibitem{DrabGirgMan} P. Dr\'abek, P. Girg, and R.~F. Man\'asevich, {\it Generic Fredholm alternative\--type results for the one\--dimensional $p$-Laplacian}, Nonlinear Diff. Equations and Appl., {\bf 8} (2001), 285--298. \bibitem{DrabGirgTakac} P. Dr\'abek, P. Girg, and P. Tak\'a\v{c}, {\it Nonlinear perturbations of homogeneous quasilinear operators: bifurcation from infinity, existence and multiplicity}, J. Differential Equations, {\bf 204}(2) (2004), 265--291. \bibitem{DGTU} P. Dr\'abek, P. Girg, P. Tak\'a\v{c}, and M. Ulm, {\it The Fredholm alternative for the $p$-Laplacian: bifurcation from infinity, existence and multiplicity}, Indiana Univ. Math. J., {\bf 53}(2) (2004), 433-482. \bibitem{DrabHolub} P. Dr\'abek and G. Holubov\'a, {\it Fredholm alternative for the $p$-Laplacian in higher dimensions}, J. Math. Anal. Appl., {\bf 263} (2001), 182--194. \bibitem{DrabTakac} P. Dr\'abek and P. Tak\'a\v{c}, {\it Poincar\'e inequality and Palais\--Smale condition for the $p$-Laplacian}, Calc. Variations, {\bf 29} (2007), 31--58. {\it Online}:\hfil\break doi: 10.1007/s00526-006-0055-8. \bibitem{LCEvans} L.~C. Evans, \emph{``Partial Differential Equations''}, Graduate Studies in Mathematics, Vol.~{\bf 19}, American Math.\ Society, Providence, R.~I., 1998. \bibitem{FGTT-1} J. Fleckinger, J.-P. Gossez, P. Tak\'a\v{c}, and F. de~Th\'elin, {\it Existence, nonexistence et principe de l'antimaximum pour le $p$-laplacien}, Comptes Rendus Acad. Sc. Paris, S\'erie I, {\bf 321} (1995), 731--734. \bibitem{FGTT-2} J. Fleckinger, J.-P. Gossez, P. Tak\'a\v{c}, and F. de~Th\'elin, {\it Nonexistence of solutions and an anti-maximum principle for cooperative systems with the $p$-Laplacian}, Mathematische Nachrichten, {\bf 194} (1998), 49--78. \bibitem{FHTT} J. Fleckinger, J. Hern\'andez, P. Tak\'a\v{c}, and F. de~Th\'elin, {\it Uniqueness and positivity for solutions of equations with the $p$-Laplacian}, in G.~Caristi and E.~Mitidieri; eds., \emph{``Proceedings of the Conference on Reaction-Diffusion Equations''}, 1995, Trieste, Italy. Lecture Notes in Pure and Applied Math., Vol.~{\bf 194}, pp.~141--155. Marcel Dekker, New York and Basel, 1998. \bibitem{FleckTakac} J. Fleckinger and P. Tak\'a\v{c}, {\it An improved Poincar\'e inequality and the $p$-Laplacian at resonance for $p>2$}, Advances Diff. Equations, {\bf 7}(8) (2002), 951--971. \bibitem{GilbargTrud} D.~Gilbarg and N.~S. Trudinger, \emph{``Elliptic Partial Differential Equations of Second Order''}, 2nd Ed., in \emph{Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften}, Vol.~{\bf 224}. Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1977. \bibitem{GirgTakac} P. Girg and P. Tak\'a\v{c}, {\it Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems}, Ann.~Henri Poincar\'e, {\bf 9} (2008), 275--327. \bibitem{Kato} T. Kato, \emph{``Perturbation Theory for Linear Operators''}, in \emph{Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften}, Vol.~{\bf 132}. Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1980. \bibitem{Lieberman} G.~M. Lieberman, {\it Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Anal., {\bf 12}(11) (1988), 1203--1219. \bibitem{Lindqvist} P. Lindqvist, {\it On the equation $\mathrm{div}(\vert\nabla u\vert^{p-2} \nabla u)$ $+$ $\lambda \vert u\vert^{p-2} u$ $= 0$}, Proc. Amer. Math. Soc., {\bf 109}(1) (1990), 157--164. \bibitem{ManTakac} R.~F. Man\'asevich and P. Tak\'a\v{c}, {\it On the Fredholm alternative for the $p$-Laplacian in one dimension}, Proc. London Math. Soc., {\bf 84} (2002), 324--342. \bibitem{PinoDrabMan} M.