\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2020 (2020), No. 60, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2020 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2020/60\hfil Superlinear elliptic problem] {Existence and multiplicity for a superlinear elliptic problem under a non-quadradicity condition at infinity} \author[L. Rec\^ova, A. Rumbos \hfil EJDE-2020/60\hfilneg] {Leandro Rec\^ova, Adolfo Rumbos} \address{Leandro L. Rec\^ova \newline T-Mobile Inc., Ontario, CA 91761, USA} \email{leandro.recova3@t-mobile.com} \address{Adolfo J. Rumbos \newline Department of Mathematics, Pomona College, Claremont, CA 91711, USA} \email{arumbos@pomona.edu} \thanks{Submitted February 28, 2020. Published June 16, 2020.} \subjclass{35J20} \keywords{Semilinear elliptic boundary value problem; \hfill\break\indent superlinear subcritical growth; infinite dimensional Morse theory; critical groups} \begin{abstract} In this article, we study the existence and multiplicity of solutions of the boundary-value problem \begin{gather*} -\Delta u = f(x,u), \quad \text{in } \Omega, \\ u = 0, \quad \text{on } \partial\Omega, \end{gather*} where $\Delta$ denotes the $N$-dimensional Laplacian, $\Omega$ is a bounded domain with smooth boundary, $\partial\Omega$, in $\mathbb{R}^N$ $(N\geqslant 3)$, and $f$ is a continuous function having subcritical growth in the second variable. Using infinite-dimensional Morse theory, we extended the results of Furtado and Silva \cite{FurtadoSilva} by proving the existence of a second nontrivial solution under a non-quadradicity condition at infinity on the non-linearity. Assuming more regularity on the non-linearity $f$, we are able to prove the existence of at least three nontrivial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Furtado and Silva \cite{FurtadoSilva} studied the existence and multiplicity of solutions for the boundary-value problem (BVP) \begin{equation}\label{prob1} \begin{gathered} -\Delta u = f(x,u), \quad \text{in } \Omega, \\ u = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Delta$ denotes the $N$-dimensional Laplacian, $\Omega$ is a bounded domain with smooth boundary, $\partial\Omega$, in $\mathbb{R}^N$ $(N\geqslant 3)$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies the following conditions: \begin{itemize} \item[(H1)] There exist constants $a_1>0$ and $p\in(2,2^{*})$ such that \begin{equation} |f(x,s)|\leq a_1(1+|s|^{p-1}),\quad\text{for } (x,s)\in\Omega\times\mathbb{R}, \label{condf0} \end{equation} where $2^{*}=2N/(N-2)$ is the critical Sobolev exponent. \item[(H2)] For $F(x,s)= \int_0^{s}f(x,\xi)\,d\xi$, for all $s\in\mathbb{R}$, \begin{equation} \lim_{|s|\to \infty}\left(f(x,s)s-2F(x,s)\right)=+\infty\quad\text{uniformly for } x\in\Omega. \label{condf1} \end{equation} \item[(H3)] The following limit holds \begin{equation} \lim_{|s|\to +\infty}\frac{2F(x,s)}{s^{2}}=+\infty\quad\text{ uniformly for }x\in\Omega\,. \label{condf3} \end{equation} \end{itemize} Furtado and Silva \cite{FurtadoSilva} proved the following existence and multiplicity result for problem \eqref{prob1}. \begin{theorem}[{\cite[Theorem 1.2]{FurtadoSilva}}] \label{theofurtado} Suppose $f$ satisfies {\rm (H1)--(H4)}. Then problem \eqref{prob1} has at least one nontrivial solution provided that \begin{equation}\label{condorigin2} \limsup_{s\to 0}\frac{F(x,s)}{s^2}=0,\quad\text{uniformly for } x\in\Omega. \end{equation} If $f(x,s)$ is odd in $s$, condition \eqref{condorigin2} can be dropped and and problem \eqref{prob1} has infinitely many weak solutions. \end{theorem} \begin{remark}\label{rmk1} \rm We remark first that condition \eqref{condorigin2} implies that \begin{equation}\label{f0at0} f(x,0) = 0, \quad\text{for all } x\in\Omega. \end{equation} Thus, the first part of Theorem \ref{theofurtado} asserts that problem \eqref{prob1} has at least two solutions, one of them being the trivial solution. \end{remark} \begin{remark}\label{rmk2}\rm Condition (H2) is denoted (NQ) in \cite{FurtadoSilva}; this is the non-quadraticity condition introduced by Costa and Magalh\~{a}es in \cite{CostaMag}. Condition (H3) is denoted (SL) in \cite{FurtadoSilva}; it imposes superlinear growth in the second variable of the nonlinearity $f$ of problem \eqref{prob1}. \end{remark} To prove Theorem \ref{theofurtado}, the authors of \cite{FurtadoSilva} first showed that the energy functional associated with problem \eqref{prob1} satisfies the Cerami condition. They then showed that the energy functional satisfies the conditions of the mountain pass theorem of Ambrosetti and Rabinowitz (see \cite{AmbRab,Rabinowitz1}). For the second part of the theorem, assuming that $f$ is odd, they showed that the conditions of the symmetric mountain pass theorem of Rabinowitz \cite{Rabinowitz1} were satisfied. Furtado and Silva also presented many examples in the literature that could be included in their framework, such as a double resonance problems. The authors of \cite{FurtadoSilva} also showed how to obtain a positive and negative solution of problem \eqref{prob1} by using a cutoff-technique presented in \cite{AmbRab}. For more details, see \cite{FurtadoSilva} and references therein. Condition (H2) was first introduced by Costa and Magalh\~{a}es in \cite{CostaMag}. It allowed the authors to