\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 117, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/117\hfil Subcritical perturbations] {Subcritical perturbations of resonant linear problems with sign-changing potential} \author[T.-L. Dinu\hfil EJDE-2005/117\hfilneg] {Teodora-Liliana Dinu} \address{Teodora-Liliana Dinu \hfill\break Department of Mathematics, ``Fra\c tii Buze\c sti" College, 200352 Craiova, Romania} \email{tldinu@gmail.com} \date{} \thanks{Submitted September 28, 2005. Published October 24, 2005.} \subjclass[2000]{35A15, 35J60, 35P30, 58E05} \keywords{Eigenvalue problem; semilinear elliptic equation; existence result; \hfill\break\indent critical point} \begin{abstract} We establish existence and multiplicity theorems for a Dirichlet boundary-value problem at resonance. This problem is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta, and includes a sign-changing potential. We obtain solutions using the Mountain Pass lemma and the Saddle Point theorem. Our paper extends some recent results of Gon\c calves, Miyagaki, and Ma. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction and main results} Let $\Omega$ be an arbitrary open set in $\mathbb{R}^{N}$, $N\geq 2$, and let $V:\Omega\to\mathbb{R}$ be a variable potential. Then we consider the eigenvalue problem \begin{equation} \label{eig} -\Delta u=\lambda V(x)u \quad \mbox{in } \Omega\,,\quad u\in H^1_0(\Omega). \end{equation} Problems of this type have a long history. If $\Omega$ is bounded and $V\equiv 1$, problem \eqref{eig} is related to the Riesz-Fredholm theory of self-adjoint and compact operators (see, e.g., Brezis \cite[Theorem VI.11]{b1}). The case of a non-constant potential $V$ was first considered in the pioneering papers of Bocher \cite{bo}, Hess and Kato \cite{hk}, Minakshisundaran and Pleijel \cite{mp} and Pleijel \cite{p}. Minakshisundaran and Pleijel \cite{mp}, \cite{p} studied the case where $\Omega$ is bounded, $V\in L^\infty (\Omega)$, $V\geq 0$ in $\Omega$ and $V>0$ in $\Omega_0\subset\Omega$ with $|\Omega_0|>0$. An important contribution in the study of Problem \eqref{eig} if $\Omega$ and $V$ are not necessarily bounded has been given recently by Cuesta \cite{c1} (see also Szulkin and Willem \cite{sw}) under the assumption that the sign-changing potential $V$ satisfies \begin{equation} V^+\not=0\quad \mbox{and}\quad V\in L^s(\Omega)\,,\label{eH} \end{equation} where $s>N/2$ if $N\geq 2$ and $s=1$ if $N=1$. As usual, we have denoted $V^{+}(x)=\max\{V(x),0\}$. Obviously, $V=V^+-V^-$, where $V^{-}(x)=\max\{-V(x),0\}$. To study the main properties (isolation, simplicity) of the principal eigenvalue of \eqref{eig}, Cuesta \cite{c1} proved that the minimization problem $$ \min\left\{\int_\Omega |\nabla u|^2dx;\ u\in H^1_0(\Omega) ,\ \int_\Omega V(x)u^2dx=1\right\} $$ has a positive solution $\varphi_1=\varphi_1 (\Omega)$ which is an eigenfunction of \eqref{eig} corresponding to the eigenvalue $\lambda_1:=\lambda_1(\Omega)=\int_\Omega |\nabla\varphi_1 |^2dx$. Our purpose in this paper is to study the existence of solutions of the perturbed nonlinear boundary-value problem \begin{equation}\label{et1} \begin{gathered} -\Delta u=\lambda_1V(x)u+g(x,u) \quad \mbox{in } \Omega, \\ u=0 \quad \mbox{on } \partial \Omega,\\ u\not\equiv 0\quad \mbox{in } \Omega, \end{gathered} \end{equation} where $V$ satisfies \eqref{eH} and $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'eodory function such that $g(x,0)=0$ with subcritical growth, that is, \begin{equation} |g(x,s)|\le a_{0}\cdot |s|^{r-1} +b_{0}, \quad \mbox{for all } s \in\mathbb{R}, \mbox{ a.e. } x \in\Omega, \end{equation} for some constants $a_{0}$, $b_{0}>0$, where $2\le r<2^*$. We recall that $2^*$ denotes the critical Sobolev exponent; that is, $2^*:=2N/(N-2)$ if $N\geq 3$ and $2^*=+\infty$ if $N\in\{1,2\}$. Problem \eqref{et1} is resonant at infinity and has been first studied by Landesman and Lazer \cite{ll} in connection with concrete problems arising in Mechanics. By multiplication with $\varphi_1$ in \eqref{et1} and integration over $\Omega$ we deduce that this problem has no solution if $g\not\equiv 0$ does not change sign in $\Omega$. The main purpose of this paper is to establish sufficient conditions on $g$ in order to obtain the existence of one or several solutions of the nonlinear Dirichlet problem \eqref{et1}. Set $G(x,s)=\int_{0}^{s}g(x,t)dt$. For the rest of this paper, we assume that there exist $k,\ m\in L^{1}(\Omega)$, with $m\geq 0$, such that \begin{gather} \label{et5} |G(x,s)|\le k(x),\quad\mbox{for all } s\in\mathbb{R}, \mbox{ a.e. } x\in\Omega\,\\ \label{et7} \liminf_{s\to 0}\frac{G(x,s)}{s^{2}}=m(x),\quad \mbox{a.e. } x\in\Omega\,. \end{gather} The energy functional associated to Problem \eqref{et1} is $$ F(u)=\frac 12\int_\Omega(|\nabla u|^2-\lambda_1V(x)u^2)dx -\int_\Omega G(x,u)dx\,, $$ for all $u\in H^1_0(\Omega)$. From the variational