~A. del~Pino, P. Dr\'abek, and R.~F. Man\'asevich, {\it The Fredholm alternative at the first eigenvalue for the one\--dimensional $p$-Laplacian}, J. Differential Equations, {\bf 151} (1999), 386--419. \bibitem{PinoElgMan} M.~A. del~Pino, M. Elgueta, and R.~F. Man\'asevich, {\it A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')' + f(t,u) = 0$, $u(0) = u(T) = 0$, $p>1$}, J. Differential Equations, {\bf 80}(1) (1989), 1--13. \bibitem{Pohozaev} S.~I. Pohozaev, {\it The fibering method in nonlinear variational problems}, in P.~Dr\'abek; ed., \emph{``Topological and Variational Methods for Nonlinear Boundary Value Problems''}, Pitman Research Notes in Mathematics Series, Vol.~{\bf 365}, pp.~35--88. Addison Wesley Longman, Essex, U.K., 1997. \bibitem{Rabin} P.~H. Rabinowitz, \emph{``Minimax Methods in Critical Point Theory with Applications to Differential Equations''}, CBMS Series in Mathematics, Vol.~{\bf 65}, Amer. Math. Soc., Providence, R.I., 1986. \bibitem{Takac-1} P. Tak\'a\v{c}, {\it Degenerate elliptic equations in ordered Banach spaces and applications}, in P.~Dr\'abek, P.~Krej\v{c}\'{\i}, and P.~Tak\'a\v{c}; eds., \emph{``Nonlinear Differential Equations''}, Chapman {\&} Hall/CRC Research Notes in Mathematics Series, Vol.~{\bf 404}, pp.~111--196. CRC Press LLC, Boca Raton, FL, U.S.A., 1999. (Formerly Pitman Mathematics Series.) \bibitem{Takac-2} P. Tak\'a\v{c}, {\it On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue}, Indiana Univ. Math. J., {\bf 51}(1) (2002), 187--237. \bibitem{Takac-3} P. Tak\'a\v{c}, {\it On the number and structure of solutions for a Fredholm alternative with the $p$-Laplacian}, J. Differential Equations, {\bf 185} (2002), 306--347. \bibitem{Takac-4} P. Tak\'a\v{c}, {\it Nonlinear Spectral Problems for Degenerate Elliptic Operators}, in M.~Chipot and P.~Quittner; eds., \emph{``Handbook of Differential Equations: Stationary Partial Differential Equations''}, Vol.~{\bf 1}, pp.~385--489. Elsevier Science B.V., Amsterdam, The Netherlands, 2004. \bibitem{Takac-5} P. Tak\'a\v{c}, {\it A variational approach to the Fredholm alternative for the $p$-Laplacian near the first eigenvalue}, J. Dynamics Diff. Equations, {\bf 18}(3) (2006), 693--765. {\it Online}:\hfil\break doi: 10.1007/s10884-006-9017-2. \bibitem{TakacUT} P. Tak\'a\v{c}, L. Tello, and M. Ulm, {\it Variational problems with a $p$-homogeneous energy}, Positivity, {\bf 6}(1) (2001), 75--94. \bibitem{Tolksdorf-1} P. Tolksdorf, {\it On the Dirichlet problem for quasilinear equations in domains with conical boundary points}, Comm. P.D.E., {\bf 8}(7) (1983), 773--817. \bibitem{Tolksdorf-2} P. Tolksdorf, {\it Regularity for a more general class of quasilinear elliptic equations}, J. Differential Equations, {\bf 51} (1984), 126--150. \bibitem{Vazquez} J.~L. V\'azquez, {\it A strong maximum principle for some quasilinear elliptic equations}, Appl. Math. Optim., {\bf 12} (1984), 191--202. \bibitem{Veron-1} L. V\'eron, {\it Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemanniean manifolds}, Colloq. Math. Soc. J\'anos Bolyai, {\bf 62} (1991), 317--352. \bibitem{Veron-2} L. V\'eron, {\it Premi\`ere valeur propre non nulle du $p$-laplacien et \'equations quasi lin\'eaires elliptiques sur une vari\'et\'e riemannienne}, Comptes Rendus Acad. Sc. Paris, S\'erie I, {\bf 314} (1992), 271--276. \end{thebibliography} \end{document}