treat resonant and double resonant problems without a restriction on the quotient $f(x,s)/s$. By considering some additional assumptions on the function $f$ and its primitive, Costa and Magalh\~{a}es proved that the associated energy functional satisfies the geometric conditions of the mountain-pass theorem and the saddle-point theorem of Rabinowitz (see \cite{Rabinowitz1}), and, consequently, proved the existence of a nontrivial solution for problem \eqref{prob1}. Furtado and Silva \cite{FurtadoSilva} assumed the non-quadradicity condition (H2) for the superlinear problem proposed in this work. One of the main motivations for (H2) and \eqref{condorigin2} is that there are many non-linearities that do not satisfy the Ambrosetti-Rabinowitz condition: \begin{itemize} \item[(H4)] There exist constants $\mu > 2$ and $R>0$ such that $$0 < \mu F(x,s) < sf(x,s),\quad\text{for } |s|> R, \text{ and } x\in\Omega,$$ \end{itemize} (see \cite{AmbRab,Rabinowitz1}). Such problems were also studied by Miyagaki and Souto \cite{MiSot}, Liu \cite{Liu5}, and Li and Wang \cite{LiWang}. In this article, we use infinite-dimensional Morse theory to extend the results of Furtado and Silva for the case in which $f$ is not assumed to be odd. We prove that, under the same hypotheses in Theorem \ref{theofurtado}, problem \eqref{prob1} has at least two nontrivial solutions; this is the content of Theorem \ref{ourthm} in Section \ref{secfinal}. To prove Theorem \ref{ourthm}, we compute the critical groups at the origin and at infinity of the energy functional associated with problem \eqref{prob1}. To compute the critical groups at the origin, we show that the trivial solution of problem \eqref{prob1} is a local minimum of the associated energy functional. To compute the critical groups at infinity, we use an argument similar to that presented in \cite[Section $3$]{RecRumb5} by using a standard argument involving a long exact sequence of reduced homology groups. Using the same techniques, we prove the existence of three nontrivial solutions of problem \eqref{prob1} for the case in which the nonlinearity $f$ is differentiable with continuous derivative, and the condition on its primitive, $F$, at the origin in \eqref{condorigin2} is stated in terms of the limit as $s$ approaches $0$, and not the limit superior. This article is organized as follows: In Section \ref{secprel}, we present the variational framework that will be used throughout this work. In Section \ref{secorigin}, we present the computation of the critical groups at the origin. In Section \ref{secinfty}, we compute the critical groups at infinity. In Section \ref{secfinal}, we use a standard argument involving the Morse relation to show the existence of a second nontrivial solution of problem \eqref{prob1} under the assumptions of Theorem \ref{theofurtado}. Finally, in Section \ref{secfinal2}, we prove the existence of three nontrivial solutions for problem \eqref{prob1} by assuming that the nonlinearity $f$ is $C^1$ and a strengthening of condition \eqref{condorigin2}. \section{Variational framework}\label{secprel} Let $X$ denote the Sobolev space $H_0^1(\Omega)$ obtained through completion of $C_{c}^{\infty}(\Omega)$ with respect to the metric induced by the norm $$\|u\|=\Big(\int_{\Omega}|\nabla u|^{2}dx\Big)^{1/2},\quad \text{for all }u\in X.$$ Weak solutions of \eqref{prob1} are critical points of the functional $J\colon X\to\mathbb{R}$ given by \begin{equation}\label{func1} J(u) = \frac{1}{2} \int_\Omega |\nabla u|^2 \,dx -\int_{\Omega}F(x,u)\,dx,\quad \text{for }u\in X. \end{equation} The functional $J$ belongs to $C^1(X,\mathbb{R})$ and its Fr\'{e}chet derivative at $u\in X$ is given by \begin{equation} \langle J'(u),v\rangle = \int_{\Omega}\nabla u\cdot\nabla v\,dx -\int_{\Omega}f(x,u)v\,dx, \quad\text{for any } v\in X. \label{func2} \end{equation} We say that a functional $J\in C^1(X,\mathbb{R})$ satisfies the Cerami condition at level $c$ (denoted $(C)_{c}$), if every sequence $(u_j)$ in $X$ such that $$J(u_{j})\to c,\quad (1+\|u_{j}\|)J'(u_{j})\to 0,\quad\text{as }j\to\infty,$$ called a $(C)_{c}$ sequence, has a convergent subsequence. We say that $J$ satisfies $(C)$ if it satisfies $(C)_{c}$ for every $c$. This condition was introduced by Cerami \cite{CER}. It is a weaker condition compared to the Palais-Smale condition, but the main deformation lemmas used in critical point theory are still valid assuming the Cerami condition (see \cite[Chapter $1$]{PerSch}). For a full exposition of the various compactness conditions used in critical point theory, we refer the reader to Mawhin and Willem \cite{MW33} and references therein. Based on conditions (H2) and (H3), the authors in \cite{FurtadoSilva} proved that the energy functional given in \eqref{func1} associated with problem \eqref{prob1} satisfies a Cerami condition at any level $c\in\mathbb{R}$ (see \cite[Theorem 1]{FurtadoSilva}). This condition is needed in the use of infinite-dimensional Morse theory, which is an important tool in the arguments presented in this paper. Let $A,B$ be two topological spaces with $B\subset A$. Denote by $H_q(A,B)$ the $q$-singular relative homology group of the pair $(A,B)$ with coefficients in a field $\mathbb{F}$. Let $c=J(u_0)$, where $u_0$ is an isolated critical point of $J$, and set $$J^{c}=\{u\in X : J(u)\leqslant c\}.