characterization of $\lambda_{1}$ and using \eqref{et5} we obtain $$ F(u)\ge-\int_{\Omega}G(x,u(x))dx\ge-|k|_{1}>-\infty\,, $$ for all $u\in H^1_0(\Omega)$ and, consequently, $F$ is bounded from below. Let us consider $u_{n}=\alpha_{n}\varphi_{1}$, where $\alpha_n\to\infty$. Then the estimate $\int_{\Omega}|\nabla\varphi_{1}|^{2}dx =\lambda_{1}\int_{\Omega}V(x)\varphi_{1}^{2}dx$ yields $F(u_{n})=-\int_{\Omega}G(x,\alpha_{n}\varphi_{1})dx\le|k|_{1}<\infty$. Thus, $\lim_{n\to\infty}F(u_{n})<\infty$. Hence the sequence $(u_{n})_{n}\subset H^1_0(\Omega)$ defined by $ u_{n}=\alpha_{n}\varphi_{1}$ satisfies $\|u_{n}\|\to\infty$ and $F(u_{n})$ is bounded. In conclusion, if we suppose that \eqref{et5} holds, then the energy functional $F$ is bounded from below and is not coercive. Our first result is the following. \begin{theorem}\label{theo1} Assume that for all $\omega\subset\Omega$ with $|\Omega\setminus\omega|>0$ we have \begin{equation}\label{et8} \int_{\omega}\limsup_{|s|\to\infty}G(x,s)dx\le 0\quad\mbox{and}\quad \int_{\Omega\setminus\omega}G(x,s)dx\le 0 \end{equation} and \begin{equation}\label{et9} \int_{\Omega}\limsup_{|s|\to\infty}G(x,s)dx\le 0\,. \end{equation} Then Problem \eqref{et1} has at lest one solution. \end{theorem} Denote $V:=\mathop{\rm Sp}(\varphi_{1})$. Since $1=\dim V<\infty$, there exists a closed complementary subspace $W$ of $V$, that is, $W\cap V=\{0\}$ and $H^1_0(\Omega)=V\oplus W$. For such a closed complementary subspace $W\subset H^1_0(\Omega)$, denote $$ \lambda_{W}:=\inf\left\{\frac{\int_{\Omega}|\nabla w|^{2}dx} {\int_{\Omega}V(x)w^{2}dx};\; w\in W,\; w\neq 0 \right\}\,. $$ The following result establishes a multiplicity result, provided $G$ satisfies a certain subquadratic condition. \begin{theorem}\label{theo2} Assume that the conditions of Theorem \ref{theo1} are fulfilled and that \begin{equation}\label{et10} G(x,s)\le\frac{\lambda_{W}-\lambda_{1}}{2}\,V(x)\,s^{2},\quad\mbox{for all } s\in\mathbb{R},\ %%@ a.e.\ x\in\Omega\,. \end{equation} Then Problem \eqref{et1} has at least two solutions. \end{theorem} In the next two theorems, we prove the existence of a solution if $V\in L^\infty (\Omega)$ and under the following assumptions on the potential $G$: \begin{gather} \limsup_{|s|\to\infty}\frac{G(x,s)}{|s|^{q}} \le b<\infty \quad\hbox{uniformly a.e. } x \in \Omega\,,\ q>2;\label{G1q}\\ \liminf_{|s|\to\infty} \frac{g(x,s)s-2G(x,s)}{|s|^{\mu}}\ge a>0\quad\hbox{uniformly a.e. } x \in \Omega;\label{G2+mu}\\ \limsup_{|s|\to\infty} \frac{g(x,s)s-2G(x,s)}{|s|^{\mu}}\le -a<0\quad\hbox{uniformly a.e. } x \in \Omega\,.\label{G2-mu} \end{gather} \begin{theorem}\label{theo3} Assume that conditions \eqref{G1q}, \eqref{G2+mu} [or \eqref{G2-mu}] and \begin{equation} \limsup_{s\to 0}\frac{2G(x,s)}{s^{2}}\le\alpha<\lambda_{1}<\beta\le\liminf _{|s|\to\infty}\frac{2G(x,s)}{s^{2}}\quad\mbox{uniformly a.e. } x\in\Omega\,,\label{G3} \end{equation} with $\mu>2N/(q-2)$ if $N\ge 3$ or $\mu>q-2$ if $1\le N\le 2$. Then Problem \eqref{et1} has at least one solution. \end{theorem} \begin{theorem}\label{theo4} Assume that \eqref{G2-mu} [or \eqref{G2+mu}] is satisfied for some $\mu>0$, and \begin{equation} \lim_{|s|\to\infty}\frac{G(x,s)}{s^{2}}=0\quad\mbox{uniformly a.e. } x\in\Omega\,.\label{G4} \end{equation} Then Problem \eqref{et1} has at least one solution. \end{theorem} The above theorems extend to the anisotropic case $V\not\equiv\,\mbox{const.}$ some results of Gon\c calves and Miyagaki \cite{gonca} and Ma \cite{tfma}. \section{Compactness conditions and auxiliary results} Let $E$ be a reflexive real Banach space with norm $\|\cdot\|$ and let $I:E\to \mathbb{R}$ be a $C^{1}$ functional. We assume that there exists a compact embedding $E\hookrightarrow X$, where $X$ is a real Banach space, and that the following interpolation type inequality holds: \begin{equation} \|u\|_{X}\le\psi (u)^{1-t}\|u\|^{t}\,, \quad\mbox{for all } u\in E\,, \label{H1} \end{equation} for some $t\in (0,1)$ and some homogeneous function $\psi :E\to\mathbb{R}_{+}$ of degree one. An example of such a framework is the following: $E=H^1_0(\Omega)$, $X=L^{q}(\Omega)$, $\psi (u)=|u|_{\mu}$, where $0<\mu 0$ and $\mu >0$ such that \begin{equation} |\langle N'(u),u\rangle-2N(u)|\ge a\psi (u)^{\mu}-c\,, \quad\mbox{for all } u\in E\,.