\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 149, pp. 1--27.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/1149\hfil Superlinear problem under strong resonance] {Asymmetric superlinear problems under strong resonance conditions} \author[L. Recova, A. Rumbos \hfil EJDE-2017/149\hfilneg] {Leandro Recova, Adolfo Rumbos} \address{Leandro L. Recova \newline Ericsson Inc., Irvine, California 92609, USA} \email{leandro.recova@ericsson.com} \address{Adolfo J. Rumbos \newline Department of Mathematics, Pomona College, Claremont, California 91711, USA} \email{arumbos@pomona.edu} \thanks{Submitted February 19, 2017. Published June 23, 2017.} \subjclass[2010]{35J20} \keywords{Strong resonance; Palais-Smale condition; Ekeland's principle} \begin{abstract} We study the existence and multiplicity of solutions of the problem \begin{equation}\label{dde1} \begin{gathered} -\Delta u = -\lambda_1 u^- + g(x,u),\quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq 2)$, $u^-$ denotes the negative part of $u\colon\Omega\to\mathbb{R}$, $\lambda_1$ is the first eigenvalue of the $N$-dimensional Laplacian with Dirichlet boundary conditions in $\Omega$, and $g:\Omega\times\mathbb{R}\to\mathbb{R}$ is a continuous function with $g(x,0) =0$ for all $x\in\Omega$. We assume that the nonlinearity $g(x,s)$ has a strong resonant behavior for large negative values of $s$ and is superlinear, but subcritical, for large positive values of $s$. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of problem \eqref{dde1} as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin \cite{PucSer}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Let $\Omega$ denote a bounded, connected, open subset of $\mathbb{R}^N$, for $N\geqslant 2$, with smooth boundary $\partial\Omega$. We are interested in the existence and multiplicity of solutions of the semilinear elliptic boundary value problem (BVP): \begin{equation}\label{de1} \begin{gathered} -\Delta u = -\lambda_1 u^- + g(x,u), \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $u^-$ denotes the negative part of $u\colon\Omega\to\mathbb{R}$, $\lambda_1$ is the first eigenvalue of the $N$-dimensional Laplacian with Dirichlet boundary conditions in $\Omega$, and $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ and its primitive \begin{equation}\label{Gdfn05} G(x,s)=\int_{0}^{s}g(x,\xi)d\xi, \quad\text{for } x\in\overline{\Omega} \text{ and }s\in\mathbb{R}, \end{equation} satisfy the following conditions: \begin{itemize} \item [(A1)] $g\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ and $g(x,0) = 0$ for all $x\in\overline{\Omega}$. \item [(A2)] $\lim_{s\to -\infty} g(x,s) = 0$, uniformly for a.e. $x\in\Omega$. \item [(A3)] There exists a constant $\sigma$ such that $1\leqslant\sigma<(N+2)/(N-2)$ for $N\geqslant 3$, or $1\leqslant \sigma < \infty$ for $N=2$, and $$ \lim_{s\to +\infty}\frac{g(x,s)}{s^{\sigma}}=0, $$ uniformly for a.e. $x\in\Omega$. \item [(A4)] There are constants $\mu > \max\big\{ 2, \frac{2N\sigma}{N+2}\big\}$ and $s_o>0$ such that $$ 0<\mu G(x,s)\leqslant sg(x,s), \quad\text{for } s\geqslant s_o \text{ and } x\in\overline{\Omega}. $$ \item [(A5)] $\lim_{s\to-\infty} G(x,s) \equiv G_{-\infty}$, uniformly in $x$, where $G_{-\infty} \in\mathbb{R}$. \end{itemize} Writing \begin{equation}\label{qDef} q(x,s) = -\lambda_1 s^- + g(x,s), \quad\text{for } (x,s)\in\overline{\Omega}\times\mathbb{R}, \end{equation} we assume further that \begin{itemize} \item [(A6)] $q\in C^1(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ and $q(x,0)=0$; and \item [(A7)] $ \frac{\partial q}{\partial s}(x,0) = a$, for all $x\in\overline{\Omega}$, where $a > \lambda_1$. \end{itemize} We determine conditions under which the BVP in \eqref{de1} has nontrivial solutions. By a solution of \eqref{de1} we mean a weak solution; i.e, a function $u\in H_{0}^{1}(\Omega)$ satisfying \begin{equation}\label{dweakcondd} \int_{\Omega}\nabla u\cdot\nabla v \,dx + \lambda_{1}\int_\Omega u^- v\,dx - \int_{\Omega}g(x,u)v\,dx = 0, \quad\text{for all } v\in H_{0}^{1}(\Omega), \end{equation} where $H_{0}^{1}(\Omega)$ is the Sobolev space obtained through completion of $C_{c}^{\infty}(\Omega)$ with respect to the metric induced by the norm $$ \|u\|=\Big(\int_{\Omega}|\nabla u|^{2}dx\Big)^{1/2},\quad \text{for all }u\in H_{0}^{1}(\Omega). $$ The weak solutions of \eqref{de1} are the critical points of the functional $J\colon H_{0}^{1}(\Omega)\to\mathbb{R}$ given by \begin{equation}\label{SResEqn00005} J(u) = \frac{1}{2} \int_\Omega |\nabla u|^2 \,dx -\frac{\lambda_1}{2}\int_\Omega (u^-)^2\,dx - \int_\Omega G(x,u(x))\,dx, \end{equation} for $u\in H_{0}^{1}(\Omega)$. Indeed, the functional $J$ given in \eqref{SResEqn00005} is in $C^1(H_{0}^{1}(\Omega),\mathbb{R})$ with Fr\'echet derivative at every $u\in H_{0}^{1}(\Omega)$ given by \begin{equation}\label{SResEqn00010} J'(u) v = \int_{\Omega}\nabla u\cdot\nabla v \,dx + \lambda_1\int_\Omega u^- v\,dx - \int_{\Omega}g(x,u)v\,dx , \quad\text{for all } v\in H_{0}^{1}(\Omega). \end{equation} Thus, comparing \eqref{dweakcondd} with \eqref{SResEqn00010}, we see that critical points of $J$ are weak solutions of \eqref{de1}. In many problems, the following condition, known as the Palais-Smale condition, is usually needed to prove the existence of critical points of a functional. \begin{definition}[Palais-Smale Sequence]\label{PSseq05} \rm Let $J\in C^1(X,\mathbb{R})$, where $X$ is a Banach space with norm $\|\cdot\|$. A sequence $(u_m)$ in $X$ satisfying $$ J(u_m) \to c \quad\text{and } \quad \|J'(u_m)\|\to 0 \quad\text{as } m\to\infty, $$ is said to be a Palais-Smale sequence for $J$ at $c$. If $(u_m)$ is a sequence satisfying \begin{itemize} \item [(i)] $|J(u_m)|\leqslant M$ for all $m=1,2,3,\ldots$ and some $M>0$; \item [(ii)] $\|J'(u_m)\|\to 0$ as $m\to\infty$, \end{itemize} we say that $(u_m)$ is a Palais-Smale sequence for $J$. \end{definition} \begin{definition}[Palais-Smale Condition]\label{PScondition05} \rm A functional $J\in C^1(X,\mathbb{R})$, where $X$ is a Banach space with norm $\|\cdot\|$, is said to satisfy the the Palais-Smale condition at $c$, denoted $\text{(PS)}_c$, if every Palais-Smale sequence for $J$ at $c$ has a convergent subsequence. In particular, if $J$ has a Palais-Smale sequence at $c$, and $J$ satisfies the $\text{(PS)}_c$ condition, then $c$ is a critical value of $J$. We say that $J$ satisfies the $\text{(PS)}$ condition if every $\text{(PS)}$ sequence for $J$ has a convergent subsequence. \end{definition} It follows from condition (A2) and \eqref{qDef} that \begin{equation}\label{SResEqn00017} \lim_{s\to-\infty} \frac{q(x,s)}{s} = \lambda_1, \quad\text{for all } x\in\overline{\Omega}. \end{equation} The condition in \eqref{SResEqn00017} makes the BVP in \eqref{de1} into a problem at resonance. Existence for problems at resonance is sometimes obtained by imposing a Landesman-Lazer type condition on the nonlinearity. The authors of this article obtained existence and multiplicity for the BVP \eqref{de1} in \cite{RecRumb2} for the case in which $$ \lim_{s\to-\infty} g(x,s) = g_{-\infty}(x) $$ exists for all $x\in\overline{\Omega}$, and \begin{equation}\label{LLc} \int_\Omega g_{-\infty}(x)\varphi_1(x)\,dx > 0, \end{equation} where $\varphi_{1}$ is an eigenfunction of the $N$-dimensional Laplacian over $\Omega$ corresponding to the eigenvalue $\lambda_{1}$, with $\varphi_{1}(x)>0$ for all $x\in\Omega$. In the case in which the Landesman-Lazer condition \eqref{LLc} holds, the authors were able to prove that the functional $J$ defined in \eqref{SResEqn00005} satisfies the $\text{(PS)}$ condition. Note that the assumption in (A2) prevents condition \eqref{LLc} from holding true. So that, a Landesman-Lazer type condition does not hold for the problem at hand. As a consequence, we will not be able to prove that the functional $J$ satisfies the $\text{(PS)}$ condition. We will, however, be able to show that $J$ satisfies the $\text{(PS)}_c$ condition at values of $c$ that are not in an exceptional set, $\Lambda$. In the case in which conditions (A1)--(A5) hold true, we will prove in the next section that the functional $J\in C^1(H_{0}^{1}(\Omega),\mathbb{R})$ given in \eqref{SResEqn00005} satisfies the $\text{(PS)}_c$ condition provided that \begin{equation}\label{SResEqn00015} c \neq - G_{-\infty}|\Omega|, \end{equation} where $|\Omega|$ denotes the Lebesgue measure of $\Omega$; thus, the exceptional set in this case is \begin{equation}\label{LambdaSet} \Lambda = \{ - G_{-\infty}|\Omega|\}. \end{equation} It is not hard to see that the functional $J$ defined in \eqref{SResEqn00005} does not satisfy the $\text{(PS)}_c$ condition at $c = c_{-\infty} \equiv - G_{-\infty}|\Omega|$. Indeed, the sequence of functions $(u_m)$ given by $$ u_m = -m \varphi_1, \quad\text{for } m = 1,2,3,\ldots, $$ is a $\text{(PS)}_{c_{-\infty}}$ sequence, as a consequence of assumptions (A2) and (A5). However, $$ \|u_{m+1} - u_m\| = \|\varphi_1\|, \quad\text{for all } m = 1,2,3,\ldots; $$ so that $(u_m)$ has no convergent subsequence. This lack of compactness is typical of problems at \emph{strong resonance}. The term strong resonance refers to the situation described by the assumptions in (A2) and (A5) and was introduced by Bartolo, Benci and Fortunato in \cite{BBF}. In \cite{BBF}, the authors consider problems similar to \eqref{de1} in which $g$ is bounded, and in which the exceptional set is a singleton as in \eqref{LambdaSet}; more precisely, the authors of \cite{BBF} consider the class