$$ The $q$-critical groups of $J$ at $u_0$, with coefficients in $\mathbb{F}$, are given by \begin{equation} C_q(J,u_0) = H_q(J^{c}\cap U,(J^{c}\cap U)\backslash\{u_0\}),\quad q\in\mathbb{Z}, \label{cgroup00} \end{equation} (see \cite[Definition $4.1$, p. 32]{KC}), where $U$ is an open neighborhood of $u_0$ such that $u_0$ is the unique critical point of $J$ in $U$. The critical groups of isolated critical points are well--defined and they do not depend on the choice of the neighborhood $U$. This follows from the excision property of homology theory. Assume that $J$ satisfies the Cerami condition and let $\mathcal{K}=\{u\in X:J'(u)=0\}$ be the set of critical points of $J$ and $-a < \inf_{u\in\mathcal{K}} J(\mathcal{K})$. The critical groups at infinity were first introduced by Barstch and Li \cite{BLi} and are $$C_q(J,\infty) = H_q(X,J^{-a}),\quad\text{for all }q\in\mathbb{Z}.$$ By the second deformation theorem (see \cite[Theorem $3.2$, Chapter I]{KC}), these critical groups are well-defined. We will also denote by $\widetilde{H}_q(A)$ the reduced homology groups of the topological space $A\subset X$ with coefficients in a field $\mathbb{F}$ defined by \begin{gather*} \widetilde{H}_q(A)=H_q(A)\quad\text{for }q>0, \\ H_0(A) =\widetilde{H}_0(A)\oplus\mathbb{F}. \end{gather*} The reduced homology groups for pair $(A,B)$ are defined in a similar manner. For more details on these definitions, we refer the reader to Hatcher \cite[Chapter $2$, page $110$]{AH}. \section{Critical groups at the origin}\label{secorigin} Note that in view of \eqref{f0at0} and the definition of the Fr\'echet derivative of $J$ in \eqref{func2}, the origin of $X$ is a critical point of $J$. In this section, we compute the critical groups at the origin of the functional $J$ defined in \eqref{func1}. Before we prove the main results of this section, we need some estimates on the function $F$. By condition \eqref{condorigin2}, given any $\varepsilon > 0$, there exists a $\delta >0$ such that \begin{equation} |s|<\delta \Rightarrow F(x,s)<\frac{\varepsilon}{2} s^{2}, \quad\text{for } x\in\Omega. \label{condf5} \end{equation} Next, condition (H1) implies that there exists a constant $A=A(\delta)$ such that \begin{equation} |F(x,s)|\leqslant A|s|^{p},\quad\text{for all }|s|\geqslant\delta \text{ and a.e. } x\in\Omega. \label{condf6} \end{equation} In fact, assume $s \geqslant \delta$, and use (H1) to obtain the estimate $|F(x,s)| \leqslant \int_0^{s}|f(x,\xi)|\,d\xi \\ \leqslant a_1s + \frac{a_1}{p}s^{p} ,$ for $x\in\Omega$; so that, \begin{equation}\label{AReqn0005} \begin{aligned} |F(x,s)| & \leqslant a_1\big[\delta\left(\frac{s}{\delta}\right) +\frac{\delta^{p}}{p}\left(\frac{s}{\delta}\right)^{p}\big] , \quad\text{for } x\in\Omega. \end{aligned} \end{equation} Consequently, since we are assuming that $s \geqslant \delta$; so that $\frac{s}{\delta} \geqslant 1$, it follows from \eqref{AReqn0005} that \begin{aligned} |F(x,s)| & \leqslant a_1\big[\delta\left(\frac{s}{\delta}\right)^{p}+\frac{\delta^{p}}{p} \left(\frac{s}{\delta}\right)^{p}\big], \quad\text{for } x\in\Omega \text{ and } s \geqslant \delta, \end{aligned} from which we obtain that \begin{equation}\label{AReqn0010} |F(x,s)| \leqslant \frac{a_1}{\delta^p}\left(\delta+\delta^{p}\right)s^{p}, \quad\text{for } x\in\Omega \text{ and } s \geqslant \delta, \end{equation} where we have used the fact that $p>1$, in view of that assumption $p\in (2,2^\ast)$ in (H1). Setting $A = A(\delta)=\frac{a_1}{\delta^p}\left(\delta+\delta^{p}\right),\$ we see that \eqref{condf6} follows from \eqref{AReqn0010} for the case $s \geqslant \delta$. The case for $s \leqslant-\delta$ is analogous. Therefore, the estimate \eqref{condf6} is valid for all $|s| \geqslant \delta$. Combining the estimates \eqref{condf5} and \eqref{condf6}, we obtain that \begin{equation} F(x,s)\leqslant \frac{\varepsilon}{2} s^{2}+A|s|^{p},\quad\text{for } x\in\Omega\text{ and } s\in\mathbb{R}. \label{condf7} \end{equation} Next, use this estimate in \eqref{condf7} to obtain $$\int_{\Omega}F(x,u)\,dx \leqslant \frac{\varepsilon}{2}\int_{\Omega}|u|^{2}\,dx + A\int_{\Omega}|u|^{p};$$ so that, using the Poincar\'{e} and Sobolev inequalities, \begin{equation} \int_{\Omega}F(x,u)\,dx \leqslant C\left(\frac{\varepsilon}{2} + A\|u\|^{p-2}\right)\|u\|^{2}, \label{condf99} \end{equation} for some positive constant $C$. Setting $\rho= (\frac{\varepsilon}{2A})^{1/(p-2)}$, we see from \eqref{condf99} that \begin{equation} \|u\| < \rho \implies \int_{\Omega}F(x,u)\,dx \leqslant C\varepsilon \|u\|^{2}. \label{condf10} \end{equation} \begin{lemma} \label{minlemma} Assume that $f$ satisfies {\rm (H1)} and \eqref{condorigin2}. Then, the critical groups of $J$ at the origin are $C_q(J,0)=H_q(J^{0}\cap B_{\rho}(0),(J^{0}\cap B_{\rho}(0))\backslash\{0\}) \cong \delta_{q,0}\mathbb{F}\quad\text{for }q\in\mathbb{Z}.