\label{H2} \end{equation} We introduce the following additional hypotheses on the functionals $J$ and $N$: \begin{gather} J(u)\ge k\|u\|^{2}\,, \quad\mbox{for all } u\in E\label{H3}\\ |N(u)|\le b\|u\|_{X}^{q}+d\,, \quad\mbox{for all } u\in E\,,\label{H4} \end{gather} for some constants $k$, $b$, $d>0$ and $q>2$. \begin{theorem}\label{auxi} Assume that \eqref{H1}, \eqref{H2}, \eqref{H3}, \eqref{H4} are fulfilled, with $qt<2$. Then the functional $I$ satisfies condition $(\hat C)_{c}$, for all $c\in\mathbb{R}$. \end{theorem} \begin{proof} Let $(u_{n})_{n}\subset E$ such that $I(u_{n})\to c$ and $(1+\|u_{n}\|)\|I'(u_{n})\|_{E^{*}}\to 0$. We have \begin{align*} |\langle I'(u),u\rangle-2I(u)| &=|\langle J'(u)-N'(u),u\rangle-2J(u)+2N(u)|\\ &=|\langle J'(u),u\rangle-2J(u)-(\langle N'(u),u\rangle-2N(u))|\,. \end{align*} However, $J$ is $2$-homogeneous and $$ \frac{J(u+tu)-J(u)}{t}=J(u)\,\frac{(1+t)^{2}-1}{t}\,. $$ This implies $\langle J'(u),u\rangle=2J(u)$ and $$ |\langle I'(u),u\rangle-2I(u)|=|\langle N'(u),u\rangle-2N(u)|\,. $$ From \eqref{H2} we obtain $$ |\langle I'(u),u\rangle-2I(u)|=|\langle N'(u),u\rangle-2N(u)| \ge a\psi (u)^{\mu}-c\,. $$ Letting $u=u_{n}$ in the inequality from above we have: $$ a\psi(u_{n})^{\mu}\le c+\|I'(u_{n})\|_{E^{*}}\|u_{n}\|+2|I(u_{n})|\,. $$ Thus, by our hypotheses, for some $c_{0}>0$ and all positive integer $n$, $\psi(u_{n})\le c_{0}$ and hence, the sequence $\{\psi(u_n)\}$ is bounded. Now, from $(H_{1})$ and $(H_{4})$ we obtain $$ J(u_{n})=I(u_{n})+N(u_{n})\le b\|u_{n}\|_{X}^{q}+d_{0} \le b\psi(u_{n})^{(1-t)q} \|u_{n}\|^{qt}+d_{0}\,. $$ Hence $$ J(u_{n})\le b_{0}\|u_{n}\|^{qt}+d_{0}\,, \quad\mbox{for all } n\in\mathbb{N}\,, $$ for some $b_{0}$, $d_{0}>0$. Finally, $(H_{3})$ implies $$ c\|u_{n}\|^{2}\le b_{0}\|u_{n}\|^{qt}+d_{0}\,,\quad\mbox{for all } n\in\mathbb{N}\,. $$ Since $qt<2$, we conclude that $(u_{n})_{n}$ is bounded in $E$. \end{proof} \begin{proposition}\label{eqqq} Assume that $I(u)=J(u)-N(u)$ is as above, where $N':E\to E^{*}$ is a compact operator and $J':E\to E^{*}$ is an isomorphism from $E$ onto $J'(E)$. Then conditions $(C)_{c}$ and $(\hat C)_{c}$ are equivalent. \end{proposition} \begin{proof} It is sufficient to show that $(\hat C)_{c}$ implies $(C)_{c}$. Let $(u_{n})_{n}\subset E$ be a sequence such that $I(u_{n})\to c$ and $(1+\|u_{n}\|)\|I'(u_{n})\|_{E^{*}}\to 0$. From $(\hat C)_{c}$ we obtain a bounded subsequence $(u_{n_{k}})_{k}$ of $(u_{n})_{n}$. But $N'$ is a compact operator. Then $N'(u_{n_{k_{l}}})\stackrel{l}{\to}f'\in E^{*}$, where $(u_{n_{k_{l}}})$ is a subsequence of $(u_{n_{k}})$. Since $ (u_{n_{k_{l}}})$ is a bounded sequence and $(1+\|u_{n_{k_{l}}}\|)\|I'(u_{n_{k_{l}}})\|_{E^{*}}\to 0$, it follows that $ \|I'(u_{n_{k_{l}}})\|\to 0$. Next, using the relation $$ u_{n_{k_{l}}}=J^{'^{-1}}(I'(u_{n_{k_{l}}})+N'(u_{n_{k_{l}}}))\,,$$ we obtain that $(u_{n_{k_{l}}})$ is a convergent subsequence of $(u_{n})_{n}$.\end{proof} \section{Proof of Theorem \ref{theo1}} We first show that the energy functional $F$ satisfies the Palais-Smale condition at level $c<0$: any sequence $(u_{n})_{n}\subset H^1_0(\Omega)$ such that $F(u_{n})\to c$ and $\|F'(u_{n})\|_{H^{-1}}\to 0$ possesses a convergent subsequence. Indeed, it suffices to show that such a sequence $(u_{n})_{n}$ has a bounded subsequence (see the Appendix). Arguing by contradiction, we suppose that %%@ $\|u_{n}\|\to\infty$. We distinguish the following two distinct situations. \smallskip \noindent\textbf{Case 1:} $|u_{n}(x)|\to\infty$ a.e. $x\in\Omega$. Thus, by our hypotheses, \begin{align*} c& =\liminf_{n\to\infty}F(u_{n})\\ &=\liminf_{n\to\infty} \left\{\frac{1}{2}\int_{\Omega}|\nabla u_{n}|^{2}dx-\frac{\lambda_{1}}{2} \int_{\Omega}V(x)\,u_{n}^{2}dx-\int_{\Omega}G(x,u_{n}(x))dx\right\}\\ & \ge\liminf_{n\to\infty}\left(-\int_{\Omega}G(x,u_{n}(x)))dx\right)\\ &=-\limsup_{n\to\infty}\int_{\Omega}G(x,u_{n}(x))dx\\ & =-\limsup_{|s|\to\infty}\int_{\Omega}G(x,s)dx. \end{align*} Using Fatou's lemma we obtain $$ \limsup_{|s|\to\infty}\int_{\Omega}G(x,s)dx\le \int_{\Omega}\limsup_{|s|\to\infty}G(x,s)dx\,. $$ Our assumption \eqref{et9} implies $c\ge 0$. This is a contradiction because $c<0$. Therefore, $(u_{n})_{n}$ is bounded in $H^1_0(\Omega)$. \smallskip \noindent\textbf{Case 2:} There exists $\omega\subset\subset\Omega$ such that $|\Omega\setminus\omega|>0$ and $|u_{n}(x)|\not\to\infty\ \ \mbox{for all } x\in\Omega\setminus\omega$. It follows that there exists a subsequence, still denoted by $(u_{n})_{n}$, which is bounded in $\Omega\setminus\omega$. So, there exists $k>0$ such that $|u_{n}(x)|\le k$, for all $x\in\Omega\setminus\omega$. Since $I(u_{n})\to c$ we obtain some $M$ such that $I(u_{n})\le M$, for all $n$. We have $$ \frac{1}{2}\,\|u_{n}\|^{2}-\frac{\lambda_{1}}{2}\int_\Omega V(x)u_{n}^{2}dx-|k|_{1}\le I(u_{n})\le M\quad\mbox{as }\|u_{n}\|\to\infty\,. $$ It follows that $\int_\Omega V(x)u_{n}^{2}dx\to\infty$. We have $$ \int_\Omega V(x)u_{n}^{2}dx= \int_{\Omega\setminus\omega}V(x)u_{n}^{2}dx+\int_{\omega}V(x)u_{n}^{2}dx \le k^{2}|\Omega\setminus\omega|\,\| V\|_{L^1}+\int_\omega V(x)u_{n}^{2}dx\,. $$ This shows that $\int_\omega V(x)u_{n}^{2}dx\to\infty$. If $(u_{n})_{n}$ is bounded in $\omega$, this yields a contradiction. Therefore, $u_n\not\in L^\infty (\omega)$. So, by Fatou's lemma and our assumptions \eqref{et8} and \eqref{et9}, \begin{align*} c& =\liminf_{n\to\infty}F(u_{n})\\ &\ge -\limsup_{n\to\infty}\int_{\Omega}G(x,u_{n}(x))dx\\ & =-\limsup_{n\to\infty}\left(\int_{\Omega\setminus\omega}G(x,u_{n}(x))dx+ \int_{\omega}G(x,u_{n}(x))dx\right)\\ & \ge-\limsup_{n\to\infty}\int_{\Omega\setminus\omega}G(x,u_{n}(x))dx- \limsup_{n\to\infty}\int_{\omega}G(x,u_{n}(x))dx\\ & \ge-\limsup_{n\to\infty}\int_{\Omega\setminus\omega}G(x,u_{n}(x))dx- \int_{\omega}\limsup_{|s|\to\infty}G(x,s)dx\ge 0\,. \end{align*} This implies $c\ge 0$ which contradicts our hypothesis $c<0$. This contradiction shows that $(u_{n})_{n}$ is bounded in $H^1_0(\Omega)$, and hence $F$ satisfies the Palais-Smale condition at level $c<0$. The assumption \eqref{et7} is equivalent with: there exist $\delta_{n}\searrow 0$ and $\varepsilon_{n}\in L^{1}(\Omega)$ with $|\varepsilon_{n}|_{1}\to 0$ such that \begin{equation}\label{et12} \int_{\Omega}\frac{G(x,s)}{s^{2}}dx\ge\int_{\Omega}m(x)dx- \int_{\Omega}\varepsilon_{n}(x)dx\,,\quad\mbox{for all } 0<|s|\le\delta_{n}\,. \end{equation} However, $|\varepsilon_{n}|_{1}\to 0$ implies that for all $\varepsilon>0$ there exists $n_{\varepsilon}$ such that for all $n\ge n_{\varepsilon}$ we have $|\varepsilon_{n}|_{1}<\varepsilon$. Set $\varepsilon=\int_{\Omega}m(x)\varphi_{1}^{2}dx/\|\varphi_{1}\|_{L^\infty}^{2}$ and fix $n$ large enough so that $$ L:=\int_{\Omega}m(x)\varphi_{1}^{2}(x)dx- |\varepsilon_{n}|_{1}\|\varphi_{1}\|_{L^\infty}^{2}>0\,. $$ Take $v\in V$ such that $\|v\|\le\delta_{n}/\|\varphi_{1}\|_{L^{\infty}}$. We have $F(v)=-\int_{\Omega}G(x,v(x))dx$. The inequality \eqref{et12} is equivalent to $$ \int_{\Omega}G(x,s)dx\ge\int_{\Omega}m(x)s^{2}dx-\int_{\Omega} \varepsilon_{n}(x)s^{2}dx $$ and therefore, \begin{equation}\label{et13} F(v) = -\int_{\Omega}G(x,v(x))dx \le -\int_{\Omega}m(x)v^{2}(x)dx+ \int_{\Omega}\varepsilon_{n}(x)v^{2}(x)dx\,. \end{equation} By our choice of $v\in V=\mathop{\rm Sp}(\varphi_{1})$ we have $$ |v(x)|=|\alpha|\,|\varphi_1(x)|\le|\alpha|\|\varphi_{1}\|_{L^{\infty}}\le |\alpha|\frac{\delta_{n}}{\|v\|}\,. $$ However, from \eqref{et13}, \begin{align*} F(v)& \le-\int_{\Omega}mv^{2}dx+\int_{\Omega}\varepsilon_{n}v^{2}dx\le -\int_{\Omega}m|\alpha|^{2}\varphi_1^{2}dx +|\alpha|^{2}\int_{\Omega}\varepsilon_{n}| |\varphi_1\|_{L^{\infty}}^{2}dx\\ & =|\alpha|^{2}\left(-\int_{\Omega}m\varphi_1^{2}dx+ |\varepsilon_{n}|_{1}\|\varphi_1\|_{L^{\infty}}^{2}\right) =-L|\alpha|^{2}=-L\|v\|^{2}\,. \end{align*} Therefore we obtain the existence of some $v_{0}\in V$ such that $F(v_{0})<0$. This implies $l=\inf_{H^1_0(\Omega)}F<0$. But the functional $F$ satisfies the Palais-Smale condition (P-S)$_{c}$, for all $c<0$. This implies that there exists $u_{0}\in H^1_0(\Omega)$ such that $F(u_{0})=l$. In conclusion, $u_{0}$ is a critical point of $F$ and consequently it is a solution to \eqref{et1}. Our assumption $g(x,0)=0$ implies $F(0)=0$ and we know that $F(u_{0})=l<0$, that is, $u_{0}\not\equiv 0$. Therefore $u_{0}\in H^1_0(\Omega)$ is a nontrivial solution of \eqref{et1} and the proof of Theorem \ref{theo1} is complete. \section{Proof of Theorem \ref{theo2}} Let $X$ be a real Banach space and $F:X\to\mathbb{R}$ be a $C^{1}$-functional. Denote \begin{gather*} K_{c}:=\{u\in X;\ F'(u)=0\mbox{ and }F(u)=c\},\\ F^{c}:=\{u\in X;\ F(u)\le c\}\,. \end{gather*} The proof of Theorem \ref{theo2} uses the following deformation lemma (see Ramos and Rebelo \cite{r2}). \begin{lemma} \label{deformm} Suppose that $F$ has no critical values in the interval $(a,b)$ and that $F^{-1}(\{a\})$ contains at most a finite number of critical points of $F$. Assume that the Palais-Smale condition $(P-S)_{c}$ holds for every $c\in[a,b)$. Then there exists an $F$-decreasing homotopy of homeomorphism $h:[0,1]\times F^{b}\setminus K_{b}\to X$ such that \begin{gather*} h(0,u)=u\,,\quad\mbox{for all } u\in F^{b}\setminus K_{b},\\ h(1,F^{b}\setminus K_{b})\subset F^{a}, \\ h(t,u)=u\,,\quad\mbox{for all } u\in F^{a}\,. \end{gather*} \end{lemma} We are now in position to give the proof of Theorem \ref{theo2}. Fix $n$ large enough so that $$ F(v)\le-L\|v\|^{2}\,,\quad\mbox{for all } v\in V \mbox{ with }\|v\|\le\frac{\delta_{n}}{\|\varphi_1\|_{L^{\infty}}}\,. $$ Denote $d:=\sup_{\partial B}F$, where $B=\{v\in V;\ \|v\|\le R\}$ and $R=\delta_{n}/\|\varphi_1\|_{L^{\infty}}$. We suppose that 0 and $u_{0}$ are the only critical points of $F$ and we show that this yields a contradiction. For any $w\in W$ we have $$ F(w)=\frac{1}{2}\left(\int_{\Omega}|\nabla w|^{2}dx-\lambda_{1}\int_{\Omega}V(x)w^{2}dx\right) -\int_{\Omega}G(x,w(x))dx\,. $$ Integrating in \eqref{et10}, we find \begin{equation}\label{et14} -\int_{\Omega}G(x,w(x))dx\ge\frac{\lambda_{1}-\lambda_{W}}{2} \int_{\Omega}V(x)w^{2}dx\,. \end{equation} Combining the definition of $\lambda_{W}$ with relation \eqref{et14} we obtain \begin{equation}\label{et15} \begin{aligned} F(w)& \ge\frac{1}{2}\int_{\Omega}|\nabla w|^{2}dx-\frac{\lambda_{1}}{2}\int_{\Omega}V(x)w^{2}dx+ \frac{\lambda_{1}-\lambda_{W}}{2}\int_{\Omega}V(x)w^{2}dx\\ & =\frac{1}{2}\left(\int_{\Omega}|\nabla w|^{2}dx-\lambda_{W}\int_{\Omega}V(x)w^{2}dx\right)\ge 