\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 216, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/216\hfil Nontrivial solutions] {Nontrivial solutions for nonlinear problems with one sided resonance} \author[G. Smyrlis \hfil EJDE-2012/216\hfilneg] {George Smyrlis} % in alphabetical order \address{George Smyrlis \newline Technological Educational Institute of Athens \\ Department of Mathematics \\ Athens 12210, Greece} \email{gsmyrlis@teiath.gr, gsmyrlis@syros.aegean.gr} \thanks{Submitted March 21, 2012. Published November 29, 2012.} \subjclass[2000]{35J80, 35J85, 58E05} \keywords{One sided resonance; principal spectral interval; Cerami-condition; \hfill\break\indent critical groups; Morse theory} \begin{abstract} We find nontrivial smooth solutions for nonlinear elliptic Dirichlet problems driven by the $p$-Laplacian ($1 -\infty$. Let $c < \inf \varphi(K_{\varphi})$. The critical groups of $\varphi$ at infinity are defined by \[ C_k(\varphi, \infty)=H_k(X, \varphi^c) \quad\text{for all } k\geq 0 \] (see \cite{Ba-Li}). The second deformation theorem (see, e.g. \cite{Gas-Pap}) implies that this definition is independent of the particular choice of the level $c < \inf \varphi(K_{\varphi})$. If $C_k(\varphi, \infty)\neq 0$, for some $k\geq 0$, then there exists a critical point $x\in X$ of $\varphi$, such that $C_k(\varphi, x) \neq 0$. Finally, let us recall some basic facts about the spectrum of the negative Dirichlet $p$-Laplacian with weight $m$, denoted by $(-\Delta^D_p, m)$. So, let \[ L^{\infty}( \Omega)_{+}=\{ m \in L^{\infty}( \Omega): m(z) \geq 0 \text{ a.e. in } \Omega \}, \] let $m\in L^{\infty}( \Omega)_{+} \setminus \{0\}$ and consider the weighted nonlinear eigenvalue problem \begin{equation} \label{eigen} \begin{gathered} -\Delta_p u(z) = \widehat{\lambda} m(z) |u(z)|^{p-2}u(z),\quad \text{a.e. in } \Omega,\\ u |_{\partial \Omega} =0, \quad \widehat{\lambda} \in \mathbb{R} . \end{gathered} \end{equation} By an eigenvalue of $(-\Delta^D_p,m)$ we mean a real number $\widehat{\lambda}$, such that \eqref{eigen} has a nontrivial solution $u$. Nonlinear regularity theory (see e.g. \cite[pp. 737-738]{Gas-Pap}) implies that $u \in C_0^1(\overline{ \Omega})$. The least $\widehat{\lambda} \in \mathbb{R}$ for which \eqref{eigen} has a nontrivial solution is the first eigenvalue of $(-\Delta^D_p,m)$ and it is denoted by $\widehat{\lambda}_1(m)$. We recall some basic properties of $\widehat{\lambda}_1(m)$: \begin{itemize} \item $\widehat{\lambda}_1(m) >0$. \item $\widehat{\lambda}_1(m)$ is isolated (i.e., there exists $\varepsilon>0$ such that $(\widehat{\lambda}_1(m), \widehat{\lambda}_1(m)+\varepsilon)$ contains no eigenvalues). \item $\widehat{\lambda}_1(m)$ is simple (i.e., the corresponding eigenspace is one-dimensional). \item $\widehat{\lambda}_1(m)$ is characterized by the Rayleigh quotient: \[ \widehat{\lambda}_1(m) =\inf \big\{ \frac{\|Du\|^p_p }{\int_ \Omega m |u|^p \,dz }: u\in W_0 ^{1,p}( \Omega), u \not\equiv 0 \big\}. \] \end{itemize} The above is attained on the one dimensional eigenspace of $\widehat{\lambda}_1(m)$. Let $\widehat{u}_1$ be a normalized eigenfunction of $\widehat{\lambda}_1(m)$, i.e., \[ \int_{ \Omega} m |\widehat{u}_1|^p dz =1. \] We already know that $\widehat{u}_1 \in C_0^1(\overline{ \Omega})$ and from the Rayleigh quotient, it is clear that $\widehat{u}_1$ does not change sign, so we may assume that $\widehat{u}_1(z) \geq 0$, for all $z\in \overline{ \Omega}$. Using the nonlinear maximum principle of V\'{a}zquez \cite{Vaz}, we obtain that $\widehat{u}_1(z)>0, $ for all $z\in \overline{ \Omega}$. It turns out that for each $\widehat{\lambda}_1(m)-$ eigenfunction $u$ we have that $u(z)\neq 0, $ for all $z\in \overline{ \Omega}$. For more details we refer for example to \cite{An,Gas-Pap,Lind1, Lind2}. Since $-\Delta^D_p$ is $(p-1)$-homogeneous operator, the Ljusternik-Schnirelmann theory implies that we have a whole strictly increasing sequence $\{\widehat{\lambda}_k (m) \}_{k \geq 1} $ of eigenvalues such that \[ \widehat{\lambda}_k (m) \to +\infty, \quad\text{as } k \to +\infty \] (see \cite{Ga-Az-Pe-Al}). These eigenvalues are called the ``LS-eigenvalues'' of $(-\Delta^D_p, m)$. We know that $\widehat{\lambda}_2 (m)$ is the second eigenvalue of $(-\Delta^D_p, m)$; i.e., $\widehat{\lambda}_2 (m) >\widehat{\lambda}_1 (m)$ and there are no eigenvalues between $\widehat{\lambda}_1(m)$ and $\widehat{\lambda}_2 (m)$. Viewed as functions of the weight $m \in L^{\infty}( \Omega)_{+}\setminus \{0\}$, the eigenvalues $\widehat{\lambda}_1(m)$ and $\widehat{\lambda}_2 (m)$ are continuous functions and exhibit certain monotonicity properties, namely: \begin{itemize} \item If $m(z) \leq \widetilde{m}(z)$, a.e. on $ \Omega$, with strict inequality on a set of positive measure, then $\widehat{\lambda}_1 (\widetilde{m}) < \widehat{\lambda}_1 (m)$. \item If $m(z) < \widetilde{m}(z)$, a.e. on $ \Omega$, then $\widehat{\lambda}_2 (\widetilde{m}) < \widehat{\lambda}_2 (m)$ (see \cite{An-Tsou}). \end{itemize} Special cases: If $m\equiv 1$, then we write $\widehat{\lambda}_k (m)=\lambda_k , k\geq 1$ and $\lambda_k$ is the $k$-th eigenvalue of the negative Dirichlet $p$-Laplacian $-\Delta^D_p$. If $m \equiv \lambda_k$ for some $k \geq 1$, then clearly $\widehat{\lambda}_k (\lambda_k)=1$. \section{Main result} In this section we establish the existence of at least one nontrivial smooth solution of the problem \eqref{P}, when one-sided resonance occurs at the principal spectral interval $[\lambda_1, \lambda_2)$ of $-\Delta^D_p$. The hypotheses on the reaction $f(z,x)$ are: \noindent\textbf{(H)} $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a Carath\'{e}odory function such that $f(z,0)=0$ for a.a. $z\in \Omega$, \begin{itemize} \item[(i)] \[ |f(z,x)| \leq \alpha(z) +c_1|x|^{p-1} \] for a.a. $z\in \Omega$, all $x\in \mathbb{R}$, with $\alpha \in L^{\infty}( \Omega)_{+}, c_1>0$. \item[(ii)] \[ \lambda_1 \leq \liminf_{|x| \to \infty} \frac{f(z,x)}{|x|^{p-2}x} \leq \limsup_{|x| \to \infty} \frac{f(z,x)}{|x|^{p-2}x} < \lambda_2 , \quad\text{uniformly for a.a.} z\in \Omega. \] \item[(iii)] If $F(z,x)= \int_0^x f(z,s)ds$, then \[ \lim_{|x| \to \infty} [f(z,x)x -pF(z,x) ] =+\infty, \quad \text{uniformly for a.a.} z\in \Omega . \] \item[(iv)] There exist $\tau, \sigma \in (1,p)$, $\delta_0 >0$, $c_2 >0 $ such that for almost all $z\in \Omega$ and for all $|x| \leq \delta_0$, we have \[ F(z,x) \geq c_2 |x|^{\tau}, \quad \sigma F(z,x) \geq f(z,x)x. \] \end{itemize} Note that Hypothesis H(ii) implies that we have one-sided resonance at the principal spectral interval $[\lambda_1, \lambda_2)$ of $-\Delta^D_p$. On the other hand, hypothesis H(iii) enables us to avoid conditions of Landesman-Lazer type which are usually imposed on the nonlinearity when one deals with problems at resonance. \begin{remark} \label{rmk2} \rm Each weak solution $u \in W_0^{1,p}( \Omega)$ of problem \eqref{P} is smooth; i.e., $u \in C^1_0(\overline{ \Omega})$. This follows from the nonlinear regularity theory (see \cite{Lad}, \cite{Lieb}) and from the fact that the function $\alpha$ in hypothesis H(i) lies in $L^{\infty}( \Omega)_{+} .$ \end{remark} \begin{example} \label{examp1} \rm The following function satisfies H(i)-(iv) (for the sake of simplicity, we drop the $z$-dependence): \[ f(x)=\begin{cases} \lambda_{1}|x|^{p-2}x -|x|^{\tau-2}x, &\text{if } |x|> 1 \\ \lambda_{1} |x|^{\tau-2}x - |x|^{p-2}x, &\text{if } |x| \leq 1 \end{cases} \] with $1 < \tau

1$ and for some $c_3>0$ we have \[ xf(x) -pF(x) =(\frac{p}{\tau} -1) |x|^{\tau} -c_3 \to +\infty, \quad\text{as } |x| \to \infty \] and thus, H(iii) also holds. Finally, to obtain H(iv) choose \[ \sigma \in (\tau, p), \quad c_2 \in (0,\frac{\lambda_1}{\tau}) \quad\text{and}\quad \delta_0 \in (0,1) \quad\text{with} \quad \delta_0^{p-\tau} < p (\frac{\lambda_1}{\tau} -c_2 ). \] Then for $|x| \leq \delta_0$ we have \begin{gather*} \sigma F(x) -xf(x) =\lambda_1 (\frac{\sigma}{\tau}-1 )|x|^{\tau} + (1 -\frac{\sigma}{p}) |x|^p \geq 0,\\ F(x)= \frac{\lambda_1}{\tau}|x|^{ \tau} - \frac{|x|^p}{p} =|x|^{ \tau} (\frac{\lambda_1}{\tau}-\frac{|x|^{p-\tau}}{p} ) \geq c_2 |x|^{\tau}. \end{gather*} \end{example} In \cite{R-R-Shap}, $f(x)$ is unbounded for $x <0$ and bounded for $x \geq 0$. For the function $f$ defined above we have that $f(+\infty)=+\infty$. Now we set $g(x)=f(x)-\lambda_1 |x|^{p-2}x, x\in \mathbb{R}$. Under the classic versions of the LL - conditions, the limits $g(\pm \infty)$ are real numbers (see for example \cite{Arc-Ors}, \cite{L-L}). Unlike these works, the above defined function $g$ satisfies $g(\pm \infty)=\mp \infty$. Moreover, \emph{generalized} LL - conditions are used in \cite{L-R-R,Rob,Rumb,Su} in the semilinear case ($p=2$). In all these works, the function $g$ satisfies the following condition: \begin{quote} For each sequence $\{w_n\} \subseteq W^{1,2}_0( \Omega)$ with \[ \|w_n\| \to \infty, \quad \frac{\|P_1 w_n\|}{\|w_n\|} \to 1, \] we have that \[ \limsup_n \int_{ \Omega} g(w_n(z)) \frac{P_1 w_n(z)}{\|P_1 w_n\|}dz >0, \] where $P_1$ is the projection operator from $W^{1,2}_0( \Omega)$ onto the principal eigenspace of $-\Delta^D$. \end{quote} In our example this condition fails. To see this, let $\widehat{u}_1$ be the normalized positive smooth principal eigenfunction of $-\Delta_p^D$ and set $w_n=n \widehat{u}_1$, $n \geq 1$. Clearly, $\|w_n\| \to \infty $ and $\|P_1 w_n\|/ \|w_n\|=1$, for all $n\geq 1$ (note that for $p\neq 2$, the projection $P_1$ is still well defined). On the other hand, for all $n> 1/ \min_{\overline{ \Omega}} \widehat{u}_1$ we have \[ \int_{ \Omega} g(w_n(z)) \frac{P_1 w_n(z)}{\|P_1 w_n\|}dz =-n^{\tau -1} \int_{ \Omega} \widehat{u}_1(z)^{\tau} dz \to -\infty, \quad\text{as } n \to \infty. \] We introduce the energy functional \[ \varphi(u) =\frac{1}{p} \|Du\|_p^p - \int_{ \Omega}F(z,u(z))dz, \quad u\in W_0^{1,p}( \Omega). \] Under hypothesis H(i), $\varphi\in C^1(W_0^{1,p}( \Omega))$ and each weak solution of the problem \eqref{P} is a critical point of $\varphi$. Since $f(z,0)=0, $ a.e. in $ \Omega$, the origin $0$ is trivially a critical point of $\varphi$. We search for nontrivial critical points of $\varphi$. For this purpose, we are going to compute the critical groups \[ C_k(\varphi, \infty), \quad C_k(\varphi, 0), \quad k \geq 0. \] First, we compute the critical groups of $\varphi$ at infinity. In this direction, we prove an auxiliary result which slightly extends \cite[Lemma 2.4]{Pe-Sch} (the latter is formulated in Hilbert spaces). \begin{proposition} \label{homotopy} Let $X$ be a Banach space and $(t,u) \to h_t (u) $ be a homotopy which belongs to $C^1([0,1] \times X)$ and it is bounded. Suppose that \begin{itemize} \item[(i)] there exists $R>0$ s.t. for all $t\in [0,1]$, \[ K_{h_t} \subseteq \overline{B}_R =\{ x\in X: \|x\| \leq R\} \] \item[(ii)] the maps $u \to \partial_t h_t (u) $ and $u \to h'_t(u) $ are both locally Lipschitz \item[(iii)] $h_0$ and $h_1$ both satisfy the C-condition \item[(iv)] there exist $\beta \in \mathbb{R}$ and $\delta >0$ s.t. \[ h_t(u) \leq \beta \Rightarrow (1+\|u\|)\| h'_t(u)\|_{*} \geq \delta \quad\text{for all } t\in [0,1]. \] \end{itemize} Then $C_k(h_0, \infty)=C_k (h_1, \infty)$, for all $k \geq0$. \end{proposition} \begin{proof} By the hypothesis $h \in C^1([0,1] \times X)$, we know that it admits a pseudogradient vector field $\widehat{v} =(v_0, v): [0,1] \times (X\setminus \overline{B}_R ) \to [0,1] \times X$. Moreover, taking into account the construction of the pseudogradient vector field, we know that $v_0 =\partial_t h_t$. Also, by definition $(t,u) \to v_t (u) $ is locally Lipschitz and in fact for every $t\in [0,1]$, $v_t(\cdot)$ is a pseudogradient vector field for the functional $h_t(\cdot)$. So, for every $(t,u) \in [0,1] \times (X \setminus \overline{B}_R) $ we have \begin{equation} \label{39} \langle h'_t(u), v_t(u) \rangle \geq \| h'_t(u)\|^2_{*}. \end{equation} The map \[ X \setminus \overline{B}_R \ni u \to -\frac{|\partial_t h_t(u)|}{\| h'_t(u)\|^2_{*}} v_t(u)=w_t(u) \in X \] is well defined and locally Lipschitz. Since by hypothesis $(t,u) \to h_t (u) $ is bounded, we can find $\eta \leq \beta $ s.t. \[ \eta < \inf [h_t (u): t\in [0,1], \|u\| \leq R]. \] We choose $\eta \leq \beta$ s.t. $h^{\eta}_0 \neq\emptyset $ or $h^{\eta}_1 \neq\emptyset, $ (if no such $\eta$ can be found, then $C_k(h_0, \infty)=C_k(h_1, \infty) =H_k(X,\emptyset)=\delta_{k,0} \mathbb{Z} $ for all $k \geq 0$ and so we are done). To fix things, we assume that $h^{\eta}_0 \neq\emptyset $ and choose $y \in h^{\eta}_0$. We consider the following Cauchy problem \begin{equation} \label{40} \frac{d\xi}{dt}=w_t(\xi) \quad t \in [0,1], \quad \xi(0)=y. \end{equation} Since $w_t$ is locally Lipschitz, this Cauchy problem admits a unique local flow (see \cite[p. 618]{Gas-Pap}). We have \begin{align*} \frac{d}{dt} h_t(\xi) &=\langle h'_t (\xi), \frac{d\xi}{dt} \rangle + \partial_t h_t(\xi) \\ &= \langle h'_t (\xi), w_t(\xi) \rangle + \partial_t h_t(\xi) \quad\text{(see \eqref{40})} \\ &\leq -|\partial_t h_t(\xi)| + \partial_t h_t(\xi) \leq 0 \end{align*} (see \eqref{39}). This implies that the mapping $ t \mapsto h_t(\xi(t,y)) $ is non-increasing. Therefore, \begin{gather*} h_t(\xi(t,y)) \leq h_0(\xi(0,y)) =h_0 (y) \leq \eta \leq \beta,\\ \Rightarrow (1+\|\xi(t,y)\| ) \|h'_t(\xi(t,y))\|_{*} \geq \delta \end{gather*} (by hypothesis); therefore, $h'_t(\xi(t,y))\neq 0$. This shows that the flow $\xi(\cdot, y)$ is global on $[0,1]$. Then $\xi(1,y)$ is a homeomorphism between $h^{\eta}_0$ and a subset of $h^{\eta}_1 .