$ \end{lemma} \begin{proof} It follows from \eqref{condf10} and the definition of $J$ in \eqref{func1} that $$J(u) \geqslant \Big(\frac{1}{2}-C\varepsilon\Big)\|u\|^{2},$$ so that, since $\varepsilon$ is arbitrary, we can choose $\varepsilon = 1/(4C)$ to obtain \begin{equation}\label{Ru20200127} J(u) \geqslant \frac{1}{4}\|u\|^{2} > J(0), \quad\text{for } 0 < \|u\| < \rho, \end{equation} where $\rho > 0$ is sufficiently small. Consequently, $u = 0$ is a local minimum of $J$ in $B_{\rho}(0)$. Then, by \eqref{cgroup00} with $U=B_{\rho}(0)$, it follows from \cite[Example $1$, page $33$]{KC} that \begin{equation} C_q(J,0)\cong \delta_{q,0}\mathbb{F},\quad\text{for }q\in\mathbb{Z}. \label{coriginfinal} \end{equation} \end{proof} \section{Critical groups at infinity} \label{secinfty} In this section, we compute the critical groups at infinity of the functional $J$ given in \eqref{func1}. We assume that the functions $f$ and $F$ satisfy the conditions in (H1) and (H2). Let $\mathcal{K}=\{u\in X: J'(u)=0\}$ be the critical set of $J$. We first show that the functional $J$ is bounded from below in $\mathcal{K}$. It follows from \eqref{func2} that \begin{equation} \|u_0\|^{2}=\int_{\Omega}f(x,u_0)u_0\,dx,\quad\text{for }u_0\in \mathcal{K}. \label{condf8} \end{equation} Substituting \eqref{condf8} into the definition of $J$ in \eqref{func1} yields \begin{equation} J(u_0)=\frac{1}{2}\int_{\Omega}\left(f(x,u_0)u_0-2F(x,u_0)\right)\,dx, \quad\text{for } u_0 \in\mathcal{K}. \label{condf9} \end{equation} Now, by condition (H2), there exists $R_1>0$ such that \begin{equation} |s| > R_1\implies f(x,s)s-2F(x,s)> 1,\quad \text{for }x\in\overline{\Omega}. \label{f2cond} \end{equation} Next, denote $f(x,s)s-2F(x,s)$ by $H(x,s)$, for $(x,s)\in\Omega\times\mathbb{R}$, to rewrite \eqref{condf9} as follows $$J(u_0) = \frac{1}{2}\int_{|u_0|\leqslant R_1}H(x,u_0)\,dx + \frac{1}{2}\int_{|u_0|>R_1}H(x,u_0)\,dx , \quad\text{for } u_0\in\mathcal{K};$$ so that, in view of \eqref{f2cond}, \begin{equation} J(u_0) \geqslant \frac{1}{2}\int_{|u_0|\leqslant R_1}H(x,u_0)\,dx , \quad\text{for } u_0\in\mathcal{K}. \label{f3cond05} \end{equation} Thus, letting \begin{equation}\label{RuEqn0001} C_0=\max_{x\in\overline{\Omega},\, |s|\leqslant R_1}|H(x,s)|, \end{equation} from \eqref{f3cond05} we obtain \begin{equation} J(u_0) \geqslant -\frac{C_0}{2}|\Omega|,\quad\text{for }u_0\in\mathcal{K}. \label{f4cond} \end{equation} It follows from \eqref{f4cond} that the set of critical values of $J$ is bounded below. Thus, the critical groups of $J$ at infinity are well-defined. In what follows, let $a_0>0$ be such that $-a_0< \inf_{u\in \mathcal{K}}J(u)$. \begin{lemma} \label{lemmacinf2} Let $J$ be the $C^1(X,\mathbb{R})$ functional defined in \eqref{func1}, and assume that {\rm (H1)} and {\rm (H2)} are satisfied. There exists a constant $M \geqslant a_0$ such that any compact subset of the sub--level set $J^{-M}$ is contractible in $J^{-M}$. \end{lemma} \begin{proof} We show that we can deform the level set $J^{-M}$ to the level set $J^{-2M}$ for an $M \geqslant a_0$ that will be chosen shortly. The rest of the proof of Lemma \ref{lemmacinf2} will follow the same steps shown in the proof of \cite[Proposition $2.1$]{LiuS1}. Let $u\in J^{-M}$ and $A$ denote a compact subset of $J^{-M}$. Using the definition of the functional $J$ given in \eqref{func1} we have that \begin{equation}\label{RuEqn0005} J(tu) = \frac{t^2}{2} \|u\|^2 -\int_{\Omega}F(x,tu)\,dx,\quad\text{for }t\in \mathbb{R}. \end{equation} Consequently, $$\frac{d}{dt}[ J(tu)] = t \|u\|^2 - \int_{\Omega}f(x,tu)u\,dx, \quad\text{for }t\in \mathbb{R},$$ which we can rewrite as $$\frac{d}{dt}[ J(tu)] = \frac{2}{t}\Big[\frac{t^2}{2}\|u\|^{2}-\frac{1}{2}\int_{\Omega}f(x,tu)tu\,dx\Big], \quad\text{for }t\neq 0;$$ so that, by \eqref{RuEqn0005}, \begin{equation}\label{eq2} \frac{d}{dt}[J(tu)] = \frac{2}{t}\Big[J(tu) -\frac{1}{2}\int_{\Omega}(f(x,tu)tu\,dx-2F(x,u))\Big], \quad\text{for }t\neq 0. \end{equation} Next, define \begin{equation} \Omega_1^{t}=\{x\in\Omega: |tu(x)|\leqslant R_1\}\quad\text{and}\quad \Omega_2^{t}=\Omega\backslash\Omega^t_1, \label{defOmega} \end{equation} where $R_1>0$ is the constant from \eqref{f2cond}. Then, denoting $f(x,s)s-2F(x,s)$ by $H(x,s)$, for $(x,s)\in\Omega\times\mathbb{R}$, we can write \eqref{eq2} as \begin{equation}\label{eq3} \frac{d}{dt}[J(tu)] = \frac{2}{t}\Big[J(tu)-\frac{1}{2}\int_{\Omega_1^{t}}H(x,tu)\,dx -\frac{1}{2}\int_{\Omega_2^{t}}H(x,tu)\,dx\Big], \end{equation} for $t\neq 0$. Now, in view of \eqref{f2cond} and the definition of $\Omega_2^t$ in \eqref{defOmega} we have that \begin{equation} \int_{\Omega_2^{t}}H(x,tu)\,dx \geqslant 0 ,\quad\text{for all } t. \label{f6cond} \end{equation} Combining \eqref{eq3} and \eqref{f6cond} yields \begin{equation}\label{RuEqn0010} \frac{d}{dt}[J(tu)] \leqslant \frac{2}{t}\Big[J(tu) -\frac{1}{2}\int_{\Omega_1^{t}}H(x,tu)\,dx \Big], \quad\text{for } t\neq 0. \end{equation} On the other hand, \begin{equation} \big|\int_{\Omega_1^{t}}H(x,tu)\,dx\big| \leq C_0|\Omega|,\quad\text{for }x\in\overline{\Omega}, \label{f5cond} \end{equation} where $C_0$ is the constant given in \eqref{RuEqn0001}. It then follows from \eqref{RuEqn0010} and \eqref{f5cond} that \begin{equation}\label{RuEqn0012} \frac{d}{dt}[J(tu)] \leqslant \frac{2}{t}\Big[J(tu)+\frac{C_0}{2}|\Omega|\Big], \quad\text{for } t\neq 0, \end{equation} which we can rewrite as \begin{equation}\label{f7cond} \frac{d}{dt}[J(tu)] - \frac{2}{t}J(tu) \leqslant \frac{C_0|\Omega|}{t}, \quad\text{for } t\neq 0. \end{equation} Multiplying \eqref{f7cond} by the integrating factor $1/t^{2}$, and integrating from $1$ to $t>1$, we obtain $$\int_1^{t}\frac{d}{d\xi}\big[\frac{1}{\xi^{2}}J(\xi u)\big]\,d\xi \leqslant C_0|\Omega|\int_1^{t}\frac{1}{\xi^{3}}\,d\xi, % \label{derv333}$$ from which we obtain \begin{equation} J(tu) \leqslant t^{2}J(u)- \frac{C_0|\Omega|}{2}(t^2-1), \quad\text{for } t\geqslant 1. \label{derv33} \end{equation} Since we are assuming that $u\in J^{-M}$,from \eqref{derv33} we obtain \begin{equation}\label{RuEqn0015} J(tu) \leqslant -t^{2} M, \quad\text{for all } t\geqslant 1. \end{equation} It then follows from \eqref{RuEqn0015} that \begin{equation}\label{RuEqn0017} J(tu) \leqslant -M, \quad\text{for all } t\geqslant 1 \text{ and all } u \in J^{-M}. \end{equation} It also follows from \eqref{RuEqn0015} that \begin{equation}\label{RuEqn0020} J(tu) \to -\infty \quad \text{as } t\to\infty, \text{ for all } u\in J^{-M}. \end{equation} Next, observe that, in view of \eqref{RuEqn0012} and \eqref{RuEqn0017}, $$\frac{d}{dt}[J(tu)] \leqslant\frac{2}{t}\Big[-M+\frac{C_0}{2}|\Omega|\Big], \quad\text{for all } t\geqslant 1 \text{ and } u\in J^{-M},$$ which we can rewrite as \begin{equation}\label{RuEqn0025} \frac{d}{dt}[J(tu)] \leqslant-\frac{2}{t}\Big[M-\frac{C_0}{2}|\Omega|\Big], \quad\text{for all } t\geqslant 1 \text{ and } u\in J^{-M}. \end{equation} Setting $a_1= C_0|\Omega|/2,\$ we see that, if $M > \max\{a_0,a_1\}$, then from \eqref{RuEqn0025} it follows that \begin{equation}\label{RuEqn0030} \frac{d}{dt}[J(tu)] < 0, \quad\text{for all } t\geqslant 1 \text{ and } u\in J^{-M}. \end{equation} This determines our choice of $M$ in the statement of Lemma \ref{lemmacinf2}. Now, it follows from \eqref{RuEqn0020}, the intermediate value theorem, and the estimate in \eqref{RuEqn0030} that there exists $t^\ast \geqslant1$ such that $J(t^\ast u)\leqslant -2M$. As a consequence of \eqref{RuEqn0030} and the implicit function theorem, we also get that $t^\ast$ is a continuous function of $u$, for $u\in J^{-M}$. Thus, for any compact subset, $A$, of $J^{-M}$, we can define a continuous map $\eta_1:[0,1]\times A\to X$ by \begin{equation}\label{derv4} \eta_1(t,u)=\left[(1-t) + tt^\ast(u)\right]u,\quad\text{for }(t,u)\in [0,1]\times A. \end{equation} Hence, in view of \eqref{RuEqn0017}, $\eta_1$ defines a continuous map from $[0,1]\times A$ to $J^{-M}$. Set $A_1=\eta_1(1,A)$. Then, $A_1$ is also a compact set and $A_1\subset J^{-2M}$. Thus, any compact subset, $A$, of $J^{-M}$ can be deformed in $J^{-M}$ to a compact subset of $J^{-2M}$. The rest of the proof follows the same steps outlined in the proof of \cite[Proposition $2.1$]{LiuS1}, or the proof of \cite[Proposition $7.1$]{RecRumb4}. \end{proof} As a consequence of Lemma \ref{lemmacinf2}, we conclude that, for $M > \max\{a_0,a_1\}$, \begin{equation}\label{RuEqn0035} \widetilde{H}_q(J^{-M})\cong 0,\quad\text{for }q\in\mathbb{Z}. \end{equation} The computation of the critical groups of $J$ at infinity follows by using a standard argument with the following long exact sequence of reduced homology groups \begin{equation}\label{RuEqn0040} \ldots{\to}{\widetilde H}_q(J^{-M})\stackrel{i_{*}}{\to}{\widetilde H}_q(X) \stackrel{j_{*}}\to {\widetilde H}_q(X,J^{-M})\stackrel{\partial_{*}} \to {\widetilde H}_{q-1}(J^{-M})\stackrel{i_{*}}\to\ldots \end{equation} where $i_{*}$ and $j_{*}$ are the induced homomorphisms of the inclusion maps $$i\colon J^{-M}\to X,\quad j\colon(X,\emptyset)\to (X,J^{-M}),$$ respectively, and $\partial_{*}:{\widetilde H}_q(X,J^{-M}) \to {\widetilde H}_{q-1}(J^{-M})$ is a homomorphism. Using the fact that $X$ is contractible and the assertion in \eqref{RuEqn0035}, we deduce from the long exact sequence in \eqref{RuEqn0040} and the definition of reduced homology groups that \begin{equation} C_q(J,\infty)=H_q(X,J^{-M})\cong \delta_{q,0}\mathbb{F},\quad\text{for }q\in\mathbb{Z}. \label{cinftygroup} \end{equation} For more details on this calculation, we refer the reader to \cite[Section $3$]{RecRumb5}. \section{Existence of a second nontrivial solution}\label{secfinal} In this section, we prove the existence of a second nontrivial solution of problem \eqref{prob1} under the assumptions of Theorem \ref{theofurtado}. To do that, we will use an argument by contradiction involving the Morse relation. First, let $u_1$ denote the nontrivial solution of problem \eqref{prob1} found in \cite[Theorem $1.2$]{FurtadoSilva} by means of the mountain-pass theorem. Assume, by way of contradiction, that $0$ and $u_1$ are the only critical points of $J$. Then, the critical groups $C_q(J,u_1)$ are given by \begin{equation} C_q(J,u_1)\cong\delta_{q,1}\mathbb{F},\quad\text{for }q\in\mathbb{Z}, \label{mttype} \end{equation} (see \cite[Proposition $6.101$]{MMP}). Before presenting the final argument, we briefly discuss the Morse relation. Let $J\in C^1(X,\mathbb{R})$ be a functional that satisfies the Cerami condition. If $J$ has a finite number of critical points, we define the Morse--type numbers of the pair $(X,J^{-M})$ by \begin{equation} M_q:=M_q(X,J^{-M}) = \sum_{u\in \mathcal{K}}\dim C_q(J,u),\quad q=0,1,2,\ldots, \label{Morsetype} \end{equation} where $-M< \inf_{u\in \mathcal{K}}J(u).