0\,. \end{aligned} \end{equation} Using $0\in W$, $F(0)=0$ and relation \eqref{et15} we find $\inf_{W}F=0$. If $v\in\partial B$ then $F(v)\le-LR<0$ and, consequently, $$ d=\sup_{\partial B}F<\inf_{W}F=0\,. $$ Obviously, $$ l=\inf_{H^1_0(\Omega)}F\le\inf_{\partial B}Ft\quad\mbox{implies}\quad F(h(s,u))From these two cases we obtain $F(\gamma_{0}(v))\le d$, for all $v\in B$ and from the definition of $\alpha$ we have $0\le\alpha\le d<0$. This is a contradiction. We conclude that $F$ has a another critical point $u_{1}\in H^1_0(\Omega)$ and, consequently, Problem \eqref{et1} has a second nontrivial weak solution. \section{Proof of Theorems \ref{theo3} and \ref{theo4}} We will use the following classical critical point theorems. \begin{theorem}[Mountain Pass, Ambrosetti and Rabinowitz \cite{a1}] Let $E$ be a real Banach space. Suppose that $I\in C^{1}(E,\mathbb{R})$ satisfies condition $(C)_{c}$, for all $c\in\mathbb{R}$ and, for some $\rho>0$ and $u_{1}\in E$ with $\|u_{1}\|>\rho$, $$ \max\{I(0),I(u_{1})\}\le\hat\alpha<\hat\beta\le\inf_{\|u\|=\rho}I(u)\,. $$ Then $I$ has a critical value $\hat c\ge\hat\beta$, characterized by $$\hat c=\inf_{\gamma\in\Gamma}\max_{0\le\tau\le1}I(\gamma(\tau))\,,$$ where $\Gamma :=\{\gamma\in C([0,1],E);\ \gamma(0)=0,\gamma(1)=u_{1}\}$. \end{theorem} \begin{theorem}[Saddle Point, Rabinowitz \cite{r1}] Let $E$ be a real Banach space. Suppose that $I \in C^{1}(E,\mathbb{R})$ satisfies condition $(C)_{c}$, for all $c\in\mathbb{R}$ and, for some $R>0$ and some $E=V\oplus W$ with $\dim V<\infty$, $$ \max_{v\in V,\|v\|=R}I(v)\le\hat\alpha<\hat\beta\le \inf_{w\in W}I(w)\,. $$ Then $I$ has a critical value $\hat c\ge\hat\beta$, characterized by $$ \hat c=\inf_{h\in\Gamma}\max_{v\in V,\|v\|\le R}I(h(v))\,, $$ where $\Gamma=\{h\in C(V\bigcap\bar B_{R},E);\ h(v)=v,\,\mbox{for all } v\in\partial B_{R}\}$. \end{theorem} \begin{lemma}\label{lema41} Assume that $G$ satisfies conditions \eqref{G1q} and \eqref{G2+mu} [or \eqref{G2-mu}], with $\mu >2N/(q-2)$ if $N\geq 3$ or $\mu > q-2$ if $1 \le N \le 2$. Then the functional $F$ satisfies condition $(C)_{c}$ for all $c\in\mathbb{R}$. \end{lemma} \begin{proof} Let $$N(u)=\frac{\lambda_{1}}{2}\int_{\Omega}V(x)u^{2}dx+\int_{\Omega}G(x,u)dx \quad \mbox{and}\quad J(u)=\frac{1}{2}\|u\|^{2}\,. $$ Obviously, $J$ is homogeneous of degree $2$ and $J'$ is an isomorphism of $E=H^1_0(\Omega)$ onto $ J'(E)\subset H^{-1}(\Omega)$. It is known that $N':E\to E^{*}$ is a compact operator. Proposition \ref{eqqq} ensures that conditions $(C)_{c}$ and $(\hat C)_{c}$ are equivalent. So, it suffices to show that $(\hat C)_{c}$ holds for all $c\in\mathbb{R}$. Hypothesis \eqref{H3} is trivially satisfied, whereas \eqref{H4} holds true from \eqref{G1q}. Condition \eqref{G1q} implies that $$ \inf_{|s|>0}\sup_{|t|>|s|}\frac{G(x,t)}{|t|^{q}}\le b\,. $$ Therefore, there exists $s_{0}\not=0$ such that $$ \sup_{|t|>|s_{0}|}\frac{G(x,t)}{|t|^{q}}\le b\quad\mbox{and}\quad G(x,t)\le b|t|^{q},\quad\mbox{for all } t\mbox{ with }|t|>|s_{0}|\,. $$ The boundedness is provided by the continuity of the application $[-s_{0},s_{0}]\ni t\longmapsto G(x,t)$. It follows that $\int_{\Omega}G(x,u)dx\le b|u|_{q}^{q}+d$. By the definition of $N(u)$ and since $q>2$, we deduce that \eqref{H4} holds true, provided $|u|_{q}\le 1$ then we obtain \eqref{H4}. Indeed, we have $|u|_{2}\le k|u|_{q}$ because $\Omega$ is bounded. Therefore, $|u|_{2}^{2}\le k|u|_{q}^{2}\le k|u|_{q}^{q}$ and finally \eqref{H4} is fulfilled. Hypothesis \eqref{H1} is a direct consequence of the Sobolev inequality. It remains to show that hypothesis \eqref{H2} holds true, that is, the functional $N$ is not $2$-homogeneous at infinity. Indeed, using assumption \eqref{G2+mu} (a similar argument works if \eqref{G2-mu} is %%@ fulfilled) together with the subcritical condition on $g$ yields $$ \sup_{|s|>0}\inf_{|t|>|s|}\frac{g(x,t)t-2G(x,t)}{|t|^{\mu}}\ge a>0\,. $$ It follows that there exists $s_{0}\not=0$ such that $$ \inf_{|t|>|s_{0}|} \frac{g(x,t)t-2G(x,t)}{|t|^{\mu}}\ge a\,. $$ Hence $$ g(x,t)t-2G(x,t)\ge a|t|^{\mu}\,,\quad\mbox{for all }|t|>|s_{0}|\,. $$ The application $t\mapsto g(x,t)t-2G(x,t)$ is continuous in $[-s_{0},s_{0}]$, therefore it is bounded. We obtain $g(x,t)-2G(x,t)\ge a_{1}|t|^{\mu}-c_{1}$, for all $s\in\mathbb{R}$ and a.e. %%@ $x\in\Omega$. We deduce that \begin{align*} |\langle N'(u),u\rangle-2N(u)| &=\left|\int_{\Omega}(g(x,u)u-2G(x,u))dx\right|\\ & \ge a_{1}\|u\|_{\mu}^{\mu}-c_{2}\,, \quad\mbox{for all } u\in H^1_0(\Omega)\,. \end{align*} Consequently, the functional $N$ is not $2$-homogeneous at infinity. Finally, when $N\ge 3$, we observe that condition $\mu>N(q-2)/2$ is equivalent with $\mu>2^{*}(q-2)/{2^{*}-2}$. From $1/q=(1-t)/\mu+t/2^{*}$ we obtain $(1-t)/\mu=(2^{*}-qt)/(2^{*}q)$. Hence $(2^{*}-qt)/q<(1-t)(2^{*}-2)/(q-2)$ and, consequently, $(q-2^{*})(2-tq)<0$. But $q<2^{*}$ and this implies $2>tq$. Similarly, when $1\le N\le 2$, we choose some $2^{**}>2$ sufficiently large so that $\mu>2^{**}(q-2)/(2^{**}-2)$ and $t\in(0,1)$ be as above. The proof of Lemma is complete in view of Theorem \ref{auxi}. \end{proof} Our next step is to show that condition \eqref{G3} implies the geometry of the Mountain Pass theorem for the functional $F$. The below assumptions have been introduced in %%@ Cuesta and Silva \cite{costad}. \begin{lemma}\label{dinnoulema} Assume that $G$ satisfies the hypotheses \begin{gather} \limsup_{|s|\to\infty}\frac{G(x,s)}{|s|^{q}}\le b<\infty \quad\mbox{uniformly }a.e.