of BVPs of the form \begin{equation}\label{BBFprobk} \begin{gathered} -\Delta u = q_k (u), \quad\text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $$ q_k(s) = \lambda_k s - g(s), $$ with $\lambda_k$ an eigenvalue of the Laplacian, and $g\colon\mathbb{R}\to\mathbb{R}$ a bounded, continuous function with $$ \lim_{|s|\to\infty} s g(s) = 0. $$ Furthermore, the authors of \cite{BBF} assume that the function $$ G(s) = \int_{-\infty}^{s} g(\xi)\,d\xi $$ is defined for all $s\in\mathbb{R}$, and satisfies $G(s)\geqslant 0$ for all $s\in\mathbb{R}$, and $$ \lim_{s\to\infty} G(s) = 0. $$ The authors of \cite{BBF} proved existence of weak solutions of BVP \eqref{BBFprobk} by introducing a compactness condition (Condition (C)) that replaces the (PS) condition, and using the new condition to prove a variant of the deformation lemma. In \cite{CostaSilva1} and \cite{CostaSilva2}, Costa and Silva are able to obtain some of the existence and multiplicity results of Bartolo, Benci and Fortunato \cite{BBF} by establishing that the associated functional $J$ satisfies the $\text{(PS)}_c$ condition for values of $c$ that are not in an exceptional set. More recently, Hirano, Li and Wang \cite{HLW} have used Morse Theory to obtain multiplicity results for this type of problems with strong resonance. In \cite{HLW}, the exceptional set, $\Lambda$, consists of a finite number of values. They are able to compute critical groups around the values in $\Lambda$; that is, critical groups are computed at values where the (PS) condition fails. These critical groups are then incorporated into a new version of the Morse inequality, which allowed the authors of \cite{HLW} to obtain multiplicity results. In all the articles cited so far, the nonlinearity $g$ is assumed to be bounded. In the present work, we relax that assumption by allowing $g(x,s)$ to grow superlinearly, but subcritically, in $s$, for positive values of $s$ (see (A3) and (A4)), while $g(x,s)$ is bounded for negative values of $s$ (see (A2)). For additional information on problems at strong resonance in the context of critical point theory, the reader is referred to the works of Arcoya and Costa \cite{ArcoyaCosta}, Li \cite{ShJ}, and Chang and Liu \cite{CL}, and the bibliographies found in those papers. After establishing that the functional $J$ defined in \eqref{SResEqn00005} satisfies the $\text{(PS)}_c$ condition for $c\neq -G_{-\infty}|\Omega|$ in Section \ref{PSproofSec05}, under assumptions (A1)--(A6), we then proceed to show in Section \ref{ExistenceSec06} that $J$ has a local minimizer distinct from $0$, provided that (A5) holds with $G_{-\infty} \leqslant 0$, and (A7) also holds. In subsequent sections, we introduce an additional condition on the nonlinearity that will allow us to prove the existence of more critical point of $J$. In particular, we will assume the following: \begin{itemize} \item [(A8)] there exists $s_1 >0$ such that $g(x,s_1) = 0$ for all $x\in\Omega$. \end{itemize} In Section \ref{ExistenceSec16}, we prove that if, in addition to (A1)--(A5), with $G_{-\infty} \leqslant 0$, (A6) and (A7), we also assume (A8), then $J$ has a second local minimizer distinct from $0$. Finally, in Section \ref{ExistenceSec17}, we prove the existence of a third nontrivial critical point of $J$ by means of a variant of the mountain-pass theorem proved by Pucci and Serrin in \cite{PucSer}. \section{Proof of the Palais-Smale Condition}\label{PSproofSec05} In this section we prove that the functional $J$ defined in \eqref{SResEqn00005}, where $g$ and its primitive $G$ satisfy the conditions in (A1)--(A5), satisfies the $\text{(PS)}_c$ condition provided that $c \neq - G_{-\infty}|\Omega|$. \begin{proposition}\label{PScondJ} Assume that $g$ and $G$ satisfy {\rm (A1)--(A5)}, and define ${J}$ as in \eqref{SResEqn00005}. Then, ${J}$ satisfies the $(PS)_{c}$ for $c\ne -G_{-\infty}|\Omega|$. \end{proposition} \begin{proof} Assume that $c\ne -G_{-\infty}|\Omega|$ and let $(u_m)$ be a sequence in $H_{0}^{1}(\Omega)$ satisfying \begin{equation}\label{SResEqn00020} J(u_m) \to c \quad\text{and} \quad \|J'(u_m)\|\to 0 \quad\text{as } m\to\infty. \end{equation} Thus, according to \eqref{SResEqn00005} and \eqref{SResEqn00010}, \begin{equation}\label{SResEqn00025} \frac{1}{2} \int_\Omega |\nabla u_m|^2 \,dx -\frac{\lambda_1}{2}\int_\Omega (u_m^-)^2\,dx - \int_\Omega G(x,u_m(x))\,dx \to c, \quad\text{as } m\to\infty, \end{equation} and \begin{equation}\label{SResEqn00030} \Big| \int_{\Omega}\nabla u_m\cdot\nabla v \,dx + \lambda_1\int_\Omega u_m^- v\,dx - \int_{\Omega}g(x,u_m)v\,dx \Big| \leqslant \varepsilon_m \|v\|, \end{equation} for all $m$ and all $v\in H_{0}^{1}(\Omega)$, where $(\varepsilon_m)$ is a sequence of positive numbers that tends to $0$ as $m\to\infty$. We will show that $(u_m)$ has a subsequence that converges in $H_{0}^{1}(\Omega)$. It follows from (A2) that there exists $s_1>0$ such that \begin{equation}\label{SResEqn00033} -1 \leqslant g(x,s) \leqslant 1, \quad\text{for } s < -s_1, \text{ and all } x\in\overline{\Omega}. \end{equation} Consequently, \begin{equation}\label{SResEqn00035} -|s| \leqslant sg(x,s) \leqslant |s|, \quad\text{for } s < -s_1, \text{ and all } x\in\overline{\Omega}, \end{equation} and \begin{equation}\label{SResEqn00040} -C_1 -|s| \leqslant G(x,s) \leqslant C_1 + |s|, \quad\text{for } s < -s_1, \text{ and all } x\in\overline{\Omega}, \end{equation} for some positive constant $C_1$. Combining \eqref{SResEqn00035} and \eqref{SResEqn00040}, and using the continuity of $g$, we can find a positive constant $C_2$ such that \begin{equation}\label{SResEqn00045} -C_2 -3|s| \leqslant sg(x,s) - 2G(x,s) \leqslant C_2 + 3|s|, \quad\text{for } s \leqslant 0, \text{ and all } x\in\overline{\Omega}. \end{equation} Similarly, we obtain from (A3) that there exists a positive constant $C_3$ such that \begin{equation}\label{SResEqn00050} |g(x,s)| \leqslant C_3 + |s|^{\sigma}, \quad\text{for } s\geqslant 0 \text{ and } x\in\overline{\Omega}. \end{equation} Finally, we obtain from (A4) that there exist positive constants $C_4$ and $C_5$ such that \begin{equation}\label{SResEqn00055} G(x,s) \geqslant C_4 s^{\mu} - C_5, \quad\text{for } s\geqslant 0\text{ and } x\in\overline{\Omega}. \end{equation} Now, it follows from \eqref{SResEqn00025} that there exists a positive constant $C_6$ such that \begin{equation}\label{SResEqn00060} \Big|\int_\Omega |\nabla u_m|^2 \,dx -\lambda_1\int_\Omega (u_m^-)^2\,dx - \int_\Omega 2G(x,u_m(x))\,dx\Big| \leqslant C_6, \quad\text{for all } m. \end{equation} Taking $v = u_m$ in \eqref{SResEqn00030}, we obtain \begin{equation} \Big|\int_\Omega |\nabla u_m|^2 \,dx -\lambda_1\int_\Omega (u_m^-)^2\,dx - \int_\Omega g(x,u_m(x))u_m(x)\,dx\Big| \leqslant \varepsilon_m\|u_m\|, \label{SResEqn00065} \end{equation} for all $m$. Combining \eqref{SResEqn00060} and \eqref{SResEqn00065} we then obtain that \begin{equation}\label{SResEqn00070} \Big|\int_\Omega [g(x,u_m(x))u_m(x) - 2G(x,u_m(x))] \,dx\Big| \leqslant C_6 + \varepsilon_m\|u_m\|, \quad\text{for all } m. \end{equation} Next, define the sets \begin{gather*} \Omega_m^- = \{ x\in\Omega\mid u_m(x) < 0\};\quad \Omega_m^+ = \{ x\in\Omega\mid u_m(x) \geqslant 0\};\\ \Omega_m^o = \{ x\in\Omega\mid 0\leqslant u_m(x) \leqslant s_o\};\quad \Omega_m^{s_o} = \{ x\in\Omega\mid u_m(x) > s_o\}. \end{gather*} Then, using the estimate in \eqref{SResEqn00045}, \begin{equation}\label{SResEqn00075} \Big|\int_{\Omega_{m}^-} [g(x,u_m(x))u_m(x) - 2G(x,u_m(x))] \,dx\Big| \leqslant C + 3\|u_m^-\|_{L^1}, \quad\text{for all } m. \end{equation} \medskip\noindent \textbf{Note}: From this point on in this paper, the symbol $C$ will be used to represent any positive constant. Thus, $C$ might represent different constants in various estimates, even within the same inequality. It follows from \eqref{SResEqn00070} and \eqref{SResEqn00075} that \begin{equation}\label{SResEqn00077} \Big|\int_{\Omega_{m}^+} [g(x,u_m(x))u_m(x) - 2G(x,u_m(x))] \,dx\Big| \leqslant C + \varepsilon_m\|u_m\| + 3\|u_m^-\|_{L^1}, \end{equation} for all $m$. Using the continuity of $g$ and $G$ we deduce the existence of a positive constant $C$ such that \begin{equation}\label{SResEqn00080} \int_{\Omega_{m}^o} [g(x,u_m(x))u_m(x) - 2G(x,u_m(x))] \,dx\leqslant C, \quad\text{for all } m. \end{equation} On the other hand, using (A4) we obtain that $$ (\mu-2) \int_{\Omega_m^{s_o}} G(x,u_m(x))\,dx \leqslant \int_{\Omega_m^{s_o}} [g(x,u_m(x))u_m(x) - 2G(x,u_m(x))] \,dx, $$ for all $m$; so that, using this estimate in conjunction with \eqref{SResEqn00080}, \eqref{SResEqn00075}, \eqref{SResEqn00070} and the assumption that $\mu > 2$, we obtain that \begin{equation}\label{SResEqn00085} \int_{\Omega_m^{s_o}} G(x,u_m(x))\,dx \leqslant C + 3\|u_m^-\|_{L^1} + \varepsilon_m\|u_m\|, \quad\text{for all } m. \end{equation} Noting that $\Omega_m^+ = \Omega_m^o\cup\Omega_{m}^{s_o}$, we obtain from \eqref{SResEqn00085} that \begin{equation}\label{SResEqn00087} \Big|\int_{\Omega_m^{+}} G(x,u_m(x))\,dx\Big| \leqslant C + 3\|u_m^-\|_{L^1} + \varepsilon_m\|u_m\|, \quad\text{for all } m. \end{equation} Next, we take $v = - u_m^-$ in \eqref{SResEqn00030} to obtain \begin{equation}\label{SResEqn00090} \Big| \int_{\Omega} |\nabla u_m^-|^2 \,dx - \lambda_1\int_\Omega (u_m^- )^2\,dx - \int_{\Omega_m^-}g(x,u_m)u_m\,dx \Big| \leqslant \varepsilon_m \|u_m^-\|, \end{equation} We get from \eqref{SResEqn00035} and \eqref{SResEqn00040} that \begin{equation}\label{SResEqn00095} \Big|\int_{\Omega_m^-} g(x,u_m(x))u_m(x)\,dx\Big| \leqslant