$ Reversing the time $(t \to 1-t)$, we show that $h^{\eta}_1$ is a homeomorphism to a subset of $h^{\eta}_0 .$ Therefore $h^{\eta}_0$ and $h^{\eta}_1$ are homotopy equivalent and so \begin{gather*} H_k (X, h^{\eta}_0 )=H_k (X, h^{\eta}_1 )\quad\text{for all } k \geq 0, \\ \Rightarrow C_k(h_0 , \infty)=C_k (h_1 ,\infty) \quad\text{for all } k \geq 0. \end{gather*} \end{proof} To proceed, let $\widehat{u}_1$ be a $\lambda_1$-eigenfunction of $-\Delta_p^D$ with $\|\widehat{u}_1\|_p=1$. Consider the set \[ V=\{ u\in W_0^{1,p}( \Omega): \int_{ \Omega} \widehat{u}_1^{p-1} u dz =0 \}. \] Then $V$ is a closed linear subspace of $W_0^{1,p}( \Omega)$ and we have \[ W_0^{1,p}( \Omega)= \mathbb{R} \widehat{u}_1 \oplus V. \] We introduce the quantity \[ \lambda_V =\inf \big\{ \frac{\|Du\|^p_p}{\|u\|_p^p} : u\in V, u\neq 0 \big\}. \] We know that $\lambda_1 < \lambda_V \leq \lambda_2 $ (see \cite[Lemma 3.3]{Gas-Pap new}). Let $\mu \in (\lambda_1, \lambda_{V} )$ and consider the $C^1$-functional $\psi: W_0^{1,p}( \Omega) \to \mathbb{R}$ defined by \[ \psi(u)=\frac{1}{p}\|Du\|_p^p - \frac{\mu}{p} \|u\|_p^p \quad \text{for all } u \in W_0^{1,p}( \Omega). \] Using standard arguments we may show that $\psi$ has the following properties: \begin{itemize} \item $0$ is the unique critical point of $\psi$. \item $\psi$ satisfies the Palais-Smale condition. \item $\psi _{\mid \mathbb{R} \widehat{u}_1} \text{is anticoercive}, \quad \psi _{\mid V} \text{ is coercive}.$ \end{itemize} The last two properties yield \begin{equation} \label{psi 1} C_1(\psi, \infty) \neq 0 \end{equation} (see \cite[Proposition 3.8]{Ba-Li}). We intend to prove the following statement. \begin{proposition} \label {infinity} Under hypotheses {\rm H(i), (ii), (iii)}, we have \[ C_k (\varphi, \infty) \simeq C_k (\psi, \infty), \quad k \geq 0. \] \end{proposition} For the proof of Proposition \ref{infinity} we shall need the following result. \begin{proposition} \label{basic} Assume that hypotheses {\rm H(i), (ii), (iii)} hold. We consider the homotopy $h: [0,1] \times W_0^{1,p}( \Omega) \to \mathbb{R}$ defined by \[ h(t,u) =(1-t)\varphi(u) + t\psi(u) \quad\text{for all } (t,u) \in [0,1] \times W_0^{1,p}( \Omega). \] Let $\{u_n\}_{n \geq 1} \subseteq W_0^{1,p}( \Omega),\, \{t_n \}_{n\geq 1} \subseteq [0,1] $ be sequences such that \[ t_n \to t, \quad (1+\|u_n\|) \|h_u'(t_n , u_n)\|_{*} \to 0, \quad \|u_n\| \to +\infty. \] Then by passing to subsequences, we obtain \[ t_n \to 0, \quad |u_n(z)| \to +\infty, \text{ a.e. in } \Omega, \quad h(t_n , u_n) \to +\infty. \] \end{proposition} \begin{proof} By the convergence \[ (1+\|u_n\|)\|h_u'(t_n , u_n)\|_{*} \to 0 \] we have \begin{equation} \label{42} \big| \langle A(u_n), h\rangle -(1-t_n) \int_{ \Omega} f(z,u_n)hdz -t_n \mu \int_{ \Omega} |u_n|^{p-2}u_n h dz \big| \leq \frac{\varepsilon_n \|h\|}{1+\|u_n\|} \end{equation} for all $h \in W_0^{1,p}( \Omega)$, with $\varepsilon_n \to 0^{+}$. We set $ y_n =\frac{u_n}{\|u_n\|}$, $n \geq 1$. Then $\|y_n\|=1$ for all $n \geq 1$ and so we may assume that \begin{equation} \label{43} y_n \stackrel{w}{\to} y \text{ in } W_0^{1,p}( \Omega), \quad y_n \to y \text{ in } L^p( \Omega), \quad y_n(z) \to y(z), \text{ a.e. in } \Omega. \end{equation} Dividing both sides of \eqref{42} by $\|u_n\|^{p-1}$ we have \begin{equation} \label{44} \begin{aligned} & \big| \langle A(y_n), h\rangle -(1-t_n) \int_{ \Omega} \frac{f(z,u_n)}{\|u_n\|^{p-1}} hdz -t_n\mu \int_{ \Omega} |y_n|^{p-2} y_n h dz \big| \\ &\leq \frac{\varepsilon_n \|h\|}{(1+\|u_n\|)\|u_n\|^{p-1}}, \quad \text{for all } n \geq 1. \end{aligned} \end{equation} Hypothesis H(i) implies that the sequence \[ \Big\{ \frac{f(\cdot, u_n(\cdot))}{\|u_n\|^{p-1}} \Big\}_{n \geq 1} \subseteq L^{p'}( \Omega), \quad 1/p +1/p'=1, \] is bounded. Thus, we may assume that it is weakly convergent in $L^{p'}( \Omega)$. Using hypothesis H(iii) and reasoning as in \cite[Proposition 5]{M-M-Pa}, we may find $\xi \in L^{\infty}( \Omega)_{+}$ such that \begin{equation} \label{45} \frac{f(\cdot, u_n(\cdot))}{\|u_n\|^{p-1}} \stackrel{w}{\to} \xi |y|^{p-2}y \text{ in } L^{p'}( \Omega) \quad\text{and}\quad \lambda_1 \leq \xi(z) < \lambda_{2} \text{ a.e. in } \Omega. \end{equation} In \eqref{44} we choose $h=y_n -y \in W_0^{1,p}(\Omega)$, pass to the limit as $n \to \infty$ and use \eqref{43}. Then \[ \lim_{n \to \infty} \langle A(y_n), y_n -y \rangle =0, \] which implies $y_n \to y $ in $W_0^{1,p}( \Omega)$ (since $A$ is of type $(S)_{+}$). Then \begin{equation} \label{46} \|y\|=1. \end{equation} So, if in \eqref{44} we pass to the limit as $n \to \infty$ and use \eqref{45} and \eqref{46}, then \[ \langle A(y), h\rangle=(1-t) \int_{ \Omega} \xi |y|^{p-2}y hdz + t \mu \int_{ \Omega} |y|^{p-2}y h dz \quad\text{for all } h\in W_0^{1,p}( \Omega), \] which implies \[ A(y)=\xi_t |y|^{p-2}y \quad \text{with } \xi_t=(1-t)\xi +t \mu; \] therefore, \begin{equation} \label{47'} -\Delta_p y(z)=\xi_t(z)|y(z)|^{p-2}y(z) \text{ a.e. in } \Omega, \quad u =0 \text{ on } \partial \Omega. \end{equation} Note that $\lambda_1 \leq \xi_t (z) < \lambda_{2} $ a.e. in $ \Omega$ (recall that $t\in [0,1], \lambda_1 < \mu <\lambda_2$). If $\xi_t \not\equiv \lambda_1$, then the monotonicity properties of the weighted eigenvalues (see Section 2) yield \[ \widehat{\lambda}_1 (\xi_t) < \widehat{\lambda}_1 (\lambda_1)=1, \quad \widehat{\lambda}_{2} (\xi_t) > \widehat{\lambda}_{2} (\lambda_2)=1; \] therefore, $y \equiv 0$ (see \eqref{47'}) which contradicts \eqref{46}. Thus, $\xi_t \equiv \lambda_1$, so $t=0$ and $\xi\equiv \lambda_1$. It follows from \eqref{47'} that $y$ is a $\lambda_1$-eigenfunction and hence, $y(z)\neq 0$, a.e. in $ \Omega$. Consequently, \begin{equation} \label{new} |u_n(z)| =\|u_n\| |y_n(z)| \to +\infty, \quad\text{a.e. in } \Omega. \end{equation} It remains to show that \[ h(t_n , u_n) \to +\infty. \] Indeed, the convergence \[ (1+\|u_n\|) \|h_u'(t_n , u_n)\|_{*} \to 0 \] implies that \[ \langle h'_u(t_n , u_n) , u_n \rangle \to 0. \] Moreover, \eqref{new} combined with hypothesis H(iii) and also with Fatou's lemma gives \[ \int_{ \Omega} [ u_n(z) f(z, u_n(z)) -pF(z, u_n(z)) ]dz \to +\infty. \] Now the conclusion follows from the formula \[ ph(t_n , u_n)= \langle h'_u(t_n , u_n) , u_n \rangle + (1-t_n) \int_{ \Omega} [ u_n(z) f(z, u_n(z)) -pF(z, u_n(z)) ]dz, \] $n \geq 1$. \end{proof} \begin{corollary} \label{coro1} Under hypotheses {\rm H(i), (ii), (iii)}, the energy functional $\varphi$ satisfies the Cerami condition. \end{corollary} \begin{proof} Suppose that $\{u_n\}_{n\geq 1} \subseteq W_0^{1,p}( \Omega)$ satisfies \[ \sup_n |\varphi(u_n)| < \infty, \quad (1+\|u_n\|) \|\varphi'(u_n)\|_{*} \to 0. \] We claim that $\{u_n\}_{n \geq 1}$ is bounded in $W_0^{1,p}( \Omega)$. Indeed, if this is not the case, then by passing to subsequences we may assume that \[ \|u_n \| \to +\infty. \] Now we observe that $\varphi(u)=h(0, u), $ for all $u\in W_0^{1,p}( \Omega)$. Applying Proposition \ref {basic} and by passing to subsequences we deduce that $\varphi(u_n)=h(0, u_n) \to +\infty$ (false). This proves our claim, i.e., $\{u_n\}_{n \geq 1}$ is bounded in $W_0^{1,p}( \Omega)$. Therefore, we may assume that \begin{equation} \label{24} u_n \stackrel{w}{\to} u \text{ in } W^{1,p}_0( \Omega) \quad\text{and} \quad u_n \to u \text{ in } L^p( \Omega) . \end{equation} Then \eqref{24} in conjunction with hypothesis H(i) and also with the convergence $\|\varphi'(u_n)\|_{*} \to 0$ yields \[ \int_{ \Omega} f(\cdot, u(\cdot))(u_n -u) dz \to 0, \quad \langle \varphi'(u_n), u_n -u \rangle \to 0. \] But \[ \langle A(u_n), u_n -u \rangle -\int_{ \Omega} f(\cdot, u(\cdot))(u_n -u) dz = \langle \varphi'(u_n), u_n -u \rangle , \quad n \geq 1, \] so, \[ \lim_{n \to \infty} \langle A(u_n), u_n -u \rangle =0, \Rightarrow u_n \to u \quad\text{in } W^{1,p}_0( \Omega) \] (since $A$ is of type $(S)_{+}$). \end{proof} \begin{proof}[Proof of Proposition \ref{infinity}] We consider the homotopy $h: [0,1] \times W_0^{1,p}( \Omega) \to \mathbb{R}$ defined by \[ h(t,u) =(1-t)\varphi(u) + t\psi(u) \quad\text{for all } (t,u) \in [0,1] \times W_0^{1,p}( \Omega). \] Clearly, $h(0, \cdot)=\varphi, h(1, \cdot)=\psi$. By Proposition \ref{homotopy}, it suffices to show that there exist $\beta \in \mathbb{R}, \delta >0$, such that for all $t\in [0,1]$, $u \in W_0^{1,p}( \Omega)$, \[ h(t,u) \leq \beta \Rightarrow (1+\|u\|) \|h'_u (t,u)\|_{*} > \delta. \] Suppose that this is not the case. Then we may find \[ \{t_n \}_{n\geq 1} \subseteq [0,1], \quad \{u_n \}_{n\geq 1} \subseteq W_0^{1,p}( \Omega), \] such that \[ t_n \to t \in [0,1], \quad (1+\|u_n\|) \|h_u'(t_n , u_n)\|_{*}\to 0, \quad h(t_n , u_n) \to -\infty. \] Now Proposition \ref{basic} guarantees that $\{u_n \}_{n\geq 1}$ is bounded so, we may assume that \eqref{24} holds. Applying \eqref{42} for $h=u_n -u$ and passing to the limit as $n \to +\infty$, we obtain \[ \lim_{n \to \infty} \langle A(u_n), u_n -u \rangle =0 \] which implies $u_n \to u$ in $W^{1,p}_0( \Omega)$ (since $A$ is of type $(S)_{+}$). Therefore, $h(t_n , u_n) \to h(t,u)$, which is a contradiction. \end{proof} Next, we compute the critical groups of $\varphi$ at zero. Without loss of generality we may assume that $0$ is an isolated critical point of $\varphi$ (otherwise we can produce a whole sequence of distinct critical points of $\varphi$, so we are done). We start with two lemmas. \begin{lemma} \label{key 1} Let $g \in C^1([0,1])$ such that either $g(1) <0$ or $g(1)=0, g'(1)>0$. If $g(\widehat{t} )>0, $ for some $\widehat{t} \in (0,1)$, then there exists $\widehat{t}_2 \in(\widehat{t},1)$, such that \[ g(\widehat{t}_2)=0, \quad g'( \widehat{t}_2) \leq 0. \] \end{lemma} \begin{proof} We claim that \[ g(\widehat{t}_1)=0, \quad\text{for some } \widehat{t}_1 \in (\widehat{t}, 1) . \] Indeed, this is clear from Bolzano's theorem, in the case $g(1)<0$. Suppose now that $g(1)=0, g'(1) >0$. By continuity of $g'$, we may find $\theta \in (0,1)$, such that \[ 0 < \widehat{t} < \theta <1, \quad g' > 0 \quad \text{on } [\theta,1]. \] Since $g(1)=0$, we obtain that $g <0$ on $[\theta, 1)$ and the claim follows again from Bolzano's theorem. To proceed, we set \[ \widehat{t}_2 =\min \{ t\in [\widehat{t}, 1]: g(t)=0 \}. \] Then \[ \widehat{t} < \widehat{t}_2 \leq \widehat{t}_1 , \quad g(\widehat{t}_2)=0, \quad g(t) \neq 0, \quad \text{for all } t\in [\widehat{t}, \widehat{t}_2). \] But since $g(\widehat{t} )>0$, the continuity of $g$ gives $g(t) >0$ for all $ t\in [\widehat{t}, \widehat{t}_2)$. Then \[ g'(\widehat{t}_2)=\lim_{t \to \widehat{t}^{ -}_2 } \frac{g(t)}{t-\widehat{t}_2} \leq 0 , \] which completes the proof. \end{proof} \begin{lemma} \label{key2} Let $X$ be a Banach space and $\varphi \in C^1(X), \rho>0$ such that \[ \langle \varphi'(u), u\rangle >0, \quad\text{for all } u\in \overline{B}_{\rho}\setminus \{0\} \text{ with } \varphi(u)=0. \] Then \begin{itemize} \item[(i)] for each $u\in \varphi^{ 0} \cap \overline{B}_{\rho}$, we have $[0, u] \subseteq \varphi^{ 0}$, where \[ [0, u]=\{ tu: t\in [0,1] \}. \] \item[(ii)] the set $\varphi^{0} \cap \overline{B}_{\rho}$ is contractible. \end{itemize} (Here $\overline{B}_{\rho}$ is the closed ball centered at the origin with radius $\rho$ and $\varphi^{ 0}$ is the sublevel set of $\varphi$ at $0$.) \end{lemma} \begin{proof} (i) Suppose on the contrary that \[ \varphi(\widehat{t}u) >0, \quad\text{for some } u\in (\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\} ,\; \widehat{t} \in (0,1). \] Define $g(t)=\varphi(tu)$, $t\in [0,1]$. Then $g(\widehat{t} ) >0$. If $\varphi(u) <0, $ then $g(1)<0$. If $\varphi(u)=0$, then $g(1)=0$ and \[ g'(1) = \langle \varphi'(u), u\rangle >0. \] Hence, $g$ satisfies the hypotheses of Lemma \ref{key 1}, so we may find $\widehat{t}_2 \in (\widehat{t},1)$, such that \[ g(\widehat{t}_2)=0, \quad \quad g'( \widehat{t}_2) \leq 0 . \] But then \[ 0 < \langle \varphi'(\widehat{t}_2 u, \widehat{t}_2 u \rangle = \widehat{t}_2 g'( \widehat{t}_2) \leq 0, \] which is a contradiction. (ii) Define the homotopy $h: [0,1] \times ( \varphi^{ 0} \cap \overline{B}_{\rho} ) \to \varphi^{ 0} \cap \overline{B}_{\rho}$ by \[ h(t,u)=(1-t)u . \] Due to (i), $h$ is well defined whereas it is clearly continuous. Since $h(1,u)=0$ for all $u\in \varphi^{ 0} \cap \overline{B}_{\rho}$, we derive that the set $\varphi^{ 0} \cap \overline{B}_{\rho}$ is contractible in itself. \end{proof} \begin{proposition} \label{zero} Under hypotheses {\rm H(i), H(iv)}, we have \[ C_k(\varphi, 0)=0, \quad\text{for all } k \geq 0. \] \end{proposition} \begin{proof} From hypothesis H(iv), we can find $c_3 , c_4>0$ such that \begin{equation} \label{47} F(z,x) \geq c_3|x|^{\tau} - c_4|x|^r \quad\text{for all } z\in \Omega, \text{ all } x\in \mathbb{R}, \end{equation} with $p < r< p^*$ ( $p^*$ denotes the critical Sobolev exponent). \noindent\textbf{Claim 1:} There exists $\rho \in (0,1)$ small such that \[ \langle \varphi'(u), u\rangle >0, \quad\text{for all } u\in \overline{B}_{\rho}\setminus \{0\} \text{ with } \varphi(u)=0. \] To see this, choose $u\in W^{1,p}_0( \Omega)\setminus \{0\}$ such that $\varphi(u)=0$. Then \begin{equation} \label{50} \begin{aligned} \langle \varphi^{ '}(u), u \rangle &=\|Du\|_p^p - \int_ \Omega f(z,u)u dz\\ &= (1- \frac{\sigma}{p})\|Du\|_p^p + \int_ \Omega ( \sigma F(z,u) - f(z,u)u ) dz \quad (\text{since }\varphi(u)=0) \\ &= (1-\frac{\sigma}{p})\|Du\|_p^p + \int_{\{|u| \leq \delta_0\}} (\sigma F(z,u) - f(z,u)u) dz \\ &\quad + \int_{\{|u|> \delta_0\}} ( \sigma F(z,u) - f(z,u)u )dz. \end{aligned} \end{equation} By hypothesis H(iv), we have \begin{equation} \label{51} \int_{\{|u| \leq \delta_0\}} ( \sigma F(z,u) - f(z,u)u ) dz \geq 0. \end{equation} Moreover, hypothesis H(i) implies \begin{equation} \label{52} \int_{\{|u| > \delta_0\}} ( \sigma F(z,u) - f(z,u)u ) dz \geq -c_5\|u\|^r_r \end{equation} for some $c_5>0$ and with $p < r< p^*$. Returning to \eqref{50} and use \eqref{51}, \eqref{52} with the embedding $W_0^{1,p}( \Omega) \subseteq L^r( \Omega)$, to obtain \[ \langle \varphi^{ '}(u), u \rangle \geq (1- \frac{\sigma}{p})\|Du\|_p^p - c_6 \|Du\|^r_p \quad \text{for some } c_6 > 0. \] Now Claim 1 follows easily from the last inequality, because of the fact that $ \sigma < p< r$. Taking into account Claim 1 in conjunction with Lemma \ref{key2}(ii) we deduce that \[ \varphi^{ 0} \cap \overline{B}_{\rho} \quad\text{is contractible}. \] \noindent\textbf{Claim 2:} For each $u \in W_0^{1,p}( \Omega) \setminus \{0\}$, there exists $t^*=t^*(u) \in (0,1)$ small such that \[ \varphi(tu) <0 \quad\text{for all } t \in (0, t^*). \] Indeed, for $t>0$ and $u\in W^{1,p}_0( \Omega)$, we have \begin{align*} \varphi(tu) &= \frac{t^p}{p} \|Du\|_p^p - \int_{ \Omega} F(z,tu)dz \\ &\leq \frac{t^p}{p} \|Du\|_p^p - c_3 t^{\tau} \|u\|^{\tau}_{\tau} + c_4 t^{r} \|u\|^r_r \quad\text{(see \eqref{47})}. \end{align*} Then Claim 2 follows from the fact that $\tau < p < r$. \noindent\textbf{Claim 3:} Let $\rho >0$ be as postulated in Claim 1. Then for each $u\in \overline{B}_{\rho}$ with $\varphi(u) >0$, there exists a unique $t(u) \in (0,1)$ such that \[ \varphi(t(u)u)=0. \] To prove this, let $u\in \overline{B}_{\rho}$ be fixed with $\varphi(u)>0$. Then Claim 2 combined with Bolzano's theorem yield \[ \varphi(t(u)u)=0, \quad\text{for some } t(u) \in (0,1). \] We need to show that this $t(u) \in (0,1)$ is unique. We argue by contradiction. So, suppose we can find \[ 0< t_1(u) 0, \] by Claim 1. Thus, we arrived at a contradiction and the proof of Claim 3 is complete. Summarizing the above arguments we obtain the following: \begin{itemize} \item For each $u\in \overline{B}_{\rho}$ with $\varphi(u) \leq 0$, we have that $\varphi \leq 0$ on $[0, u]$. Moreover, the set $\varphi^{ 0} \cap \overline{B}_{\rho}$ is contractible. \item For each $u\in \overline{B}_{\rho} \setminus \{0\}$ with $\varphi(u) >0$, there exists a unique $t(u) \in (0,1)$ such that \[ \varphi(t(u)u)=0, \quad \varphi < 0 \text{ on } (0, t(u)u), \quad \varphi>0 \text{ on } (t(u)u, u]. \] \end{itemize} To proceed, let $q: \overline{B}_{\rho}\setminus \{0\} \to (0,1]$ be defined by \[ q(u)=\begin{cases} 1 &\text{if } u \in \overline{B}_{\rho}\setminus \{0\}, \varphi(u) \leq 0\\ t(u) &\text{if } u \in \overline{B}_{\rho}\setminus \{0\}, \varphi(u) > 0. \end{cases} \] According to the previous discussion, $q$ is well-defined and the implicit function theorem implies that $q$ is continuous. Let $Q:\overline{B}_{\rho}\setminus \{0\} \to (\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}$ be defined by \[ Q(u)= q(u)u. \] Clearly, $Q$ is continuous and $Q |_{(\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}} =\text{id}| _{(\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}}$. It follows that $(\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}$ is a retract of $\overline{B}_{\rho}\setminus \{0\}$. Since $W^{1,p}_0( \Omega)$ is infinite dimensional, the set $\overline{B}_{\rho}\setminus \{0\}$ is contractible in itself, hence so is the set $(\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}$. Finally, since both $\varphi^{ 0} \cap \overline{B}_{\rho}$ and $(\varphi^{ 0} \cap \overline{B}_{\rho})\setminus \{0\}$ are contractible, we conclude that \[ C_k(\varphi, 0) =H_k(\varphi^{ 0} \cap \overline{B}_{\rho} , (\varphi^{ 0} \cap \overline{B}_{\rho}) \setminus \{0\})=0 \quad \text{for all } k\geq 0 \] (see Granas -Dugundji \cite[p. 389]{Gr-Du}). \end{proof} Now we are ready to state and prove our existence result. \begin{theorem} \label{main} Under hypotheses {\rm H(i)-(iv)}, problem \eqref{P} has at least one nontrivial smooth solution. \end{theorem} \begin{proof} By Proposition \ref{infinity} we obtain that \[ C_1(\varphi, \infty) \simeq C_1(\psi, \infty) \neq 0 \quad \] (see \eqref{psi 1}), which implies that \[ C_1(\varphi, u)\neq 0, \quad\text{for some } u\in K_{\varphi} . \] Clearly, $u$ is a smooth weak solution to the problem (see remark \ref{rmk2}). On the other hand, Proposition \ref{zero} says that $C_1(\varphi,0)=0$. Hence, $u\not\equiv 0$. \end{proof} \begin{thebibliography}{99} \bibitem{An} A. Anane: \emph{Simplicit\`{e} et isolation de la premi\`{e}re valeur propre du $p$-laplacien avec poids} C.R. Acad. Sci. Paris S\`{e}r. I Math 305 (1987), 725-728. \bibitem{An-Tsou} A. Anane, N. Tsouli: \emph{On the second eigenvalue of the $p$- Laplacian} in A. Benikrane, J-P. Gossez, eds \emph{Nonlinear Partial Differential Equations (F\`{e}s, 1994)}, Vol. 343 of Pitman Research Notes in Math. Series, Longman, Harlow (1996), 1-9. \bibitem{Arc-Ors} D. Arcoya, L. Orsina: \emph{Landesman-Lazer conditions and quasilinear elliptic equations}, Nonlin. Anal. 28 (1997), 1623-1632. \bibitem{Ba-Li} T. Bartsch - S. Li: \emph{Critical point theory for asymptotically quadratic functionals and applications to problems with resonance}, Nonlin. Anal. 28 (1997), 419-441. \bibitem{Ch} K.-C. Chang: \emph{Infinite Dimensional Morse Theory and Multiple Solution Problems}, Birk\-h\"{a}user, Boston (1993). \bibitem{Ga-Az-Pe-Al} J. Garcia Azorero, J. Manfredi, I. Peral Alonso: \emph{Sobolev versus H\"{o}lder minimizers and global multiplicity for some quasilinear elliptic equations}, Comm. Contemp. Math. 2 (2000), 385-404. \bibitem{Gas-Pap} L. Gasinski, N. S. Papageorgiou: \emph{Nonlinear Analysis}, Chapman Hall/ CRC Press, Boca Raton(2006). \bibitem{Gas-Pap new} L. Gasinski, N. S. Papageorgiou: \emph{Multiple solutions for asymptotically $(p-1)$-homogeneous $p$-Laplacian equations}, J. Functional Anal 262 (2012), 2403-2435. \bibitem{Gr-Du} A. Granas, J. Dugundji: \emph{Fixed Point Theory} Springer-Verlag, New York (2003). \bibitem{Lad} O. Ladyzhenskaya, N. Uraltseva: \emph{Linear and Quasilinear Elliptic Equations}, Academic Press, New York (1968). \bibitem{L-L} E. M. Landersman, A.C. Lazer: \emph{Nonlinear perturbations of linear eigenvalues problem at resonance}, J. Math. Mech., 19 (1970), 609-623 \bibitem{L-R-R} E. Landesman, S. Robinson, A. Rumbos: \emph{Multiple solutions of semilinear elliptic problems at resonance}, Nonlinear Analysis 24 (1995), 1045-1059. \bibitem{Lieb} G. Lieberman: \emph{Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Analysis 12(1988), 1203-1219. \bibitem{Lind1} P. Lindqvist: \emph{On the equation div$(\|\nabla x\|^{p-2} \nabla x) +\lambda |x|^{p-2}x=0$}, Proc. Amer. Math. Soc. 109 (1990), 157-164. \bibitem{Lind2} P. Lindqvist: \emph{Addendum to ``On the equation div$(\|\nabla x\|^{p-2} \nabla x) +\lambda |x|^{p-2}x=0$''} Proc. Amer. Math. Soc. 116 (1992), 583-584. \bibitem{Maw-Mill} J. Mawhin, M. Millem: \emph{Critical Point Theory and Hamiltonian Systems}, Springer-Verlag, New York (1989). \bibitem{M-M-Pa} D. Motreanu, V. Motreanu, N. S. Papageorgiou: \emph{A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations}, Manuscripta Math. 124 (2007), 507-531. \bibitem{Pe-Sch} K. Pererra, M. Schechter: \emph{Solutions of nonlinear equations having asymptotic limits at zero and infinity}, Calc. Var. PDEs 12 (2001), 359-369. \bibitem{Rob} S. Robinson: \emph{Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains}, Nonlinear Analysis 21 (1993), 407-424. \bibitem{R-R-Shap} S. B. Robinson, A. Rumbos, V. L. Shapiro: \emph{One-sided resonance problems for quasilinear elliptic operators}, J. Math. Anal. Appl. 256 (2001), 636-649. \bibitem{Rumb} A. Rumbos: \emph{Semilinear elliptic boundary value problem at resonance where the nonlinearity may grow linearly}, Nonlinear Anal. 16 (1991), 1159-1168. \bibitem{Su} J. Su: \emph{Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues}, Nonlinear Analysis 48 (2002), 881-895. \bibitem{Vaz} J-L. Vazquez: \emph{A strong maximum principle for some quasilinear elliptic equations}, Appl. Math. Optim. 12 (1984), 191-202. \end{thebibliography} \end{document}