\ x \in \Omega\,,\label{G1qq}\\ \limsup_{s\to 0}\frac{2G(x,s)}{s^{2}}\le\alpha<\lambda_{1}<\beta\le\liminf _{|s|\to\infty}\frac{2G(x,s)}{|s|^{2}}\quad\mbox{uniformly a.e. } x\in\Omega\,.\label{G33} \end{gather} Then there exists $\rho$, $\gamma>0$ such that $F(u)\ge\gamma$ if $|u|=\rho$. Moreover, there exists $\varphi_{1}\in H^1_0(\Omega)$ such that $F(t\varphi_{1})\to -\infty$ as $t\to\infty$. \end{lemma} \begin{proof} In view of our hypotheses and the subcritical growth condition, we obtain $$ \liminf_{|s|\to\infty}\frac{2G(x,s)}{s^{2}}\ge\beta\quad \mbox{is equivalent to} \quad \sup_{s\not=0}\inf_{|t|>|s|}\frac{2G(x,t)}{t^{2}}\ge\beta\,. $$ There exists $s_{0}\not=0$ such that $\inf_{|t|>|s_{0}|}\frac{2G(x,t)}{t^{2}}\ge\beta$ and therefore $\frac{2G(x,t)}{t^{2}}\ge\beta$, for all $|t|>|s_{0}|$ or $G(x,t)\ge\frac{1}{2}\beta t^{2}$, provided $|t|>|s_{0}|$. We choose $t_{0}$ such that $|t_{0}|\le|s_{0}|$ and $G(x,t_{0})<\frac{1}{2}\beta|t_{0}|^{2}$. Fix $\varepsilon>0$. There exists $B(\varepsilon,t_{0})$ such that $G(x,t_{0})\ge\frac{1}{2}(\beta-\varepsilon)|t_{0}|^{2}-B(\varepsilon,t_{0})$. Denote $B(\varepsilon)=\sup_{|t_{0}|\le|s_{0}|}B(\varepsilon,t_{0})$. We obtain for any given $\varepsilon>0$ there exists $B=B(\varepsilon)$ such that \begin{equation} G(x,s)\ge\frac{1}{2}\,(\beta-\varepsilon)|s|^{2}-B\,, \quad\mbox{for all } s\in\mathbb{R}\,,\mbox{a.e. } x\in\Omega\,. \end{equation} Fix arbitrarily $\varepsilon>0$. In the same way, using the second inequality of \eqref{G33} and \eqref{G1qq} it follows that there exists $A=A(\varepsilon)>0$ such that \begin{equation}\label{et17} 2G(x,t)\le(\alpha+\varepsilon)t^{2}+2(b+A(\varepsilon))|t|^{q}\,,\quad \mbox{for all } t\in\mathbb{R}\,, \mbox{ a.e. } x\in\Omega\,. \end{equation} We now choose $\varepsilon>0$ so that $\alpha+\varepsilon<\lambda_{1}$ and we use \eqref{et17} together with the Poincar\'e inequality to obtain the first assertion of the lemma. Set $H(x,s)=\lambda_{1}V(x)s^{2}/2+G(x,s)$. Then $H$ satisfies \begin{gather} \limsup_{|s|\to\infty}\frac{H(x,s)}{|s|^{q}}\le b<\infty\,,\quad \mbox{uniformly a.e. } x\in\Omega\,, \label{H1q}\\ \limsup_{s\to 0}\frac{2H(x,s)}{s^{2}}\le\alpha<\lambda_{1} <\beta\le\liminf_{|s|\to\infty}\frac{2H(x,s)}{s^{2}}\,,\quad \mbox{uniformly a.e. } x \in \Omega\,.\label{H33} \end{gather} In the same way, for any given $\varepsilon>0$ there exists $A=A(\varepsilon)>0$ and $B=B(\varepsilon)$ such that \begin{equation}\label{et18} \frac{1}{2}(\beta-\varepsilon)s^{2}-B \le H(x,s) \le\frac{1}{2}(\alpha+\varepsilon)s^{2}+A|s|^{q}\,, \end{equation} for all $s\in\mathbb{R}$, a.e. $x\in\Omega$. Then we have \begin{align*} F(u)&=\frac{1}{2}\|u\|^{2}-\int_{\Omega}H(x,u)dx\\ &\ge\frac{1}{2}\|u\|^{2} -\frac{1}{2}(\alpha+\varepsilon)|u|_{2}^{2}-A|u|_{q}^{q}\\ &\ge\frac{1}{2}\left(1-\frac{\varepsilon+\alpha}{\lambda_{1}}\right) \|u\|^{2}-Ak\|u\|^{q}\,. \end{align*} We can assume without loss of generality that $q>2$. Thus, the above estimate yields $F(u)\ge\gamma$ for some $\gamma>0$, as long as $\rho>0$ is small, thus proving the first assertion of the lemma. On the other hand, choosing now $\varepsilon>0$ so that $\beta-\varepsilon>\lambda_{1}$ and using \eqref{et18}, we obtain $$ F(u)\le\frac{1}{2}\|u\|^{2}-\frac{\beta-\varepsilon}{2}\,|u|_{2}^{2} +B|\Omega|\,. $$ We consider $\varphi_{1}$ be the $\lambda_{1}$-eigenfunction with $\|\varphi_{1}\|=1$. It follows that $$ F(t\varphi_{1})\le\frac{1}{2}\left(1-\frac{\beta-\varepsilon}{\lambda_{1}}\right) t^{2}+B|\Omega|\to -\infty\quad \mbox{as }t\to\infty. $$ This proves the second assertion of our lemma. \end{proof} \begin{lemma}\label{betty} Assume that $G(x,s)$ satisfies \eqref{G2-mu} (for some $\mu>0$) and \begin{equation} \label{G44} \lim_{|s|\to\infty}\frac{G(x,s)}{s^{2}}=0\,,\quad \mbox{uniformly a.e. }\ x\in\Omega\,. \end{equation} Then there exists a subspace $W$ of $H^1_0(\Omega)$ such that $H^1_0(\Omega)=V\oplus W$ and \begin{itemize} \item[(i)] $F(v)\to -\infty$, as $\|v\|\to\infty$, $v\in V$ \item[(ii)] $F(w)\to\infty$, as $\|w\|\to\infty$, $w\in W$. \end{itemize} \end{lemma} \begin{proof} (i) The condition \eqref{G2-mu} is equivalent to: There exists $s_{0}\neq 0$ such that $$ g(x,s)s-2G(x,s)\le -a|s|^{\mu}\,,\quad\mbox{for all } |s|\ge|s_{0}|= R_{1}\,,\ \mbox{a.e.