$ Applying the infinite-dimensional Morse-theory developed in \cite{KC}, \cite{MW}, or \cite{MMP}, we can derive the Morse relation \begin{equation} \sum_{q=0}^{\infty}M_qt^q=\sum_{q=0}^{\infty}\beta_qt^q +(1+t)\sum_{q=0}^{\infty}a_qt^q, \label{morserel} \end{equation} where $\beta_q=\dim C_q(J,\infty)$ and $a_q$ are non-negative numbers. The integers $\beta_q$, for $q\in\mathbb{Z}$, are called the Betti numbers of the pair $(X,J^{-M})$. Let $M$ be the constant from Lemma \ref{lemmacinf2}. We first note that $J$ satisfies the Cerami condition as a consequence of \cite[Theorem $1,1$]{FurtadoSilva}. Hence, by \eqref{coriginfinal} and \eqref{mttype}, we obtain the Morse type numbers of the pair $(X,J^{-M})$ as \begin{equation} M_0=\dim C_0(J,0) =1 ,\quad M_1=\dim C_1(J,u_1)=1,\quad M_q=0,\quad \text{for }q>1. \label{mvalues} \end{equation} On the other hand, it follows from \eqref{cinftygroup} that the Betti numbers of the pair $(X,J^{-M})$ are given by \begin{equation} \beta_0=1\quad\text{and}\quad\beta_q=0,\quad \text{for }q>0. \label{bettinumbers} \end{equation} Therefore, by \eqref{mvalues} and \eqref{bettinumbers}, it follows from the Morse relation \eqref{morserel} with $t=-1$ that $M_0(-1)^{0}+M_1(-1)^1 = \beta_0(-1)^{0},$ that is, $0=1$, which is a contradiction. Thus, $J$ must have another critical point. Hence, assuming the same hypotheses in Theorem \ref{theofurtado}, we have proved the following result. \begin{theorem}\label{ourthm} Suppose $f$ satisfies {\rm (H1), (H2), (H4)}, Then problem \eqref{prob1} has at least two nontrivial solutions, provided that \begin{equation}\label{RuEqn0045} \limsup_{s\to 0}\frac{F(x,s)}{s^2}=0,\quad\text{uniformly for }x\in\Omega. \end{equation} \end{theorem} \section{Existence of three nontrivial solutions}\label{secfinal2} In this section we show that, under an additional regularity assumption on the nonlinearity $f$, we can obtain three nontrivial solutions of problem \eqref{prob1}. This result is motivated by the final remark in \cite{FurtadoSilva}. Furtado and Silva obtained two nontrivial solutions of mountain pass type using a cutoff technique. We will use the arguments of the previous section to prove that problem \eqref{prob1} has a third nontrivial solution, provided that $f$ is assumed to be $C^1$ and that the assumption \eqref{RuEqn0045} is replaced by \begin{equation}\label{RuEqn0047} \lim_{s\to 0}\frac{F(x,s)}{s^2}=0,\quad\text{uniformly for }x\in\Omega. \end{equation} This is the content of the following theorem \begin{theorem}\label{theo3sol} Suppose $f$ satisfies {\rm (H1)--(H3)}. Assume also that $f\in C^1(\Omega\times\mathbb{R},\mathbb{R})$ and that \eqref{RuEqn0047} holds. Then, problem \eqref{prob1} has at least three nontrivial solutions. \end{theorem} \begin{remark}\label{RmkDerf0} \rm We remark that the assumption that $f\in C^1(\Omega\times\mathbb{R},\mathbb{R})$ and condition \eqref{RuEqn0047} imply that \begin{equation}\label{RuEqn0049} \frac{\partial f}{\partial s}(x,0) = 0, \quad\text{ uniformly for } x\in\Omega, \end{equation} as a consequence of L'Hospital's rule. \end{remark} \begin{proof}[Proof of Theorem \ref{theo3sol}] We start the proof by showing the existence of two nontrivial solutions of the mountain-pass type as described in \cite{FurtadoSilva}. We present the details here for the reader's convenience. We then proceed with the argument using the Morse relation to obtain a third nontrivial solution of \eqref{prob1}. First, we obtain a positive solution $u_1$ of problem \eqref{prob1}. A negative solution, $u_2$, can be obtained in an analogous way. The assumption $f\in C^1$ will imply that weak solutions of \eqref{prob1} are also classical solutions (see Agmon \cite{Ag}). This will allow us to use the maximum principle and obtain a positive solution and a negative solution. We will use the arguments presented in \cite[Corollary $2.23$]{Rabinowitz1}. Consider the truncated version of the function $f$, \begin{equation}\label{truncf} \overline{f}(x,s)=\begin{cases} f(x,s), & \text{for } s \geqslant 0; \\ 0, & \text{for } s < 0, \end{cases} \end{equation} and its primitive \begin{equation}\label{FbarDfn05} \overline{F}(x,s)=\int_0^{s}\overline{f}(x,\xi)\ d\xi, \quad\text{for all } (x,s)\in\Omega\times\mathbb{R}. \end{equation} Define the associated functional $J^{+}: X\to \mathbb{R}$ by \begin{equation}\label{newfunc} J^{+}(u) =\frac{1}{2}\|u\|^{2}-\int_{\Omega}\overline{F}(x,u)\,dx, \quad\text{for } u\in X. \end{equation} We note that $J^+$ is Fr\'echet differentiable with derivative given by $$\langle {J^{+}}'(u),v\rangle = \int_{\Omega}\nabla u\cdot\nabla v\,dx -\int_{\Omega}\overline{f}(x,u)v\,dx, \quad\text{for all } u,v\in X,$$ which, in view of the definition of $\overline{f}$ in \eqref{truncf} is equivalent to \begin{equation}\label{DerJplus} \langle {J^{+}}'(u),v\rangle = \int_{\Omega}\nabla u\cdot\nabla v\,dx -\int_{\Omega}\chi_{_{\{u\geqslant 0\}}} f(x,u)v\,dx, \quad \text{for all } u,v\in X, \end{equation} where $\chi_{_A}$ denotes the indicator function of $A\subseteq\Omega$, and $\{u\geqslant 0\}$ denotes the set $\{ x\in\Omega \colon u(x)\geqslant 0\}$. We will verify that the the functional $J^{+}$ given in \eqref{newfunc} satisfies the conditions of the mountain-pass theorem: \begin{itemize} \item[(1)] $J^{+}(0)=0$; \item[(2)] there exist constants $\alpha > 0$ and $\rho>0$ such that $$J^+(v) \geqslant \alpha, \quad\text{for all } v\in X \text{ with } \|v\| = \rho;$$ \item[(3)] there exists $v_1\in X$ such that $\|v_1\|>\rho$ and $J^+ (v_1)\leqslant 0$; \item[(4)] $J^+$ satisfies the Cerami condition. \end{itemize} First, observe that (1) follows from the definition of $J^+$ in \eqref{newfunc} and the definition of $\overline{F}$ in \eqref{FbarDfn05}. Next, note that by conditions (H2) and (H3) in Theorem \ref{ourthm}, we can show that \begin{gather}\label{ff1} \lim_{s\to +\infty}(s\overline{f}(x,s)-2\overline{F}(x,s)) =+\infty,\quad\text{uniformly for } x\in\Omega, \\ \label{ff2} \lim_{s\to +\infty}\frac{2\overline{F}(x,s)}{s^2} =+\infty \quad\text{uniformly for } x\in\Omega. \end{gather} Consequently, $\overline{f}$ and $\overline{F}$ satisfy the non-quadraticity condition in \eqref{ff1} and the superlinearity condition in \eqref{ff2} at infinity, respectively. Therefore, the Cerami condition can be verified for $J^+$ using the arguments in the proof of \cite[Therorem $1.1$]{FurtadoSilva}. Hence, condition (4) is verified. Next, observe that the assumption \eqref{RuEqn0045} in Theorem \ref{ourthm}, together with the definition of $\overline{F}$ in \eqref{FbarDfn05}, can be used to show that there exists $\rho > 0$ such that \begin{equation}\label{RuEqn0060} \|u\|< 2\rho \implies J^+(u) \geqslant \frac{1}{4} \|u\|^2, \end{equation} using the calculations leading to \eqref{Ru20200127} in Section \ref{secorigin}. Thus, setting $\alpha = \rho^2/4$, we obtain from \eqref{RuEqn0060} that that $J^{+}(u)\geqslant \alpha$ for $u\in\partial B_{\rho}(0)$, which shows that (2) is verified. To verify condition (3) of the mountain-pass theorem, let $\varphi_1$ be an eigenfunction of the Laplacian over $\Omega$, with Dirichlet boundary conditions, associated with the first eigenvalue, $\lambda_1$, of the Laplacian, and satisfying $\varphi_1 > 0$ and $\|\varphi_1\| = 1$. Then, using the definition of $J^+$ in \eqref{newfunc}, \begin{equation}\label{RuEqn0065} J^{+}(t\varphi_1) =\frac{t^2}{2}-\int_{\Omega}F(x,t\varphi_1)\,dx, \quad\text{for } t > 0. \end{equation} Now, by conditions \eqref{ff2}, given any $M>0$ (to be chosen shortly), there exists $R_1 > 0$ such that \begin{equation}\label{RuEqn0070} s > R_1 \implies \frac{2F(x,s)}{s^2} > M , \quad\text{for all } x\in\Omega. \end{equation} With $R_1$ dictated by our choice of $M$ (to be given shortly), define the sets \begin{equation}\label{RuEqn0075} \Omega_1^{t}=\{x\in\Omega: t\varphi(x) > R_1\} \quad\text{and} \quad\Omega_2^{t}=\Omega\backslash\Omega^t_1. \end{equation} We can then rewrite \eqref{RuEqn0065} as $$J^{+}(t\varphi_1) =\frac{t^2}{2}-\int_{\Omega_1^t}F(x,t\varphi_1)\,dx -\int_{\Omega_2^t}F(x,t\varphi_1)\,dx, \quad\text{for } t > 0,$$ or \begin{equation}\label{RuEqn0080} J^{+}(t\varphi_1) =\frac{t^2}{2} \Big(1-\int_{\Omega_1^t}\frac{2F(x,t\varphi_1)}{t^2\varphi^2}\varphi_1^2\,dx\Big) -\int_{\Omega_2^t}F(x,t\varphi_1)\,dx, \end{equation} for $t > 0$. By the definition of $\Omega_2^t$ in \eqref{RuEqn0075}, $$\Omega_2^t = \big\{ x\in\Omega \colon \varphi_1 (x) \leqslant \frac{R_1}{t} \big\} \quad\text{for } t > 0.$$ Consequently, \begin{equation}\label{RuEqn0085} \lim_{t\to\infty} |\Omega_2^t| = 0, \end{equation} where $|A|$ denotes the Lebesgue measure of a measurable subset, $A$, of $\mathbb{R}^N$. Now, it follows from condition (H1) that \begin{equation}\label{RuEqn0090} |F(x,s)|\leqslant C\left(|s|+|s|^{p}\right),\quad\text{for } (x,s)\in\Omega\times\mathbb{R}, \end{equation} for some positive constant $C$. Thus, by the Sobolev embedding theorem and the assumption that $2 M \int_{\Omega_1^t} \varphi_1^2 \ dx, \quad\text{for } t > 0, \end{equation} where $$\Omega_1^t = \Big\{ x\in\Omega \colon \varphi_1 (x) > \frac{R_1}{t} \Big\} \quad\text{for } t > 0.$$ Consequently, \begin{equation}\label{RuEqn0105} \lim_{t\to\infty}\int_{\Omega_1^t} \varphi_1^2 \ dx = \int_{\Omega} \varphi_1^2 \ dx = \frac{1}{\lambda_1}, \end{equation} since we are assuming that$\|\varphi_1\| = 1$. It follows from \eqref{RuEqn0105} that there exists$R_2 > 0$such that \begin{equation}\label{RuEqn0110} \int_{\Omega_1^t} \varphi_1^2 \ dx > \frac{2}{3\lambda_1}, \quad\text{for } t\geqslant R_2. \end{equation} Combining \eqref{RuEqn0100} and \eqref{RuEqn0110} we get \begin{equation}\label{RuEqn0115} \int_{\Omega_1^t}\frac{2F(x,t\varphi_1)}{t^2\varphi^2} \varphi_1^2\,dx > \frac{2M}{3\lambda_1}, \quad\text{for } t\geqslant R_2. \end{equation} Thus, choosing $$M = \frac{9\lambda_1}{2},$$ from \eqref{RuEqn0115} there exists$R_2 > 0$such that \begin{equation}\label{RuEqn0120} \int_{\Omega_1^t}\frac{2F(x,t\varphi_1)}{t^2\varphi^2} \varphi_1^2\,dx > 3, \quad\text{for } t\geqslant R_2. \end{equation} Using estimate \eqref{RuEqn0120} in \eqref{RuEqn0080} yields \begin{equation}\label{RuEqn0125} J^{+}(t\varphi_1) < -t^2 - \int_{\Omega_2^t}F(x,t\varphi_1)\,dx, \quad\text{for } t \geqslant R_2. \end{equation} The estimate in \eqref{RuEqn0125}, together with limit fact in \eqref{RuEqn0095}, yields that \begin{equation}\label{RuEqn0130} J^{+}(t\varphi_1)\to -\infty \quad \text{as } t\to \infty. \end{equation} To complete the verification of (3), use \eqref{RuEqn0130} to find$R_3 > \rho$such that$J^{+}(R_3\varphi_1)\leqslant 0$and set$v_1 = R_3\varphi_1$. Therefore, the conditions for the mountain-pass theorem have been verified for$J^+$. Hence,$J^{+}$has a nontrivial critical point,$u_1$, which corresponds to a