C +\|u_m^-\|_{L^1}, \quad\text{for all } m, \end{equation} and \begin{equation}\label{SResEqn00100} \Big|\int_{\Omega_m^-} G(x,u_m(x))\,dx\Big| \leqslant C +\|u_m^-\|_{L^1}, \quad\text{for all } m. \end{equation} Taking $v = u_m^+$ in \eqref{SResEqn00030} we then get \begin{equation}\label{SResEqn00105} \Big| \int_{\Omega} |\nabla u_m^+|^2 \,dx - \int_{\Omega_m^+}g(x,u_m)u_m\,dx \Big| \leqslant \varepsilon_m \|u_m^+\|, \quad\text{for all } m. \end{equation} It follows from \eqref{SResEqn00105}, \eqref{SResEqn00087} and \eqref{SResEqn00077} that $$ \int_{\Omega} |\nabla u_m^+|^2 \,dx \leqslant C + \varepsilon_m \|u_m^+\| + 2\varepsilon_m \|u_m\| + 6\|u_m^-\|_{L^1} , \quad\text{for all } m, $$ which can be rewritten as \begin{equation}\label{SResEqn00110} \int_{\Omega} |\nabla u_m^+|^2 \,dx \leqslant C + 3\varepsilon_m \|u_m^+\| + 2\varepsilon_m \|u_m^-\| + 6\|u_m^-\|_{L^1} , \quad\text{for all } m, \end{equation} by the triangle inequality. We claim that, if \eqref{SResEqn00015} holds true, then $(u_m^-)$ is bounded. We argue by contradiction. Suppose, passing to a subsequence if necessary, that \begin{equation}\label{SResEqn00115} \|u_m^-\|\to\infty, \quad\text{as } m\to\infty. \end{equation} It follows from \eqref{SResEqn00110}, the Cauchy-Schwarz inequality, and the Poincar\'e inequality that \begin{equation}\label{SResEqn00120} \|u_m^+\|\leqslant C + C \sqrt{1+\|u_m^-\|}, \quad\text{for all } m. \end{equation} Combining \eqref{SResEqn00120} and \eqref{SResEqn00115} we then deduce that \begin{equation}\label{SResEqn00125} \lim_{m\to\infty} \frac{\|u_m^+\|}{\|u_m^-\|} = 0. \end{equation} Next, define \begin{equation}\label{SResEqn00130} v_m = - \frac{u_m^-}{\|u_m^-\|}, \quad\text{for all } m; \end{equation} so that $\|v_m\| = 1$ for all $m$. We may therefore extract a subsequence $(v_{m_k})$ of $(v_m)$ such that \begin{equation}\label{SResEqn00135} v_{m_k} \rightharpoonup \overline{v}\text{ (weakly) as } k\to\infty, \end{equation} for some $\overline{v}\in H_{0}^{1}(\Omega)$. We may also assume, passing to further subsequences if necessary, that \begin{gather}\label{SResEqn00140} v_{m_k} \to \overline{v}\text{in } L^2(\Omega) \quad \text{as } k\to\infty, \\ \label{SResEqn00145} v_{m_k}(x) \to \overline{v}(x) \quad \text{for a.e. } x\in\Omega \text{ as } k\to\infty. \end{gather} Now, it follows from \eqref{SResEqn00030} and the fact that $u_{m_k} = u_{m_k}^+ - u_{m_k}^-$ that \begin{equation}\label{SResEqn00155} \begin{aligned} &\Big| -\int_\Omega \nabla u_{m_k}^-\cdot\nabla v\,dx +\lambda_1 \int_\Omega u_{m_k}^-v\,dx \Big|\\ & \leqslant \varepsilon_{m_k} \|v\| + \int_\Omega |\nabla u_{m_k}^+ \cdot\nabla v|\,dx + \int_\Omega |g(x,u_{m_k}(x))||v|\,dx , \end{aligned} \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. Using the Cauchy-Schwarz inequality, we can rewrite the estimate in \eqref{SResEqn00155} as \begin{equation}\label{SResEqn00160} \begin{aligned} &\Big| \int_\Omega \nabla u_{m_k}^-\cdot\nabla v\,dx -\lambda_1 \int_\Omega u_{m_k}^-v\,dx \Big| \\ & \leqslant \varepsilon_{m_k} \|v\| + \| u_{m_k}^+\|\| v\| + \int_\Omega |g(x,u_{m_k}(x))||v|\,dx , \end{aligned} \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. Next, we estimate the last integral on the right-hand side of \eqref{SResEqn00160} by first writing \begin{equation}\label{SResEqn00165} \begin{aligned} &\int_\Omega |g(x,u_{m_k}(x))||v|\,dx\\ &= \int_{\Omega_{m_k}^-} |g(x,u_{m_k}(x))||v|\,dx + \int_{\Omega_{m_k}^+} |g(x,u_{m_k}(x))||v|\,dx, \end{aligned} \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. To estimate the first integral on the right-hand side of \eqref{SResEqn00165}, we use \eqref{SResEqn00033}, the Cauchy-Schwarz inequality, and the Poincar\'e inequality to get that \begin{equation}\label{SResEqn00170} \big|\int_{\Omega_{m_k}^-} |g(x,u_{m_k}(x))||v|\,dx\big| \leqslant C\|v\|, \quad\text{for all } k \text{ and all } v\in H_{0}^{1}(\Omega). \end{equation} To estimate the second integral in the right-hand side of \eqref{SResEqn00165}, apply H\"{o}lder's inequality with $p=2N/(N+2)$ and $q=2N/(N-2)$ for $N\geqslant 3$. If $N=2$, take $1\leqslant p \leqslant \mu/\sigma$, which can be done because (A4) implies that $p\sigma<\mu$. Then, $$ \big|\int_{\Omega_{m_k}^+}|g(x,u_{m_k})||v|\,dx\big| \leqslant \Big(\int_{\Omega_{m_k}^+} |g(x,u_{m_k})|^{p}\Big)^{1/p} \Big(\int_\Omega |v|^{q}\Big)^{1/q}; $$ so that, in view of \eqref{SResEqn00050} and the Sobolev embedding theorem, \begin{equation}\label{SResEqn00175} \big|\int_{\Omega_{m_k}^+}|g(x,u_{m_k})||v|\,dx\big| \leqslant C \Big(\int_{\Omega} (C+|u_{m_k}^+|^\sigma)^{p}\,dx\Big)^{1/p}\|v\|, \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. We then obtain from \eqref{SResEqn00175} and Minkowski's inequality that \begin{equation}\label{SResEqn00180} \big|\int_{\Omega_{m_k}^+}|g(x,u_{m_k})||v|\,dx\big| \leqslant C(1+ \|u_{m_k}^+\|_{L^{p\sigma}}^\sigma) \|v\|, \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. Combining \eqref{SResEqn00165} with the estimates in \eqref{SResEqn00170} and \eqref{SResEqn00180}, we then obtain that \begin{equation}\label{SResEqn00185} \int_\Omega |g(x,u_{m_k}(x))||v|\,dx \leqslant C(1+ \|u_{m_k}^+\|_{L^{p\sigma}}^\sigma)\|v\|, \end{equation} for all $k$ and all $v\in H_{0}^{1}(\Omega)$. Finally, combining the estimates in \eqref{SResEqn00160} and \eqref{SResEqn00185}, \begin{equation}\label{SResEqn00190} \Big| \int_\Omega \nabla u_{m_k}^-\cdot\nabla v\,dx -\lambda_1 \int_\Omega u_{m_k}^-v\,dx \Big| \leqslant C(1+\|u_{m_k}^+\| + \|u_{m_k}^+\|_{L^{p\sigma}}^\sigma)\|v\|, \end{equation} for all $k$ and all $ v\in H_{0}^{1}(\Omega)$. Next, divide on both sides of \eqref{SResEqn00190} by $\|u_{m_k}^-\|$ and use \eqref{SResEqn00130} to get \begin{equation}\label{SResEqn00195} \begin{aligned} &\Big| \int_\Omega \nabla v_{m_k}\cdot\nabla v\,dx -\lambda_1 \int_\Omega v_{m_k}v\,dx \Big| \\ &\leqslant C\Big(\frac{1}{\|u_{m_k}^-\|}+\frac{\|u_{m_k}^+\|}{\|u_{m_k}^-\|} + \frac{\|u_{m_k}^+\|_{L^{p\sigma}}^\sigma}{\|u_{m_k}^-\|}\Big)\|v\|, \end{aligned} \end{equation} for all $k$ and all $ v\in H_{0}^{1}(\Omega)$. We will show next that \begin{equation}\label{SResEqn00200} \lim_{k\to\infty} \frac{\|u_{m_k}^+\|_{L^{p\sigma}}^\sigma}{\|u_{m_k}^-\|} = 0. \end{equation} Using the estimates in \eqref{SResEqn00055} and \eqref{SResEqn00087} we obtain \begin{equation}\label{SResEqn00205} \int_\Omega (u_{m_k}^+)^\mu \,dx \leqslant C + C\|u_{m_k}^-\| + \varepsilon_{m_k}\|u_{m_k}\|, \quad\text{for all } k, \end{equation} where we have also used the Cauchy-Schwarz and Poincar\'e inequalities. It then follows from \eqref{SResEqn00205} that \begin{equation}\label{SResEqn00210} \|u_{m_k}^+\|_{L^\mu} \leqslant C (1 + \|u_{m_k}^-\|^{1/\mu} +\|u_{m_k}\|^{1/\mu}), \quad\text{for all } k. \end{equation} Next, dividing on both sides of \eqref{SResEqn00210} by $\|u_{m_k}^-\|^{1/\sigma}$ and using the fact that $$ \|u_{m_k}\|\leqslant \|u_{m_k}^+\|+\|u_{m_k}^-\| $$ we obtain $$ \frac{\|u_{m_k}^+\|_{L^\mu} }{\|u_{m_k}^-\|^{1/\sigma}} \leqslant C \Big(\frac{1}{\|u_{m_k}^-\|^{1/\sigma}} + \frac{\|u_{m_k}^-\|^{1/\mu}}{\|u_{m_k}^-\|^{1/\sigma}} +\frac{\|u_{m_k}^+\|^{1/\mu}}{\|u_{m_k}^-\|^{1/\sigma}}\Big), \quad\text{for all } k, $$ which we can rewrite as \begin{equation}\label{SResEqn00215} \frac{\|u_{m_k}^+\|_{L^\mu} }{\|u_{m_k}^-\|^{1/\sigma}}\leqslant C \Big(\frac{1}{\|u_{m_k}^-\|^{1/\sigma}} + \frac{1}{\|u_{m_k}^-\|^{1/\sigma-1/\mu}} + \Big(\frac{\|u_{m_k}^+\|}{\|u_{m_k}^-\|} \Big)^{1/\mu} \frac{1}{\|u_{m_k}^-\|^{1/\sigma-1/\mu}}\Big), \end{equation} for all $k$. Now, in view of (A4) we see that $\mu > \sigma$; we then obtain from \eqref{SResEqn00115} and \eqref{SResEqn00125}, in conjunction with \eqref{SResEqn00215}, that \begin{equation}\label{SResEqn00220} \lim_{k\to\infty} \frac{\|u_{m_k}^+\|_{L^\mu} }{\|u_{m_k}^-\|^{1/\sigma}} = 0. \end{equation} Next, using the condition $\mu > p\sigma$ in (A4) to apply H\"{o}lder's inequality with $p_1 = \mu/p\sigma$ and $p_2$ its conjugate exponent we obtain $$ \|u_{m_k}^+\|_{L^{p\sigma}}^{p\sigma} = \int_\Omega (u_{m_k}^+)^{p\sigma}\,dx \leqslant \big(\int_\Omega (u_{m_k}^+)^{\mu}\,dx\big)^{p\sigma/\mu} |\Omega|^{1/p_2}; $$ so that, $$ \|u_{m_k}^+\|_{L^{p\sigma}}^{\sigma} \leqslant C\|u_{m_k}^+\|_{L^\mu}^\sigma, \quad\text{for all } k, $$ and, dividing on both sides by $\|u_{m_k}^-\|$, \begin{equation}\label{SResEqn00225} \frac{\|u_{m_k}^+\|_{L^{p\sigma}}^{\sigma}}{\|u_{m_k}^-\|} \leqslant C\Big(\frac{\|u_{m_k}^+\|_{L^\mu}}{\|u_{m_k}^-\|^{1/\sigma}}\Big)^\sigma, \quad\text{for all } k. \end{equation} It then follows from \eqref{SResEqn00220} and \eqref{SResEqn00225} that $$ \lim_{k\to\infty} \frac{\|u_{m_k}^+\|_{L^{p\sigma}}^{\sigma}}{\|u_{m_k}^-\|} = 0, $$ which is \eqref{SResEqn00200}. Using \eqref{SResEqn00115}, \eqref{SResEqn00125} and \eqref{SResEqn00200}, we obtain from \eqref{SResEqn00195} that \begin{equation}\label{SResEqn00230} \lim_{k\to\infty}\Big| \int_\Omega \nabla v_{m_k}\cdot\nabla v\,dx -\lambda_1 \int_\Omega v_{m_k}v\,dx \Big| = 0, \quad\text{for all } v\in H_{0}^{1}(\Omega). \end{equation} It then follows from \eqref{SResEqn00130}, \eqref{SResEqn00135} and \eqref{SResEqn00230} that $$ \int_\Omega \nabla \overline{v}\cdot\nabla v\,dx -\lambda_1 \int_\Omega \overline{v} v\,dx = 0, \quad\text{for all } v\in H_{0}^{1}(\Omega); $$ so that, $\overline{v}$ is a weak solution of the BVP \begin{equation}\label{evalProb05} \begin{gathered} -\Delta u = \lambda_1 u, \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega\,. \end{gathered} \end{equation} Now, it follows from \eqref{SResEqn00090} that \begin{equation}\label{SResEqn00235} \Big| \int_{\Omega} |\nabla u_{m_k}^-|^2 \,dx - \lambda_1\int_\Omega (u_{m_k}^- )^2\,dx \Big| \leqslant \varepsilon_{m_k} \|u_{m_k}^-\| + \Big|\int_{\Omega_{m_k}^-} g(x,u_{m_k})u_{m_k}\,dx\Big| , \end{equation} for all $k$; where, according to \eqref{SResEqn00095}, \begin{equation}\label{SResEqn00240} \big|\int_{\Omega_m^-} g(x,u_{m_k}(x))u_{m_k}(x)\,dx\big| \leqslant C(1 +\|u_{m_k}^-\|), \quad\text{for all } k. \end{equation} Thus, combining \eqref{SResEqn00235} and \eqref{SResEqn00240}, \begin{equation}\label{SResEqn00245} \Big| \int_{\Omega} |\nabla u_{m_k}^-|^2 \,dx - \lambda_1\int_\Omega (u_{m_k}^- )^2\,dx \big| \leqslant C(1 +\|u_{m_k}^-\|), \quad\text{for all } k. \end{equation} Next, divide on both sides of \eqref{SResEqn00245} by $\|u_{m_k}^-\|^2$ and use \eqref{SResEqn00130} to obtain \begin{equation}\label{SResEqn00250} \big| 1 - \lambda_1\int_\Omega (v_{m_k} )^2\,dx \big| \leqslant C\Big(\frac{1}{\|u_{m_k}^-\|^2} + \frac{1}{\|u_{m_k}^-\|}\Big), \quad\text{for all } k. \end{equation} It then follows from \eqref{SResEqn00115}, \eqref{SResEqn00140} and \eqref{SResEqn00250} that $$ \lambda_1\int_\Omega (\overline{v})^2\,dx = 1, $$ from which we conclude that $\overline{v}$ is a nontrivial solution of BVP \eqref{evalProb05}. Consequently, since $v_m \leqslant 0$ for all $m$, according to \eqref{SResEqn00130}, we obtain that \begin{equation}\label{SResEqn00255} \overline{v} = - \varphi_1, \end{equation} where $\varphi_1$ is the eigenfunction for the BVP \eqref{evalProb05} corresponding to the eigenvalue $\lambda_1$ with $$ \varphi_1 > 0 \text{ in } \Omega \quad\text{and }\quad \|\varphi_1 \| = 1. $$ We therefore obtain from \eqref{SResEqn00255} that \begin{equation}\label{SResEqn00260} \overline{v} < 0 \quad \text{ in } \Omega. \end{equation} Furthermore, \begin{equation}\label{SResEqn00265} \frac{\partial \overline{v}}{\partial \nu} > 0 \quad \text{on } \partial\Omega, \end{equation} where $\nu$ denotes the outward unit normal vector to $\partial\Omega$. We can then conclude from \eqref{SResEqn00125}, \eqref{SResEqn00145}, in conjunction with \eqref{SResEqn00260} and \eqref{SResEqn00265}, that \begin{equation}\label{SResEqn00270} u_{m_k} (x) \to -\infty \quad \text{for a.e. } x\in \Omega. \end{equation} Thus, using (A5) and the Lebesgue dominated convergence theorem, we obtain from \eqref{SResEqn00270} that \begin{equation}\label{SResEqn00275} \lim_{k\to\infty} \int_\Omega G(x,u_{m_k}(x))\,dx = G_{-\infty}|\Omega|. \end{equation} It then follows from \eqref{SResEqn00275} and the first assertion in \eqref{SResEqn00020} that \begin{equation}\label{SResEqn00280} \lim_{k\to\infty} \Big( \frac{1}{2}\int_{\Omega} |\nabla u_{m_k}|^2 \,dx - \frac{\lambda_1}{2}\int_{\Omega}(u_{m_k}^-)^2\,dx \Big) = c + G_{-\infty}|\Omega|. \end{equation} Next, we go back to the estimate in \eqref{SResEqn00030} and set $v = u_{m_k}^+$ to obtain $$ \Big| \int_{\Omega} |\nabla u_{m_k}^+|^2 \,dx - \int_{\Omega}g(x,u_{m_k}^+)u_{m_k}^+\,dx \Big| \leqslant \varepsilon_{m_k} \|u_{m_k}^+\|, \quad\text{for all } k, $$ or, dividing by $\|u_{m_k}^+\|$, \begin{equation}\label{SResEqn00285} \Big| \|u_{m_k}^+\| - \int_{\Omega}g(x,u_{m_k}^+) \frac{u_{m_k}^+}{\|u_{m_k}^+\|}\,dx \Big| \leqslant \varepsilon_{m_k} , \quad\text{for all } k. \end{equation} Now, it follows from \eqref{SResEqn00270} that $$ u_{m_k}^+ \to 0 \text{ a.e. as } k \to \infty. $$ Therefore, it follows from the assumption that $g(x,0) =0$ in (A1), together with the Lebesgue dominated convergence theorem and the estimate in \eqref{SResEqn00285}, that \begin{equation}\label{SResEqn00290} \lim_{k\to\infty} \|u_{m_k}^+\| = 0. \end{equation} Next, set $V = \operatorname{span}\{\varphi_1\}$ and $W = V^\perp$; so that, $H_{0}^{1}(\Omega) = V \oplus W$. Write $u_{m_k}^- = v_k + w_k$, for each $k$, where $v_k\in V$ and $w_k\in W$. Once again, use the estimate in \eqref{SResEqn00030}, this time with $v = w_k$, to obtain \begin{equation}\label{SResEqn00295} \Big| \int_{\Omega}|\nabla w_k|^2 \,dx - \lambda_1\int_\Omega w_k^2\,dx - \int_{\Omega}g(x,u_{m_k}(x))w_k\,dx \Big| \leqslant \varepsilon_{m_k} \|w_k\|, \end{equation} for all $k$. Now, since $w_k \in W$, we have that \begin{equation}\label{SResEqn00298} \lambda_2 \int_\Omega w_k^2 \,dx \leqslant \int_\Omega |\nabla w_k|^2 \,dx, \quad\text{for all } k, \end{equation} where $\lambda_2$ denotes the second eigenvalue of the $N$-dimensional Laplacian over $\Omega$ with Dirichlet boundary conditions. Consequently, \begin{equation}\label{SResEqn00300} \big(1-\frac{\lambda_1}{\lambda_2} \big) \|w_k\|^2 \leqslant \int_{\Omega}|\nabla w_k|^2 \,dx - \lambda_1\int_\Omega w_k^2\,dx, \quad\text{for all } k. \end{equation} Thus, setting $ \alpha = 1-\frac{\lambda_1}{\lambda_2}\ $ in \eqref{SResEqn00300}, we obtain from \eqref{SResEqn00300} and \eqref{SResEqn00295} that \begin{equation}\label{SResEqn00305} \alpha \|w_k\|^2 \leqslant \varepsilon_{m_k} \|w_k\| + \Big| \int_\Omega g(x,u_{m_k}(x))w_k\,dx \Big|, \quad\text{for all } k, \end{equation} where $\alpha > 0$. Next, we divide on both sides of \eqref{SResEqn00305} by $\|w_k\|$ to get \begin{equation}\label{SResEqn00310} \alpha \|w_k\| \leqslant \varepsilon_{m_k} + \Big| \int_\Omega g(x,u_{m_k}(x))\frac{w_k}{\|w_k\|} \,dx \Big|, \quad\text{for all } k. \end{equation} Now, it follows from \eqref{SResEqn00310}, \eqref{SResEqn00270}, assumption (A2), and the Lebesgue dominated convergence theorem that \begin{equation}\label{SResEqn00315} \lim_{k\to\infty} \|w_k\| = 0. \end{equation} Next, we observe that $$ \int_\Omega |\nabla u_{m_k}|^2 \,dx = \int_\Omega |\nabla u_{m_k}^+|^2 \,dx + \int_\Omega |\nabla v_k|^2 \,dx + \int_\Omega |\nabla w_k|^2 \,dx, \quad\text{for all } k, $$ and $$ \int_\Omega (u_{m_k}^-)^2 \,dx = \int_\Omega v_k^2 \,dx + \int_\Omega w_k^2 \,dx, \quad\text{for all } k; $$ consequently, \begin{equation}\label{SResEqn00320} \begin{aligned} &\frac{1}{2}\int_{\Omega} |\nabla u_{m_k}|^2 \,dx - \frac{\lambda_1}{2}\int_{\Omega}(u_{m_k}^-)^2\,dx \\ &= \frac{1}{2}\|u_{m_k}^+\|^2 + \frac{1}{2}\int_{\Omega} |\nabla w_k|^2 \,dx - \frac{\lambda_1}{2}\int_{\Omega}w_k^2\,dx, \end{aligned} \end{equation} for all $k$, where we have used the fact that $v_k\in V$ for all $k$. It follows from \eqref{SResEqn00290}, \eqref{SResEqn00315}, \eqref{SResEqn00298} and \eqref{SResEqn00320} that \begin{equation}\label{SResEqn00325} \lim_{k\to\infty} \Big( \frac{1}{2}\int_{\Omega} |\nabla u_{m_k}|^2 \,dx - \frac{\lambda_1}{2}\int_{\Omega}(u_{m_k}^-)^2\,dx \Big) = 0. \end{equation} Combining \eqref{SResEqn00280} and \eqref{SResEqn00325} we obtain $$ G_{-\infty}|\Omega| + c = 0, $$ which is in direct contradiction with \eqref{SResEqn00015}. We therefore conclude that $(u_m^-)$ is bounded. Since, $(u_m^-)$ is bounded, it follows from \eqref{SResEqn00110} that $(u_m^+)$ is also bounded. Consequently, $(u_m)$ is bounded. We will next proceed to show that $(u_m)$ has a subsequence that converges strongly in $H_{0}^{1}(\Omega)$. To see why this is the case, first write the functional $J\colon H_{0}^{1}(\Omega)\to\mathbb{R}$ defined in \eqref{SResEqn00005} in the form \begin{equation*} J(u) = \frac{1}{2} \int_\Omega |\nabla u|^2 \,dx - \int_\Omega \text{\rm Q}(x,u(x))\,dx, \quad\text{for all } u\in H_{0}^{1}(\Omega), \end{equation*} where $$ \text{\rm Q}(x,s) = \int_0^s q(x,\xi ) \,d\xi, \quad\text{for all } x\in\overline{\Omega}\text{ and } s\in\mathbb{R}, $$ where $q$ is as given in \eqref{qDef}. It follows from \eqref{qDef} and the assumptions in (A2) and (A3), that $q(x,s)$ has subcritical growth in $s$, uniformly in $x\in\overline{\Omega}$; so that, the derivative map of $J$, $\nabla J \colon H_{0}^{1}(\Omega)\to H_{0}^{1}(\Omega)$, is of the form \begin{equation}\label{PScPfEqn00010} \nabla J = I - \nabla\mathcal{Q}, \end{equation} where $\nabla\mathcal{Q}\colon H_{0}^{1}(\Omega)\to H_{0}^{1}(\Omega)$, given by $$ \langle \nabla\mathcal{Q}(u),v\rangle = \int_\Omega g(x, u(x)) v(x)\,dx , \quad\text{for } u,v\in H_{0}^{1}(\Omega), $$ is a compact operator. Now, it follows from the second condition in \eqref{SResEqn00020} and \eqref{PScPfEqn00010} that \begin{equation}\label{PScPfEqn00015} u_m - \nabla\mathcal{Q}(u_m) \to 0, \quad\text{as } m\to\infty. \end{equation} Since we have already seen that the $\text{(PS)}_c$ sequence $(u_m)$ is bounded, we can extract a subsequence, $(u_{m_k})$, of $(u_m)$ that converges weakly to some $u\in H_{0}^{1}(\Omega)$. Therefore, given that the map $\nabla\mathcal{Q}\colon H_{0}^{1}(\Omega)\to H_{0}^{1}(\Omega)$ is compact, we have that \begin{equation}\label{PScPfEqn00020} \lim_{k\to\infty} \nabla\mathcal{Q}(u_{m_k}) = \nabla\mathcal{Q}(u). \end{equation} Thus, combining \eqref{PScPfEqn00015} and \eqref{PScPfEqn00020}, we obtain that $$ \lim_{k\to\infty} u_{m_k} = \nabla\mathcal{Q}(u). $$ We have therefore shown that $(u_m)$ has a subsequence that converges strongly in $H_{0}^{1}(\Omega)$, and the proof of the fact that $J$ satisfies that $\text{(PS)}_c$ condition, provided that $c \neq - G_{-\infty}|\Omega|$, is now complete. \end{proof} \section{Existence of a local minimizer}\label{ExistenceSec06} Assume that $g$ and $G$ satisfy conditions (A1)--(A7) hold. In this section, we will use Ekeland's Variational Principle and a cutoff technique similar to that used by Chang, Li and Liu in \cite{ChangAl} to prove the existence of a nontrivial solution of problem \eqref{de1} for the case in which $G_{-\infty}\leqslant 0$ in (A5). To do that, we first define $\widetilde{g}\in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ by \begin{equation}\label{newg} \widetilde{g}(x,s)=\begin{cases} g(x,s),&\text{ for } s < 0, \\ 0, & \text{ for } s\geq 0. \end{cases} \end{equation} Define a corresponding functional $\widetilde{J}:H_{0}^{1}(\Omega)\to\mathbb{R}$ by \begin{equation}\label{newJ} \widetilde{J}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^{2}\,dx - \frac{\lambda_{1}}{2}\int_{\Omega}(u^{-})^{2}\,dx - \int_{\Omega}\widetilde{G}(x,u)\,dx,\quad u\in H_{0}^{1}(\Omega), \end{equation} where \begin{equation}\label{GtildeDfn} \widetilde{G}(x,s)=\int_{0}^{s}\widetilde{g}(x,\xi)\,d\xi, \quad \text{for } x\in\overline{\Omega} \text{ and } s\in\mathbb{R}. \end{equation} Then, $\widetilde{J}\in C^{1}(H_{0}^{1}(\Omega),\mathbb{R})$. We claim that $\widetilde{J}$ is bounded below. In fact, by condition (A5) and \eqref{newg}, it follows that \begin{equation} |\widetilde{G}(x,s)| \leqslant M_{o},\quad \text{for all }x\in\overline{\Omega} \text{ and } s\in\mathbb{R}, \label{Gtildebdd} \end{equation} for some $M_{o}>0$. Then, using \eqref{newg} and \eqref{Gtildebdd}, we can write \begin{equation} \begin{aligned} \widetilde{J}(u) & = \frac{1}{2}\int_{\Omega}|\nabla u|^{2}dx - \frac{\lambda_{1}}{2}\int_{\Omega}(u^{-})^{2}dx - \int_{\Omega}\widetilde{G}(x,u)dx, \\ & \geqslant \frac{1}{2}\|u^{+}\|^{2}+\frac{1}{2}\|u^{-}\|^{2} -\frac{\lambda_{1}}{2}\|u^{-}\|_{L^{2}} -M_{o}|\Omega|, \end{aligned} \label{Jbddbelow} \end{equation} for all $u\in H_{0}^{1}(\Omega)$. It then follows from \eqref{Jbddbelow} and the Poincar\'e inequality that $$ \widetilde{J}(u) \geqslant -M_{o}|\Omega|, \quad\text{for all } u\in H_{0}^{1}(\Omega); $$ so that $\widetilde{J}$ is bounded below. Thus, the infimum of $\widetilde{J}$ over $H_{0}^{1}(\Omega)$ exists; we can, therefore define \begin{equation} c_1 = \inf_{u\in H_{0}^{1}(\Omega)}\widetilde{J}(u). \label{cdeff} \end{equation} Notice that, since $\widetilde{J}(0)=0$, we must have $c_1 \leqslant 0$. In fact, we presently show that, if (A6) and (A7) hold, then \begin{equation}\label{cneg} c_1 < 0. \end{equation} To do this, first use \eqref{qDef} and (A7) to compute $$ \lim_{s\to 0^-} \frac{g(x,s)}{s} = a - \lambda_1; $$ so that $$ \lim_{s\to 0^-} \frac{g(x,s)}{s} > 0, $$ for all $x\in\overline{\Omega}$, by the assumption on $a$ in (A7). Consequently, there exists $s_1 < 0$ such that $$ g(x,s) < 0, \quad\text{for } s_1 < s < 0, $$ and all $x\in\overline{\Omega}$. It then follows from the definition of $\widetilde{G}$ in \eqref{GtildeDfn} that \begin{equation}\label{Min1Eqn0005} \widetilde{G}(x,s) > 0 \quad\text{for } s_1 < s < 0, \text{ and all } x\in\overline{\Omega}. \end{equation} Next, let $\varepsilon > 0$ be small enough so that \begin{equation}\label{Min1Eqn0010} s_1 < -\varepsilon\varphi_1(x) < 0, \quad\text{for all } x\in\Omega. \end{equation} We then have that \begin{equation}\label{Min1Eqn0015} \int_\Omega \widetilde{G}(x, -\varepsilon\varphi(x))\,dx > 0, \end{equation} by \eqref{Min1Eqn0005} and \eqref{Min1Eqn0010}. It then follows from the definition of $\widetilde{J}$ in \eqref{newJ} and \eqref{Min1Eqn0015} that $$ \widetilde{J}(-\varepsilon\varphi_1 ) = - \int_\Omega \widetilde{G}(x, -\varepsilon\varphi(x))\,dx <0. $$ Consequently, in view of the definition of $c_1$ in \eqref{cdeff}, we obtain that $c_1 < 0$, which is \eqref{cneg}. We now use \eqref{cdeff} and a consequence of Ekeland's Variational Principle (see \cite[Theorem 4.4]{DFEke}) to obtain, for each positive integer $m$, $u_{m}\in H_{0}^{1}(\Omega)$ such that \begin{equation} \widetilde{J}(u_{m}) \leqslant \inf_{u\in H_{0}^{1}(\Omega)} \widetilde{J}(u)+\frac{1}{m}, \quad\text{for all } m, \label{ekcond1} \end{equation} and \begin{equation}\label{ekcond2} \|\widetilde{J}'(u_{m})\|\leq\frac{1}{m}, \quad\text{for all } m; \nonumber \end{equation} we therefore obtain a $\text{(PS)}_{c}$ sequence for $c=c_1$. Consequently, if $\widetilde{J}$ happens to satisfy the $\text{(PS)}_{c}$ condition at $c=c_1$, we would conclude that $c_1$ is a critical value of $\widetilde{J}$. We will show shortly that this is the case if we assume that $G_{-\infty}$ given in (A5) satisfies \begin{equation}\label{GinfCondition} G_{-\infty} \leqslant 0. \end{equation} We will first establish that $\widetilde{J}$ satisfies the $(PS)_{c}$ provided that $c\ne -G_{-\infty}|\Omega|$. \begin{proposition}\label{pscondjtilde} Assume that $g$ and $G$ satisfy {\rm (A1), (A2)} and {\rm (A5)}, and define $\widetilde{J}$ as in \eqref{newJ}, where $\widetilde{G}$ is given in \eqref{GtildeDfn} and \eqref{newg}. Then, $\widetilde{J}$ satisfies the $(PS)_{c}$ condition for $c\ne -G_{-\infty}|\Omega|$. \end{proposition} \begin{proof} Assume that $c\ne -G_{-\infty}|\Omega|$ and let $(u_{m})$ be a $(PS)_{c}$ sequence for $\widetilde{J}$; that is, \begin{equation}\label{jtilde1} \frac{1}{2}\int_{\Omega}|\nabla u_{m}|^{2}\,dx - \frac{\lambda_{1}}{2}\int_{\Omega}(u_{m}^{-})^{2}\,dx - \int_{\Omega}\widetilde{G}(x,u_{m})\,dx \to c,\quad\text{as }m\to +\infty, \end{equation} and, \begin{equation} \Big|\int_{\Omega}\nabla u_{m}\cdot \nabla \varphi\,dx + \lambda_{1}\int_{\Omega}u_{m}^{-}\varphi\,dx - \int_{\Omega}\widetilde{g}(x,u_{m})\varphi\,dx\Big| \leq\varepsilon_{m}\|\varphi\|, \label{jtilde2} \end{equation} for all $m$ and all $\varphi\in H_{0}^{1}(\Omega)$, where $(\varepsilon_m)$ is a sequence of positive numbers such that $\varepsilon_{m}\to 0$ as $m\to\infty$. Write $u_{m}=u_{m}^{+} - u_{m}^{-}$. We will show that $(u_{m}^{+})$ and $(u_{m}^{-})$ are bounded sequences. First, let's see that $( u_{m}^{+})$ is bounded. Setting $\varphi=u_{m}^{+}$ in \eqref{jtilde2} we have \begin{equation} \Big|\int_{\Omega}|\nabla u_{m}^{+}|^{2}\,dx - \int_{\Omega}\widetilde{g}(x,u_{m})u_{m}^{+}\,dx\Big| \leq\varepsilon_{m}\|u_{m}^{+}\| \quad\text{for all } m. \label{jtilde3} \end{equation} By \eqref{newg} and the assumption in (A2), it can be shown that $\widetilde{g}(x,u_{m})$ is bounded for all $x\in\overline{\Omega}$. Then, using H\"{o}lder and Poincar\'{e}'s inequalities, we obtain that \begin{equation}\label{jtilde4} \big|\int_{\Omega}\widetilde{g}(x,u_{m})u_{m}^{+}\,dx\big|\leq C\|u_{m}^{+}\|, \end{equation} for some constant $C>0$. Then, from \eqref{jtilde3} and \eqref{jtilde4}, we obtain that $$ \|u_{m}^{+}\|^{2}\leq \left(C+\varepsilon_{m}\right)\|u_{m}^{+}\| , \quad\text{for all } m, $$ which shows that $(u_{m}^{+})$ is a bounded sequence. Next, let us show that $(u_{m}^{-})$ is a bounded sequence. Suppose that this is not the case; then, passing to a subsequence if necessary, we may assume that \begin{equation}\label{jtilde405} \|u_{m}^{-}\|\to\infty \quad\text{as } m\to\infty. \end{equation} Define \begin{equation}\label{jtilde5} v_{m}=-\frac{u_{m}^{-}}{\|u_{m}^{-}\|},\quad\text{for all }m. \end{equation} Then, since $\|v_{m}\|=1$ for all $m$, passing to a further subsequences if necessary, we may assume that there is $v\in H_{0}^{1}(\Omega)$ such that \begin{gather}\label{vm1} v_{m}\rightharpoonup v\quad\text{(weakly) in $H_{0}^{1}(\Omega)$, as } m\to\infty;\\ \label{vm2} v_{m}\to v \quad\text{in }L^{2}(\Omega), \quad\text{as } m\to\infty; \\ \label{vm3} v_{m}(x)\to v(x)\quad\text{for a.e. } x \text{in $\Omega$, as } m\to\infty. \end{gather} Now, writing $u_{m}=u_{m}^{+}-u_{m}^{-}$ in \eqref{jtilde2} we have \begin{equation*} \Big|\int_{\Omega}\nabla u_{m}^{+}\cdot\nabla\varphi\,dx -\int_{\Omega}\nabla u_{m}^{-}\cdot\nabla\varphi\,dx + \lambda_{1}\int_{\Omega} u_{m}^{-}\varphi\,dx - \int_{\Omega}\widetilde{g}(x,u_{m})\varphi\,dx \Big| \leqslant \varepsilon_{m}\|\varphi\|, \end{equation*} for all $\varphi\in H_{0}^{1}(\Omega)$ and all $m$, from which we obtain that \begin{equation}\label{jtilde7} \Big|-\int_{\Omega}\nabla u_{m}^{-}\cdot\nabla\varphi\,dx + \lambda_{1}\int_{\Omega} u_{m}^{-}\varphi\,dx - \int_{\Omega}\widetilde{g}(x,u_{m})\varphi\,dx \Big| \leqslant (\varepsilon_{m} + C\|u_{m}^{+}\|) \|\varphi\|, \end{equation} for all $\varphi\in H_{0}^{1}(\Omega)$, all $m$, and some constant $C>0$, by the Cauchy-Schwarz and Poincar\'{e} inequalities. Now, we divide both sides of \eqref{jtilde7} by $\|u_{m}^{-}\|$ and use \eqref{jtilde5} to obtain \begin{equation}\label{jtilde8} \Big|\int_{\Omega}\nabla v_{m}\cdot\nabla\varphi\,dx - \lambda_{1}\int_{\Omega}v_{m}\varphi\,dx - \int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{-}\|}\varphi\,dx \Big| \leqslant \Big(\frac{\varepsilon_{m}+C\|u_{m}^{+}\|}{\|u_{m}^{-}\|}\Big) \|\varphi\|, \end{equation} for all $\varphi\in H_{0}^{1}(\Omega)$ and all $m$. Since $\widetilde{g}$ is bounded, by condition (A2) and \eqref{newg}, we obtain from \eqref{jtilde405} that \begin{equation*} \lim_{m\to +\infty}\frac{\widetilde{g}(x,u_{m}(x))}{\|u_{m}^{-}\|}=0, \quad\text{for a. e. } x\in\overline{\Omega}. \end{equation*} It then follows from the Lebesgue dominated convergence theorem that \begin{equation} \lim_{m\to +\infty} \int_\Omega \frac{\widetilde{g}(x,u_{m}(x))}{\|u_{m}^{-}\|} \varphi \,d x=0, \quad\text{for all } \varphi\in H_{0}^{1}(\Omega). \label{jtilde9} \end{equation} Therefore, using \eqref{vm1}, \eqref{vm2}, \eqref{jtilde9}, \eqref{jtilde405}, the fact that the sequence $(u_{m}^{+})$ is bounded, and letting $m\to\infty$ in \eqref{jtilde8}, we obtain \begin{equation*} \int_{\Omega}\nabla v\cdot\nabla \varphi\,dx - \lambda_{1}\int_{\Omega}v\varphi\,dx = 0,\quad\text{for all } \varphi\in H_{0}^{1}(\Omega); \end{equation*} so that, $v$ is a weak