weak solution of the elliptic boundary-value problem \begin{equation}\label{probpos} \begin{gathered} -\Delta u = \overline{f}(x,u), \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega. \end{gathered} \end{equation} Since we are assuming that$f$is a$C^1$function, we can apply elliptic regularity theory (see Agmon \cite{Ag}) to conclude that$u_1$is also a classical solution of \eqref{probpos}. Next, we proceed to show that$u_1 > 0$in$\Omega$. First, we show that$u_1 \geqslant 0$in$\Omega$. To see this, let$\Omega^- = \{ x\in \Omega: u_1(x) < 0\}$. Then, by the definition of$\overline{f}$in \eqref{truncf},$u_1$is a solution of the BVP \begin{equation}\label{probpos2} \begin{gathered} -\Delta v = 0, \quad \text{in } \Omega^-; \\ v = 0, \quad \text{on } \partial\Omega^-, \end{gathered} \end{equation} which has only the trivial solution$v\equiv 0$in$\Omega^-$; this assertion can be proved, for instance, by applying the maximum principle. Consequently,$\Omega^-=\emptyset$, which proves that$u_1\geqslant 0$in$\Omega$. Thus,$u_1$is a non-negative solution of the BVP \eqref{probpos}. Hence, by the definition of$\overline{f}$in \eqref{truncf},$u_1$is also a solution of the BVP \begin{equation}\label{RuEqn0135} \begin{gathered} -\Delta u = f(x,u) , \quad \text{in } \Omega; \\ u = 0, \quad \text{on } \partial\Omega. \end{gathered} \end{equation} Define \begin{equation}\label{RuEqn0140} g(x) = \begin{cases} \frac{f(x,u_1(x))}{u_1(x)}, & \text{if } u_1(x) > 0;\\[4pt] 0, & \text{if } u_1(x) = 0; \end{cases} \end{equation} so that, in view of \eqref{RuEqn0049},$g\colon\Omega\to\mathbb{R}$is a continuous function. Thus, since$u_1$is a non-negative solution of the BVP in \eqref{RuEqn0135},$u_1$is also a solution of the linear BVP \begin{equation}\label{RuEqn0145} \begin{gathered} -\Delta v = g(x) v , \quad \text{in } \Omega; \\ v = 0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where$g$is the function defined in \eqref{RuEqn0140}. Write$g(x)= g^{+}(x)-g^{-}(x)$, for$x\in\Omega$, where$g^{+}(x)= \max\{g(x),0\}$is the positive part of the function$g$defined in \eqref{RuEqn0140}, and$g^{-}(x)= \max\{-g(x),0\}$is the negative part. Then, the BVP in \eqref{RuEqn0145} can be written as \begin{equation}\label{probpos3} \begin{gathered} -\Delta v + g^{-}(x) v = g^{+}(x)v, \quad \text{ in } \Omega; \\ v = 0, \quad \text{on } \partial\Omega; \end{gathered} \end{equation} so that, since$u_1$is a non-negative solution of \eqref{probpos3},$u_1$satisfies \begin{equation}\label{RuEqn0150} \begin{gathered} -\Delta v + g^{-}(x) v \geqslant 0, \quad \text{in } \Omega; \\ v = 0, \quad \text{on } \partial\Omega; \end{gathered} \end{equation} Therefore, we can apply Hopf's maximum principle (see, for instance, \cite[Theorem 4 on page 333]{EV}), to conclude that$u_1(x)>0$, for all$x\in\Omega$, because$u_1$is nontrivial. Since, we are assuming that$\Omega$has smooth boundary, it is also the case that$ \frac{\partial u_1}{\partial \nu} < 0$, on$\partial\Omega$, where$\nu$denotes the outward unit normal vector to$\partial\Omega$(see Hopf's Lemma on page 330 in \cite{EV}). We have therefore shown that$J^+$has a critical point,$u_1$, that is given by the mountain-pass theorem and is positive in$\Omega$. We show presently that$u_1$is also a critical point of$J$. Indeed, since$u_1>0$in$\Omega$, it follows from the definition of the Fr\'echet derivative of$J^+in \eqref{DerJplus} that \begin{align*} \langle J'(u_1),v\rangle & = \int_{\Omega}\nabla u_1\cdot\nabla v\,dx - \int_{\Omega}f(x,u_1) v\,dx \\ & = \int_{\Omega}\nabla u_1\cdot\nabla v\,dx - \int_{\Omega} \chi_{_{\{u_1\geqslant 0\}}} f(x,u_1)v\,dx \\ & = \langle J^{+\prime}(u_1),v\rangle =0, \end{align*} for anyv\in X$The existence of another non-trivial critical point,$u_2$, of$J$satisfying$u_2<0$in$\Omega$can be proved by similar arguments to those presented above. This negative solution,$u_2$, is also obtained as an application of the mountain-pass theorem. Using arguments similar to those found in \cite[Theorem$A$]{ChangAl}, it can be shown that \begin{equation} C_q(J,u_1) \cong C_q(J^{+},u_1)\cong\delta_{q,1}\mathbb{F},\quad\text{for }q\in\mathbb{Z}. \label{mtps1} \end{equation} A similar result can also be obtained for the negative solution$u_2$. Next, we show the existence of a third nontrivial critical point of$J$. Assume that$J$has only three critical points:$0,u_1$, and$u_2$. We will show that this leads to a contradiction. Since$u_1$and$u_2$are of mountain-pass type, it follows from \eqref{mtps1} that the critical groups of$J$at$u_1$and$u_2$are given by \begin{equation} C_q(J,u_1)\cong C_q(J,u_2)\cong\delta_{q,1}\mathbb{F},\quad\text{for }q\in\mathbb{Z}. \label{cmtn2} \end{equation} Hence, by \eqref{coriginfinal}, \eqref{cmtn2}, and \eqref{cinftygroup}, it follows from the Morse relation \eqref{morserel}, with$t=-1$, that \begin{equation} M_0(-1)^{0}+ M_1(-1)^1 + M_2(-1)^1 = \beta_0(-1)^{0}, \label{finalrel2} \end{equation} where the Morse type numbers are given by$M_0=\dim C_0(J,0)=1,\ M_1=\dim C_1(J,u_1)=1,\ M_2=C_1(J,u_2)=1$, and, by \eqref{bettinumbers}, the Betti number$\beta_0$is$\beta_0=1$. Then, it follows from \eqref{finalrel2} that$-1=1$which is a contradiction. Hence,$J$must have a fourth critical point. 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