solution of the BVP \begin{equation} \begin{gathered} -\Delta u = \lambda_1 u,\quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega. \end{gathered} \label{eigProb} \end{equation} Next, we set $\varphi=v_{m}$ in \eqref{jtilde8} to get \begin{equation} \Big|1 - \lambda_{1}\int_{\Omega}v_{m}^{2}\,dx - \int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{-}\|} v_{m}\,dx\Big| \leqslant \frac{\varepsilon_{m}+C\|u_{m}^{+}\|}{\|u_{m}^{-}\|}, \quad\text{for all } m, \label{jtilde11} \end{equation} where we have also used the definition of $v_m$ in \eqref{jtilde5}. Now, using the Cauchy-Schwarz and Poincar\'e inequalities, we obtain that \begin{equation}\label{jtilde1105} \Big|\int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{-}\|} v_{m}\,dx\Big| \leqslant \frac{C}{\|u_{m}^{-}\|}, \quad\text{for all } m, \end{equation} for some positive constant $C$, since $\widetilde{g}$ is bounded. We then get from \eqref{jtilde1105} and \eqref{jtilde405} that \begin{equation} \lim_{m\to\infty}\int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{-}\|} v_{m}\,dx = 0. \label{jtilde1110} \end{equation} Thus, letting $m\to\infty$ in \eqref{jtilde11} and using \eqref{vm2}, \eqref{jtilde1110}, and \eqref{jtilde405}, we obtain that \begin{equation*} \lambda_{1}\int_{\Omega}v^{2}dx = 1, \end{equation*} which shows that $v$ is a nontrivial solution of \eqref{eigProb}. Now, it follows from \eqref{jtilde5} and \eqref{vm3} that $$ v(x) \leqslant 0, \quad\text{for a. e. } x\in\Omega. $$ Consequently, since $v$ is nontrivial, it must be the case that \begin{equation}\label{jtilde1115} v = - \varphi_1, \end{equation} where $\varphi_{1}$ is the eigenfunction of the BVP problem \eqref{eigProb} corresponding to the eigenvalue $\lambda_{1}$ with $\varphi_{1}>0$, $\|\varphi_{1}\|=1$. Thus, $v<0\in \Omega$ and $\partial v/\partial \nu > 0$ on $\partial\Omega$, where $\nu$ is the outward unit normal vector to $\partial\Omega$. Next, we write $u_m = u_m^+ - u_m^-$ and use \eqref{jtilde5} to get $$ \frac{u_m}{\|u_m^-\|} = \frac{u_m^+}{\|u_m^-\|} + v_m, \quad\text{for all } m; $$ so that, by the fact that $(u_m^+)$ is bounded and \eqref{jtilde405}, we may assume, passing to a further subsequence if necessary, that \begin{equation}\label{jtilde1120} \frac{u_m(x)}{\|u_m^-\|} \to -\varphi_1(x), \quad\text{for a. e. } x\in\Omega, \quad\text{as } m\to\infty, \end{equation} where we have also used \eqref{vm3} and \eqref{jtilde1115}. It then follows from \eqref{jtilde1120} that \begin{equation}\label{jtilde19} u_{m}(x)\to -\infty\quad\text{for a. e. }x\in\Omega, \text{ as }m\to\infty. \end{equation} Then, using condition (A5) and the Lebesgue dominated convergence theorem, we conclude from \eqref{jtilde1} that \begin{equation}\label{jtilde20} \lim_{m\to\infty}\Big(\frac{1}{2}\int_{\Omega}|\nabla u_{m}|^{2}\,dx -\frac{\lambda_{1}}{2}\int_{\Omega}(u_{m}^{-})^{2}\,dx\Big) =c+G_{-\infty}|\Omega|. \end{equation} Next, we divide both sides of \eqref{jtilde3} by $\|u_{m}^{+}\|$ to obtain \begin{equation}\label{jtilde21} \Big|\|u_{m}^{+}\| - \int_{\Omega} \frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{+}\|}u_{m}^{+}\,dx\Big| \leqslant \varepsilon_{m},\quad\text{for all }m. \end{equation} Notice that $$ \int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{+}\|}u_{m}^{+}\,dx = \int_{\Omega} \widetilde{g}(x,u_{m}^+) \frac{u_{m}^{+}}{\|u_{m}^{+}\|}\,dx, \quad\text{for all } m; $$ so that, using the Cauchy-Schwarz and Poincar\'e inequalities, \begin{equation}\label{jtilde2105} \big| \int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{+}\|}u_{m}^{+}\,dx \big| \leqslant C \sqrt{\int_\Omega \widetilde{g}(x,u_{m}^+(x))^2 \,dx}, \quad\text{for all } m, \end{equation} and some positive constant $C$. Now, it follows from \eqref{jtilde19} that \begin{equation*} u_{m}^+(x)\to 0\quad\text{for a.e }x\in\Omega, \quad \text{as }m\to\infty; \end{equation*} consequently, using the assumption (A1) along with the Lebesgue dominated convergence theorem, we obtain from \eqref{jtilde2105} that \begin{equation}\label{jtilde2110} \lim_{m\to\infty} \int_{\Omega}\frac{\widetilde{g}(x,u_{m})}{\|u_{m}^{+}\|}u_{m}^{+}\,dx = 0. \end{equation} Therefore, letting $m$ tend to $\infty$ in \eqref{jtilde21} and using \eqref{jtilde2110} we obtain that \begin{equation}\label{jtilde2115} \lim_{m\to\infty}\|u_{m}^{+}\|=0. \end{equation} Thus, combining \eqref{jtilde20} and \eqref{jtilde2115} we can then write \begin{equation}\label{jtilde2120} \lim_{m\to\infty}\Big(\frac{1}{2} \int_{\Omega}|\nabla u_{m}^-|^{2}\,dx -\frac{\lambda_{1}}{2}\int_{\Omega}(u_{m}^{-})^{2}\,dx\Big) = c+G_{-\infty}|\Omega|. \end{equation} We may now proceed as in the proof of Proposition \ref{PScondJ} in Section \ref{PSproofSec05} to show that $$ \lim_{m\to\infty} \Big( \frac{1}{2}\int_{\Omega}|\nabla u_{m}^-|^{2}\,dx -\frac{\lambda_{1}}{2}\int_{\Omega}(u_{m}^{-})^{2}\,dx \Big) = 0. $$ Hence, in view of \eqref{jtilde2120}, we obtain that $c+G_{-\infty}|\Omega| =0$, which contradicts the assumption that $c \neq - G_{-\infty}|\Omega|$. We therefore conclude that $(u_m^-)$ must be a bounded sequence. Thus, since we have already seen that $(u_m^+)$ is bounded, we see that $(u_m)$ is bounded. We have therefore shown that any $\text{(PS)}_c$ sequence with $c \neq - G_{-\infty}|\Omega|$ must be bounded. The remainder of this proof now proceeds as in the proof of Proposition \ref{PScondJ} presented in Section \ref{PSproofSec05}, using in this case the fact that $\widetilde{g}$ is bounded. \end{proof} Now, if we assume that the value $G_{-\infty}$ given in (A5) satisfies the condition in \eqref{GinfCondition}, then we would have that $- G_{-\infty}|\Omega|\geqslant 0$. Consequently, in view of \eqref{cneg}, we see that the value of $c_1$ given in \eqref{cdeff} is such that $$ c_1 < - G_{-\infty}|\Omega|; $$ therefore, $\widetilde{J}$ satisfies the $\text{(PS)}_c$ condition at $c = c_1$. Hence, by the discussion preceding the statement of Proposition \ref{pscondjtilde}, $c_1$ is a critical value of $\widetilde{J}$. Thus, there exists $u_{1}\in H_{0}^{1}(\Omega)$ that is a global minimizer for $\widetilde{J}$. We note that $u_1\not\equiv 0$ in $\Omega$ by \eqref{cneg}. Now, since the function $\widetilde{g}$ defined in \eqref{newg} is locally Lipschitz (refer to assumption in (A6)), it follows that $u_{1}$ is a classical solution of the problem \begin{equation}\label{clas1} \begin{gathered} -\Delta u = -\lambda_1 u^- + \widetilde{g}(x,u), \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} (see Agmon \cite{Ag}). Let $\Omega_{+}=\{x\in\Omega\mid u_{1}(x) > 0 \}$. Then, by the definition of $\widetilde{g}$ in \eqref{newg}, $u_1$ solves the BVP \begin{equation}\label{clas2} \begin{gathered} -\Delta u = 0, \quad \text{in } \Omega_{+}; \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} which has only the trivial solution $u\equiv 0$. Thus, $\Omega_+ = \emptyset $ and therefore $u_{1}\leqslant 0$ in $\Omega$. Before we state the main result of this section, though, we will discuss some properties of the critical point $u_1$. Since we have already seen that $u_1\leqslant 0$ in $\Omega$, it follows from the definition of $\widetilde{g}$ in \eqref{newg} that $u_1$ is also a solution of the BVP \eqref{de1}; consequently, $u_1$ is also a critical point of $J$. We will show shortly that $u_1$ is a local minimizer for $J$. Since $u_1$ is a solution of the BVP \eqref{clas1}, then $u_1$ is also a solution of the BVP \begin{equation} \begin{gathered} -\Delta u - p(x) u = \lambda_1 u_1 - g^-(x,u_1(x)), \quad \text{in } \Omega;\\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \label{Hopf05} \end{equation} where \begin{equation} p(x) = \begin{cases} \frac{g^+(x,u_1(x))}{u_1(x)}, & \text{if } u_1(x) < 0; \\ 0, & \text{if } u_1(x) = 0. \end{cases} \label{Hopf10} \end{equation} Now, it follows from \eqref{Hopf05} and the fact that $u_1(x)\leqslant 0$ for all $x\in \Omega$ that $u_1$ solves \begin{equation}\label{Hopf15} \begin{gathered} -\Delta u - p(x) u \leqslant 0, \quad \text{in } \Omega;\\ u = 0, \quad \text{on } \partial\Omega. \end{gathered} \end{equation} Thus, since $p(x)\leqslant 0$, according to \eqref{Hopf10}, we can apply the Hopf's Maximum Principle (see, for instance, \cite[Theorem 4, p. 333]{EV}) to conclude that \begin{equation}\label{Min1Eqn0020} u_1(x) < 0, \quad\text{for all } x\in\Omega, \end{equation} since $u_1$ is nontrivial, and \begin{equation} \frac{\partial u_1}{\partial \nu}(x) > 0, \quad\text{for } x\in\partial\Omega, \label{Min1Eqn0025} \end{equation} where $\nu$ denotes the outward unit normal vector to the surface $\partial\Omega$. We can then use \eqref{Min1Eqn0020} and \eqref{Min1Eqn0025}, and the assumption that $\Omega$ is bounded to show that there exists $\delta > 0$ such that, if $u\in C^1(\overline{\Omega})\cap H_{0}^{1}(\Omega)$ is such that $$ \|u - u_1\|_{C^1(\overline{\Omega})} < \delta, $$ then $$ u(x) < 0, \quad\text{for all } x\in\Omega. $$ Consequently, if $u$ is in a $\delta$-neighborhood of $u_1$ in the $C^1(\overline{\Omega})$ topology, then $$ J(u) = \widetilde{J}(u) \geqslant \widetilde{J}(u_1) = J(u_1); $$ so that $u_1$ is a local minimizer of $J$ in the $C^1(\overline{\Omega})$ topology. It then follows from a result of Brezis and Nirenberg in \cite{BN2} that $u_1$ is also a local minimizer for $J$ in the $H_{0}^{1}(\Omega)$ topology. We have therefore demonstrated the following theorem. \begin{theorem}\label{NontirivialSolThm05} Assume that $g$ and $G$ satisfy conditions {\rm (A1)--(A4)}. Assume also that (A6) and (A7) are satisfied. If (A5) holds true with $G_{-\infty} \leqslant 0$, then the BVP \eqref{de1} has a nontrivial solution, $u_1$, that is a local minimizer of the functional $J\colon H_{0}^{1}(\Omega)\to\mathbb{R}$ defined in \eqref{SResEqn00005}. \end{theorem} In the next section, we will provide additional conditions on the nonlinearity, $g$, that will allow us to show that the functional $J$ defined in \eqref{SResEqn00005} has another local minimizer. \section{Existence of a second local minimizer}\label{ExistenceSec16} In addition to (A1)--(A5), with $G_{-\infty} \leqslant 0$, and (A6)--(A7), we will assume \begin{itemize} \item [(A8)] there exists $s_1 >0$ such that $g(x,s_1) = 0$ for all $x\in\Omega$. \end{itemize} In this case, we consider the truncated nonlinearity $\overline{g}\colon\Omega\times\mathbb{R}\to\mathbb{R}$ given by \begin{equation}\label{newg2} \overline{g}(x,s)=\begin{cases} g(x,s),&\text{ for } 0\leqslant s \leqslant s_1, \\ 0, & \text{ elsewhere. } \end{cases} \end{equation} The corresponding primitive, $$ \overline{G}(x,s) = \int_{0}^{s} \overline{g}(x,\xi)\,d\xi, \quad\text{for all } x\in\Omega \text{ and } s\in\mathbb{R}, $$ is then given by \begin{equation}\label{Min2Eqn0005} \overline{G}(x,s) =\begin{cases} 0, & \text{if } s\leqslant 0; \\ G(x,s), & \text{if } 0 s_1, \end{cases} \end{equation} where $G$ is as given in \eqref{Gdfn05}. In view of the definitions of $\overline{g}$ and $\overline{G}$ in \eqref{newg2} and \eqref{Min2Eqn0005}, respectively, we see that $\overline{g}$ and $\overline{G}$ are bounded functions. Thus there exist positive constants $M_1$ and $M_2$ such that \begin{gather}\label{Min2Eqn0010} |\overline{g}(x,s)|\leqslant M_1, \quad\text{for all $x\in\overline{\Omega}$ and } s\in\mathbb{R}, \\ \label{Min2Eqn0015} |\overline{G}(x,s)|\leqslant M_2, \quad\text{for all $x\in\overline{\Omega}$ and }s\in\mathbb{R}. \nonumber \end{gather} The corresponding truncated functional, $\overline{J}\colon H_{0}^{1}(\Omega)\to\mathbb{R}$ is then given by \begin{equation}\label{Min2Eqn0020} \overline{J}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^{2}\,dx - \frac{\lambda_{1}}{2}\int_{\Omega}(u^{-})^{2}\,dx - \int_{\Omega}\overline{G}(x,u)\,dx,\quad u\in H_{0}^{1}(\Omega), \end{equation} where $\overline{G}$ is given in \eqref{Min2Eqn0005}. We then get that $\overline{J}$ is Fr\'echet differentiable with continuous derivative given by \begin{equation}\label{Min2Eqn0025} \langle\nabla\overline{J}(u),\varphi\rangle = \int_{\Omega}\nabla u \cdot\nabla\varphi \,dx + \lambda_{1}\int_{\Omega}u^{-}\varphi\,dx - \int_{\Omega}\overline{g}(x,u)\varphi\,dx, \end{equation} for all $u$ and $\varphi$ in $H_{0}^{1}(\Omega)$. Next, we show that $\overline{J}$ satisfies the $\text{(PS)}_c$ condition for $c\not\in\overline{\Lambda}$, where the exceptional set, $\overline{\Lambda}$, in this case is $$ \overline{\Lambda} = \{ 0\}. $$ \begin{proposition}\label{PScondJbar} Assume that $g$ and $G$ satisfy {\rm (A1), (A2)} and {\rm (A8)}, and define $\overline{J}$ as in \eqref{Min2Eqn0020}, where $\overline{G}$ is given in \eqref{Min2Eqn0005}. Then, $\overline{J}$ satisfies the $(PS)_{c}$ for $c\ne 0$. \end{proposition} \begin{proof} Let $(u_m)$ be a $\text{(PS)}_c$ sequence for $\overline{J}$, where \begin{equation}\label{Min2Eqn0027} c \neq 0; \end{equation} so that, according to \eqref{Min2Eqn0020} and \eqref{Min2Eqn0025} \begin{equation}\label{Min2Eqn0030} \frac{1}{2}\int_\Omega |\nabla u_m|^2 \,dx - \frac{\lambda_1}{2}\int_\Omega (u_m^-)^2 \,d x - \int_\Omega \overline{G}(x, u_m) \,dx \to c, \quad\text{as } m\to\infty, \end{equation} and \begin{equation}\label{Min2Eqn0035} \Big| \int_{\Omega}\nabla u_m \cdot\nabla\varphi \,dx + \lambda_{1}\int_{\Omega}u_m^{-}\varphi\,dx - \int_{\Omega}\overline{g}(x,u_m)\varphi\,dx\Big| \leqslant \varepsilon_m \|\varphi\|, \end{equation} for all $m$ and all $\varphi\in H_{0}^{1}(\Omega)$, where $(\varepsilon_m)$ is a sequence of positive numbers that tends to $0$ as $m\to\infty$. Writing $u_m = u_m^+ - u_m^-$ for all $m$, and taking $\varphi = u_m^+$ in \eqref{Min2Eqn0035} we obtain \begin{equation}\label{Min2Eqn0040} \Big| \|u_m^+\|^2 - \int_{\Omega}\overline{g}(x,u_m^+) u_m^+\,dx\Big| \leqslant \varepsilon_m \|u_m^+\|, \quad\text{for all } m. \end{equation} Using \eqref{Min2Eqn0010} we then estimate \begin{equation}\label{Min2Eqn0045} \big| \int_{\Omega}\overline{g}(x,u_m^+) u_m^+\,dx\big| \leqslant C \|u_m^+\|, \quad\text{for all } m, \end{equation} where we have also used the Cauchy-Schwarz and Poincar\'e inequalities. Combing \eqref{Min2Eqn0040} and \eqref{Min2Eqn0045} we then get that \begin{equation}\label{Min2Eqn0050} \|u_m^+\|^2 \leqslant C \|u_m^+\|, \quad\text{for all } m, \end{equation} where we have used the fact that $\varepsilon_m \to 0$ as $m\to\infty$. It follows from \eqref{Min2Eqn0050} that the sequence $(u_m^+)$ is bounded in $H_{0}^{1}(\Omega)$. Next, we show that $(u_m^-)$ is also bounded in $H_{0}^{1}(\Omega)$. If this is not the case, we may assume, passing to a subsequence if necessary, that \begin{equation}\label{Min2Eqn0055} \|u_{m}^{-}\|\to\infty \quad\text{as } m\to\infty. \end{equation} Define \begin{equation}\label{Min2Eqn0060} v_{m}=-\frac{u_{m}^{-}}{\|u_{m}^{-}\|},\quad\text{for all }m. \end{equation} Then, since \begin{equation}\label{Min2Eqn0062} \|v_{m}\|=1, \quad\text{for all } m, \end{equation} passing to a further subsequences if necessary, we may assume that there is $v\in H_{0}^{1}(\Omega)$ such that \begin{gather}\label{Min2Eqn0065} v_{m}\rightharpoonup v\quad\text{(weakly) in }H_{0}^{1}(\Omega), \text{ as } m\to\infty; \\ v_{m}\to v \quad\text{in }L^{2}(\Omega), \quad\text{as } m\to\infty; \label{Min2Eqn0070}\\ \label{Min2Eqn0075} v_{m}(x)\to v(x)\quad\text{for a.e. } x \text{in }\Omega, \quad\text{as } m\to\infty. \end{gather} Now, writing $u_{m}=u_{m}^{+}-u_{m}^{-}$ in \eqref{Min2Eqn0035} we have \begin{equation}\label{Min2Eqn0080} \Big|\int_{\Omega}\nabla u_{m}^{-}\cdot\nabla\varphi\,dx + \lambda_{1}\int_{\Omega} u_{m}^{-}\varphi\,dx - \int_{\Omega}\overline{g}(x,u_{m})\varphi\,dx \Big| \leqslant (\varepsilon_{m}+C\|u_m^+\|) \|\varphi\|, \end{equation} for all $m$ and all $\varphi\in H_{0}^{1}(\Omega)$, where we have also used the Cauchy-Schwarz and Poincar\'{e} inequalities. Next, divide both sides of \eqref{Min2Eqn0080} by $\|u_{m}^{-}\|$ and use \eqref{Min2Eqn0060} to obtain \begin{equation}\label{Min2Eqn0085} \Big|\int_{\Omega}\nabla v_{m}\cdot\nabla\varphi\,dx - \lambda_{1}\int_{\Omega}v_{m}\varphi\,dx - \int_{\Omega}\frac{\overline{g}(x,u_{m})}{\|u_{m}^{-}\|}\varphi\,dx \Big| \leqslant \Big(\frac{\varepsilon_{m}+C\|u_{m}^{+}\|}{\|u_{m}^{-}\|}\Big)\|\varphi\|, \end{equation} for all $\varphi\in H_{0}^{1}(\Omega)$ and all $m$. Using the estimate in \eqref{Min2Eqn0010} and \eqref{Min2Eqn0055} we obtain \begin{equation}\label{Min2Eqn0090} \lim_{m\to\infty} \Big| \int_{\Omega}\frac{\overline{g}(x,u_{m})}{\|u_{m}^{-}\|}\varphi\,dx \Big| = 0, \quad\text{for all } \varphi\in H_{0}^{1}(\Omega). \end{equation} Combining \eqref{Min2Eqn0085} and \eqref{Min2Eqn0090} we then get \begin{equation}\label{Min2Eqn0095} \lim_{m\to\infty} \Big|\int_{\Omega}\nabla v_{m}\cdot\nabla\varphi\,dx - \lambda_1\int_{\Omega}v_{m}\varphi\,dx \Big| = 0, \quad\text{for all } \varphi\in H^1(\Omega), \end{equation} where we have used \eqref{Min2Eqn0055} and the facts that $(u_m^+)$ is bounded in $H_{0}^{1}(\Omega)$ and $\varepsilon_m\to 0$ as $m\to\infty$. Now, it follows from \eqref{Min2Eqn0065}, \eqref{Min2Eqn0070} and \eqref{Min2Eqn0095} that \begin{equation}\label{Min2Eqn0100} \int_{\Omega}\nabla v\cdot\nabla \varphi\,dx - \lambda_{1}\int_{\Omega}v\varphi\,dx = 0,\quad\text{for all } \varphi\in H_{0}^{1}(\Omega); \end{equation} so that, $v$ is a weak solution of the BVP %\label{Min2Eqn0105} \begin{gather*} -\Delta u = \lambda_1 u, \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega. \end{gather*} Next, we take $\varphi = v_m$ in \eqref{Min2Eqn0085} and use \eqref{Min2Eqn0062} to obtain \begin{equation} \Big|1 - \lambda_{1}\int_{\Omega}v_{m}^2\,dx - \int_{\Omega}\frac{\overline{g}(x,u_{m})}{\|u_{m}^{-}\|}v_m\,dx \Big| \leqslant \frac{\varepsilon_{m}+C\|u_{m}^{+}\|}{\|u_{m}^{-}\|}, \quad\text{for all } m, \label{Min2Eqn0110} \end{equation} where, by \eqref{Min2Eqn0010}, \eqref{Min2Eqn0055} and \eqref{Min2Eqn0070}, \begin{equation}\label{Min2Eqn0115} \lim_{m\to\infty} \big| \int_{\Omega}\frac{\overline{g}(x,u_{m})}{\|u_{m}^{-}\|}v_m\,dx \big| = 0. \end{equation} Thus, using \eqref{Min2Eqn0055}, and the facts that $(u_m^+)$ is a bounded sequence and $\varepsilon_m \to 0$ as $m\to\infty$, we obtain from \eqref{Min2Eqn0110} and \eqref{Min2Eqn0115} that $$ \lim_{m\to\infty} \Big|1 - \lambda_{1}\int_{\Omega}v_{m}^2\,dx \Big| = 0; $$ so that, in view of \eqref{Min2Eqn0070}, \begin{equation}\label{Min2Eqn0120} \lambda_{1}\int_{\Omega}v^2\,dx = 1, \end{equation} from which we conclude that $v\not\equiv 0 $. Thus, $v$ is an eigenfunction of $-\Delta$ with Dirichlet boundary conditions over $\Omega$. It follows from this observation and \eqref{Min2Eqn0100}, in conjunction with \eqref{Min2Eqn0120}, that $\|v\|=1$. Consequently, we obtain from the definition of $v_m$ in \eqref{Min2Eqn0060} and from \eqref{Min2Eqn0075} that \begin{equation}\label{Min2Eqn0125} v = - \varphi_1. \end{equation} Recall that we have chosen $\varphi_1$ so that $\varphi_1 > 0$ in $\Omega$ and $\|\varphi_1\| = 1$. Next, writing $u_m = u_m^+ - u_m^-$ and using \eqref{Min2Eqn0060}, we obtain $$ \frac{u_m}{\|u_m^-\|} = \frac{u_m^+}{\|u_m^-\|} + v_m, \quad\text{for all } m; $$ so that, by the fact that $(u_m^+)$ is bounded and \eqref{Min2Eqn0055}, we may assume, passing to a further subsequence if necessary, that \begin{equation}\label{Min2Eqn0130} \frac{u_m(x)}{\|u_m^-\|} \to -\varphi_1(x), \quad\text{for a. e. } x\in\Omega, \quad\text{as } m\to\infty, \end{equation} where we have also used \eqref{Min2Eqn0075} and \eqref{Min2Eqn0125}. It then follows from \eqref{Min2Eqn0130} that \begin{equation}\label{Min2Eqn0135} u_{m}(x)\to -\infty\quad\text{for a. e. }x\in\Omega, \text{ as }m\to\infty. \end{equation} Then, using the definition of $\overline{G}$ in \eqref{Min2Eqn0005} and the Lebesgue dominated convergence theorem, we conclude from \eqref{Min2Eqn0135} that $$ \lim_{m\to\infty} \int_\Omega \overline{G}(x, u_m(x)) \,dx =0; $$ so that, in conjunction with \eqref{Min2Eqn0030}, \begin{equation}\label{Min2Eqn0140} \lim_{m\to\infty} \Big( \frac{1}{2}\int_\Omega |\nabla u_m|^2 \,dx - \frac{\lambda_1}{2}\int_\Omega (u_m^-)^2 \,d x \Big) = c. \end{equation} We may now proceed as in the proof of Proposition \ref{PScondJ} and show that \begin{equation}\label{Min2Eqn0145} \lim_{m\to\infty} \Big( \frac{1}{2}\int_\Omega |\nabla u_m|^2 \,dx - \frac{\lambda_1}{2}\int_\Omega (u_m^-)^2 \,d x \Big) = 0. \end{equation} Note that \eqref{Min2Eqn0140} and \eqref{Min2Eqn0145} are in contradiction with \eqref{Min2Eqn0027}. Consequently, $(u_m^-)$ is also bounded in $H_{0}^{1}(\Omega)$. Therefore, as in the last portion of the proof of Proposition \ref{PScondJ}, we can show that $(u_m)$ has a convergent subsequence. We have therefore established the fact that $\overline{J}$ satisfies the $\text{(PS)}_c$ condition, provided that $c\neq 0$. \end{proof} It follows from the definition of $\overline{J}$ in \eqref{Min2Eqn0025} and the estimate in \eqref{Min2Eqn0020} that $\overline{J}$ is bounded from below in $H_{0}^{1}(\Omega)$. Indeed, we obtain the estimate $$ \overline{J}(u) \geqslant \frac{1}{2}\|u^+\|^2 + \frac{1}{2}\|u^-\|^2 - \frac{\lambda_1}{2}\|u^-\|_{L^2}^2 - M_2 |\Omega|, \quad\text{for all } u \in H_{0}^{1}(\Omega); $$ so that, using the Poincar\'{e} inequality, $$ \overline{J}(u) \geqslant \frac{1}{2}\|u^+\|^2 - M_2 |\Omega|, \quad\text{for all } u \in H_{0}^{1}(\Omega), $$ from which we obtain that $$ \overline{J}(u) \geqslant - M_2 |\Omega|, \quad\text{for all } u \in H_{0}^{1}(\Omega). $$ Set \begin{equation}\label{Min2Eqn0150} c_2 = \inf_{v\in H_{0}^{1}(\Omega)} \overline{J}(v). \end{equation} We will show that, if (A6) and (A7) hold, then \begin{equation}\label{Min2Eqn0155} c_2 < 0. \end{equation} Indeed, for $$ 0 < t < \frac{s_1}{\max_{x\in\overline{\Omega}}\varphi_1(x)}, $$ compute \[ \overline{J}(t\varphi_1) = J(t\varphi_1) = \frac{t^2}{2} - \int_\Omega G(x, t\varphi_1(x))\,dx, \] so that \begin{gather*} \frac{d}{dt}\left[\overline{J}(t\varphi_1)\right] = t - \int_\Omega g(x,t\varphi_1(x))\varphi_1(x) \,dx, \\ \label{Min2Eqn0160} \frac{d^2}{dt^2}\left[\overline{J}(t\varphi_1)\right] = 1 - \int_\Omega \frac{\partial g}{\partial s} (x, t\varphi_1 (x)) (\varphi_1(x))^2 \,dx. \end{gather*} It then follows from \eqref{Min2Eqn0160} and (A6) and (A7) that $$ \lim_{t\to 0^+} \frac{d^2}{dt^2}[\overline{J}(t\varphi_1)] = 1 - a\int_\Omega \varphi_1^2 \,dx , $$ or $$ \lim_{t\to 0^+} \frac{d^2}{dt^2}[\overline{J}(t\varphi_1)] = 1 - \frac{a}{\lambda_1} < 0 , $$ since $a > \lambda_1$ according to (A7). Consequently, there exists $t_1 > 0$ such that $$ \overline{J}(t_1 \varphi_1) < 0. $$ Thus, in view of \eqref{Min2Eqn0150}, the assertion in \eqref{Min2Eqn0155} follows. In view of \eqref{Min2Eqn0155} and the result in Proposition \ref{PScondJbar} we see that $\overline{J}$ satisfies the $\text{(PS)}_{c_2}$ condition. Thus, given the definition of $c_2$ in \eqref{Min2Eqn0150}, the argument invoking Ekeland's Variational Principle leading to Theorem \ref{NontirivialSolThm05} in Section \ref{ExistenceSec06} can now be used to obtain a minimizer, $u_2$, of $\overline{J}$. Furthermore, as was done in Section \ref{ExistenceSec06}, we can use the Maximum Principle to conclude that $$ 0 < u_2(x) < s_1, \quad\text{for all } x\in\Omega; $$ so that, $u_2$ is also a critical point of $J$. Indeed, $u_2$ is a local minimizer of $J$ by the Br\'{e}zis and Nirenberg result in \cite{BN2}. We have therefore established the following multiplicity result. \begin{theorem}\label{NontirivialSolThm10} Assume that $g$ and $G$ satisfy conditions {\rm (A1)--(A5)}, with $G_{-\infty} \leqslant 0$, and {\rm (A6)--(A8)}. Let $J\colon H_{0}^{1}(\Omega)\to\mathbb{R}$ be the $C^1$ functional defined in \eqref{SResEqn00005}. In addition to the local minimizer, $u_1$, of $J$ given by Theorem \ref{NontirivialSolThm05}, which is a negative solution of the BVP \eqref{de1}, there exists another local minimizer, $u_2$, of $J$ that yields a positive solution of the BVP \eqref{de1}. \end{theorem} \section{Existence of a third nontrivial critical point}\label{ExistenceSec17} In the previous section we saw that, if $g$ and $G$ satisfy conditions (A1)--(A5), with $G_{-\infty} \leqslant 0$, and (A6)--(A8)), then the functional $J$ defined in \eqref{SResEqn00005} has two local minimizers distinct from $0$. In this section we prove the existence of a third, nontrivial, critical point of $J$. This will follow from following variant of the Mountain-Pass Theorem first proved by Pucci and Serrin in 1985, \cite{PucSer}. \begin{theorem}[{\cite[Theorem 1]{PucSer}}] \label{theopuc} Let $X$ be a real Banach space with norm $\|\cdot\|$ and $J:X\to\mathbb{R}$ be a $C^{1}$ functional. Let $u_o$ and $u_1$ be distinct points in $X$. Assume that there are real numbers $r$ and $R$ such that $$ 0 < r < \|u_1 -u_o\| < R, $$ and a real number $a$ such that \begin{gather*} J(u_o) \leqslant a \quad J(u_1) \leqslant a, \\ J(v) \geqslant a \quad\text{for all $v$ such that } r < \|v-u_o\| 0$ such that \begin{equation} \|\nabla J(u)\|\geq \delta,\quad\text{for all }u\in M. \label{p21} \end{equation} It then follows from that assumption that $J$ is $C^1$ that there exists $\varepsilon > 0$ such that \begin{equation} \|\nabla J(v)\|\geq\frac{\delta}{2} \quad\text{for all } v\in M_{\varepsilon} =\{v\in H_{0}^{1}(\Omega)| \text{dist}(v, M ) < \varepsilon\}. \label{p31} \end{equation} Notice that $M_{\varepsilon}$ is an open neighborhood of $M$ which does not contain $u_{1}$ and $u_2$ because they are local minimizers of $J$. Next, let $\rho\in C_{c}(H_{0}^{1}(\Omega),\mathbb{R})$ denote a function with $\operatorname{supp}(\rho)\subset M_{\varepsilon}$ such that $\rho(u)=A$, for $u\in M$, and $0\leq\rho(u)\leq A$ for all $u\in H_{0}^{1}(\Omega)$, where $A$ a positive constant to be chosen shortly. Define the following deformation $\eta\colon H_{0}^{1}(\Omega)\times\mathbb{R}\to H_{0}^{1}(\Omega)$ given by \begin{equation}\label{p41} \eta(u,t)=u-t\rho(u)\nabla J(u),\quad\text{for } u\in H_{0}^{1}(\Omega)\quad\text{and }t\in\mathbb{R}. \end{equation} It then follows that \begin{equation} \begin{aligned} \frac{d}{dt}J(\eta(u,t)) & = \langle \nabla J(\eta(u,t)),\eta_{t}(u,t)\rangle \\ & = \langle \nabla J(\eta(u,t)),-\rho(u)\nabla J(\eta(u,t))\rangle \\ & = -\rho(u)\|\nabla J(\eta(u,t))\|^{2}, \end{aligned} \label{p51} \end{equation} for $u\in H_{0}^{1}(\Omega)$ and $t\in\mathbb{R}$. In particular, it follows from \eqref{p51} and \eqref{p41} that \begin{equation} \frac{d}{dt}J(\eta(u,t))\big|_{t=0} = -\rho(u)\|\nabla J(u)\|^{2},\quad\text{for all }u\in H_{0}^{1}(\Omega). \label{p61} \end{equation} Note that, for points $u\in H_{0}^{1}(\Omega)$ such that $\rho(u)>0$, it follows from \eqref{p21} and \eqref{p31} that \begin{equation} \rho(u)\|\nabla J(\eta(u,t))\|^{2}>\rho(u)\frac{\delta^{2}}{4}> 0. \label{p71} \end{equation} Hence, by \eqref{p61} and the assumption that $J\in C^{1}(H_{0}^{1}(\Omega),\mathbb{R})$, we obtain that, for each $u\in H_{0}^{1}(\Omega)$ such that $\rho(u)>0$, there exists a neighborhood $U$ of $u$ and $T_{u} > 0$ such that \begin{equation} \frac{d}{dt}J(\eta(u,t)) < -\frac{\rho(u)}{2}\|\nabla J(u)\|^{2},\quad \text{for $u\in U$ and }|t|0$ such that \begin{equation*}%\label{p92} \frac{d}{dt}J(\eta(u,t)) \leq -\frac{\rho(u)}{8}\delta^{2}, \quad\text{for }u\in H_{0}^{1}(\Omega)\text{ and }|t|\lambda_{1}$. To construct $g$, let $g_{0}\colon\Omega\times\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying the following conditions: \begin{itemize} \item [(a)] $g_{0}(x,0)=0$ and $g_{0}(x,s)=0$ for all $x\in\Omega$ and $s\geqslant s_1$; \item [(b)] the derivative of $g_{0}$ has a jumping discontinuity at $0$ prescribed by $$ \lim_{s\to 0^{-}}\frac{g_{0}(x,s)}{s}=a-\lambda_{1}\quad\text{and }\quad \lim_{s\to 0^{+}}\frac{g_{0}(x,s)}{s}=a+s_{1}^{r-1}, $$ uniformly for a.e $x\in\Omega$; \item [(c)] $\lim_{s\to -\infty} g_{0}(x,s)=0$; \item [(d)] $\lim_{s\to -\infty} G_{0}(x,s)=G_{-\infty}$ where $ G_{0}(x,s)=\int_{0}^{s}g(x,\xi)d\xi$, for $x\in\Omega$, $s\in\mathbb{R}$. \end{itemize} Then, the function $g:\Omega\times\mathbb{R}\to\mathbb{R}$ given by $$ g(x,s) = g_{0}(x,s) + (s^{+})^{r}-s_{1}^{r-1}s^{+}, \quad\text{for } x\in\Omega, \text{ and }\ s\in\mathbb{R}, $$ where $s^{+}$ denotes the positive part of $s$, satisfies the conditions (A1)--(A8). \subsection*{Acknowledgments} The authors would like to thank Professor Alfonso Castro for very helpful conversations during the time they worked on the problems discussed in this paper. 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