}\ x\in\Omega\,. $$ Integrating the identity $$ \frac{d}{ds}\frac{G(x,s)}{|s|^{2}}=\frac{g(x,s)s^{2}-2|s|G(x,s)} {s^{4}}=\frac{g(x,s)|s|-2G(x,s)}{|s|^{3}} $$ over an interval $[t,T]\subset[R,\infty)$ and using the above inequality we find $$ \frac{G(x,T)}{T^{2}}-\frac{G(x,t)}{t^{2}}\le -a\int_{t}^{T}s^{\mu-3}ds= \frac{a}{2-\mu}\left(\frac{1}{T^{2-\mu}}-\frac{1}{t^{2-\mu}}\right)\,. $$ Since we can assume that $\mu<2$ and using the above relation, we obtain $G(x,t)\ge\hat a t^{\mu}$ for all $t\ge R_{1}$, where $\hat a=\frac{a}{2-\mu}>0$. Similarly, we show that $$ G(x,t)\ge\hat a|t|^{\mu}\,,\quad\mbox{for }|t|\ge R_{1}\,. $$ Consequently, $\lim_{|t|\to\infty}G(x,t) = \infty$. Now, letting $v=t\varphi_{1}\in V$ and using the variational characterization of $\lambda_{1}$, we have $$ F(v)\ge-\int_{\Omega}G(x,v)dx\to -\infty\,,\quad\mbox{as } \|v\|=|t|\|\varphi_{1}\|\to\infty\,. $$ This result is a consequence of the Lebesgue's dominated convergence theorem. \noindent(ii) Let $V=\mathop{\rm Sp}(\varphi_{1})$ and $W \subset H^1_0(\Omega)$ be a closed complementary subspace to $V$. Since $\lambda_{1}$ is an eigenvalue of Problem \eqref{eig}, it follows that there exists $d>0$ such that $$ \inf_{0\not=w\in W} \frac{\int_{\Omega}|\nabla w|^{2}dx}{\int_{\Omega}V(x)w^{2}dx} \ge\lambda_{1}+d\,. $$ Therefore, $$ \|w\|^{2}\ge(\lambda_{1}+d)|w|_{2}^{2}\,,\quad\mbox{for all } w\in W\,. $$ Let $0 < \varepsilon < d$. From $(G_{4})$ we deduce that there exists $\delta = \delta(\varepsilon)>0$ such that for all $s$ satisfying $|s| >\delta$ we have $2G(x,s)/s^{2}\le\varepsilon$, a.e. $x\in\Omega$. In conclusion $$ G(x,s)-\frac{1}{2}\varepsilon s^{2}\le M\,,\quad\mbox{for all } s\in\mathbb{R}\,, $$ where $$M:=\sup_{|s|\le\delta}\left(G(x,s)-\frac{1}{2}\,\varepsilon\, s^{2} \right)<\infty\,. $$ Therefore, \begin{align*} F(w)& =\frac{1}{2}\|w\|^{2}- \frac{\lambda_{1}}{2}\int_\Omega V(x)w^{2}-\int_{\Omega}G(x,w)dx\\ & \ge\frac{1}{2}\|w\|^{2}-\frac{\lambda_{1}}{2}|w|_{2}^{2}- \frac{1}{2}\varepsilon|w|_{2}^{2}-M\\ & \ge\frac{1}{2}\left(1-\frac{\lambda_{1}+\varepsilon}{\lambda_{1}+d}\right) \|w\|^{2}-M=N\|w\|^{2} -M\,,\quad\mbox{for all } w\in W\,. \end{align*} It follows that $F(w)\to\infty$ as $\|w\|\to\infty$, for all $w\in W$, which completes the proof of the lemma. \end{proof} \begin{proof}[Proof of Theorem \ref{theo3}] In view of Lemmas \ref{lema41} and \ref{dinnoulema}, we may apply the Mountain Pass theorem with $u_{1}=t_{1}\varphi_{1}$, $t_{1}>0$ being such that $F(t_{1}\varphi_{1})\le 0$ (this is possible from Lemma \ref{dinnoulema}). Since $F(u)\ge\gamma$ if $\|u\|=\rho$, we have $$ \max\{F(0),F(u_{1})\}=0=\hat\alpha<\inf_{\|u\|=\rho} F(u)=\hat\beta\,. $$ It follows that the energy functional $F$ has a critical value $\hat c\ge\hat\beta>0$ and, hence, \eqref{et1} has a nontrivial solution $u\in H^1_0(\Omega)$. \end{proof} \begin{proof}[Proof of Theorem \ref{theo4}] In view of Lemmas \ref{lema41} and \ref{betty}, we may apply the Saddle Point theorem with $\hat\beta:=\inf_{w\in W}F(w)$ and $R>0$ being such that $\sup_{\|v\|=R}F(v):=\hat\alpha<\hat\beta$, for all $v\in V$ (this is possible because $F(v)\to -\infty$ as $\|v\|\to\infty$). It follows that $F$ has a critical value $\hat c\ge\hat\beta$, which is a weak solution to \eqref{et1}. \end{proof} \section{Appendix} Throughout this section we assume that $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth boundary. We start with the following auxiliary result. \begin{lemma}\label{brezislema} Let $g:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carath\'eodory function and assume that there exist some constants $a$, $b\ge 0$ such that $$ |g(x,t)|\le a+b|t|^{r/s}\,,\quad\mbox{for all } t\in\mathbb{R}\,,\ \mbox{a.e.} x\in\Omega\,. $$ Then the application $\varphi(x)\mapsto g(x,\varphi(x))$ is in $C(L^{r}(\Omega),L^{s}(\Omega))$. \end{lemma} \begin{proof} For any $u\in L^{r}(\Omega)$ we have \begin{align*} \int_{\Omega}|g(x,u(x))|^{s}dx &\le\int_{\Omega}(a+b|u|^{r/s})^{s}dx\\ &\le 2^{s}\int_{\Omega}(a^{s}+b^{s}|u|^{r})dx\\ &\le c\int_{\Omega}(1+|u|^{r})dx<\infty \,. \end{align*} This shows that if $\varphi\in L^{r}(\Omega)$ then $g(x,\varphi)\in L^{s}(\Omega)$. Let $u_{n}$, $u\in L^{r}$ be such that $|u_{n}-u|_{r}\to 0$. By Theorem IV.9 in Brezis \cite{b1}, there exist a subsequence $(u_{n_{k}})_{k}$ and $h\in L^{r}$ such that $ u_{n_{k}}\to u$ a.e. in $\Omega$ and $|u_{n_{k}}|\le h$ a.e. in $\Omega$. By our hypotheses it follows that $ g(u_{n_{k}})\to g(u)$ a.e. in $\Omega$. Next, we observe that $$ |g(u_{n_{k}})|\le a+b|u_{n_{k}}|^{r/s}\le a+b|h|^{r/s}\in L^{s}(\Omega)\,. $$ So, by Lebesgue's dominated convergence theorem, $$ |g(u_{n_{k}})-g(u)|_{s}^{s}=\int_{\Omega}|g(u_{n_{k}})-g(u)|^{s}dx \stackrel{k}{\to} 0\,. $$ This completes the proof of the lemma. \end{proof} The mapping $\varphi\mapsto g(x,\varphi(x))$ is the Nemitski operator of the function $g$. \begin{proposition} Let $g:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carath\'eodory function such that $|g(x,s)|\le a+b|s|^{r-1}$ for all $(x,s)\in\Omega\times\mathbb{R}$, with $2\le r<2N/(N-2)$ if $N> 2$ or $2\le r<\infty$ if $1\leq N\leq 2$. Denote $G(x,t)=\int_0^tg(x,s)ds$. Let $I:H^1_0(\Omega)\to\mathbb{R}$ be the functional defined by $$ I(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^{2}dx- \frac{\lambda_{1}}{2}\int_{\Omega}V(x)u^{2}dx-\int_{\Omega}G(x,u(x))dx\,, $$ where $V\in L^s(\Omega)$ ($s>N/2$ if $N\geq 2$, $s=1$ if $N=1$). Assume that $(u_{n})_{n}\subset H^1_0(\Omega)$ has a bounded subsequence and $I'(u_{n})\to 0$ as $n\to\infty$. Then $(u_{n})_{n}$ has a convergent subsequence. \end{proposition} \begin{proof} We have $$ \langle I'(u),v\rangle=\int_{\Omega}\nabla u\nabla vdx- \lambda_{1}\int_{\Omega}V(x)uvdx-\int_{\Omega}g(x,u(x))v(x)dx\,. $$ Denote by \begin{gather*} \langle a(u),v\rangle=\int_{\Omega}\nabla u\nabla vdx\,;\\ J(u)=\frac{\lambda_{1}}{2}\int_{\Omega}V(x)u^{2}dx+\int_{\Omega}G(x,u(x))dx\,. \end{gather*} It follows that $$ \langle J'(u),v\rangle=\lambda_{1}\int_{\Omega}V(x)uvdx+\int_{\Omega}g(x,u(x))v(x)dx $$ and $ I'(u)=a(u)-J'(u)$. We prove that $a$ is an isomorphism from $H^1_0(\Omega)$ onto $a(H^1_0(\Omega))$ and $J'$ is a compact operator. This assumption yields $$ u_{n}=a^{-1}\langle (I'(u_{n}\rangle)+J'(u_{n}))\to\lim_{n\to\infty} a^{-1}\langle(J'(u_{n}) \rangle)\,. $$ But $J'$ is a compact operator and $(u_{n})_{n}$ is a bounded sequence. This implies that $(J'(u_{n}))_{n}$ has a convergent subsequence and, consequently, $(u_{n})_{n}$ has a convergent subsequence. Assume, up to a subsequence, that $(u_{n})_{n}\subset H^1_0(\Omega)$ is bounded. From the compact embedding $H^1_0(\Omega)\hookrightarrow L^{r}(\Omega)$, we can assume, passing again at a subsequence, that $u_{n}\to u$ in $L^{r}(\Omega)$. We have $$ \begin{aligned} & \|J'(u_{n})-J'(u)\|\\ & \le \sup_{\|v\|\le 1} \left|\int_{\Omega}\left(g(x,u_{n}(x))-g(x,u(x))\right)v(x)dx\right| +\sup_{\|v\|\le 1}\lambda_{1}\left|\int_{\Omega}V(x)(u_{n}-u)vdx\right|\\ & \le\sup_{\|v\|\le 1}\int_{\Omega}|g(x,u_{n}(x))- g(x,u(x))||v(x)|dx +\lambda_{1}\sup_{\|v\|\le 1} \int_{\Omega}|V(x)(u_{n}-u)v|dx\\ & \leq\sup_{\|v\|\le 1}\left(\int_{\Omega}|g(x,u_{n})-g(x,u)|^{\frac{r}{r-1}}dx\right) ^{\frac{r-1}{r}}|v|_{r} +\lambda_{1}\sup_{\|v\|\le 1} \int_{\Omega}|V(x)(u_{n}-u)v|dx\\ & \leq c\sup_{\|v\|\le 1} \left(\int_{\Omega}|g(x,u_{n})-g(x,u)|^{\frac{r}{r-1}}dx\right) ^{\frac{r-1}{r}}\|v\| +\lambda_{1}|V|_{L^s}\cdot |u_{n}-u|_{\alpha}\cdot |v|_{\beta}\,, \end{aligned} $$ where $\alpha$, $\beta<2N/(N-2)$ (if $N\geq 2$). Such a choice of $\alpha$ and $\beta$ is possible due to our choice of $s$. By Lemma \ref{brezislema} we obtain $g\in C(L^{r},L^{r/(r-1)})$. Next, since $u_{n}\to u$ in $L^{r}$ and $u_{n}\to u$ in $L^{2}$, the above relation implies that $J'(u_{n})\to J'(u)$ as $n\to\infty$, that is, $J'$ is a compact operator. This completes our proof. \end{proof} Set $$ \Gamma :=\{\gamma\in C(B,H^1_0(\Omega));\ \gamma(v)=v\,,\ \mbox{for all } v\in\partial B\} $$ and $B=\{v\in {\rm Sp}\, (\varphi_1);\ \|v\|\le R\}$. The following result has been used in the proof of Lemma \ref{deformm}. \begin{proposition} We have $\gamma(B)\bigcap W\not=\emptyset$, for all $\gamma\in\Gamma$. \end{proposition} \begin{proof} Let $P:H^1_0(\Omega)\to {\rm Sp}\, (\varphi_1)$ be the projection of $H^1_0$ in ${\rm Sp}\, (\varphi_1)$. Then $P$ is a linear and continuous operator. If $v\in\partial B$ then $(P\circ\gamma)(v)=P(\gamma(v))=P(v)=v$ and, consequently, $P\circ\gamma=Id$ on $\partial B$. We have $P\circ\gamma\ ,Id\in C(B,H^1_0)$ and $0\not\in Id(\partial B)=\partial B$. Using a property of the Brouwer topological degree we obtain $\mbox{deg}\,(P\circ\gamma,{\rm Int} B,0)=\mbox{deg}\,(Id,{\rm Int}B,0)$. But $0\in{\rm Int}\,B$ and it follows that $\mbox{deg}\,(Id,{\rm Int}\,B,0)=1\not=0$. So, by the existence property of the Brouwer degree, there exists $v\in{\rm Int}\,B$ such that $(P\circ\gamma)(v)=0$, that is, $P(\gamma(v))=0$. Therefore $\gamma(v)\in W$ and this shows that $\gamma(B)\cap W\not=\emptyset$. \end{proof} \begin{thebibliography}{99} \bibitem{a1} A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical theory and application, {\it J.~Funct. Anal.} {\bf 14} (1973), 349-381. \bibitem{bo} M. Bocher, The smallest characteristic numbers in a certain exceptional case, {\it Bull. Amer. Math. Soc.} {\bf 21} (1914), 6--9. \bibitem{b1} H. Brezis, {\it Analyse fonctionelle: th\'eorie et applications}, Masson, Paris, 1987. \bibitem{costad} D. Costa and E. A. de B. e Silva, Existence of solutions for a class of resonant elliptic problems, {\it J.~Math. Anal. Appl.} {\bf 175} (1993), 411--424. \bibitem{c1} M. Cuesta, Eigenvalue problems for the $p$-Laplacian with indefinite weights, {\it Electron. J.~Differential Equations} 2001, No. 33, 9 pp. (electronic). \bibitem{gonca} J. V. A. Gon\c calves and O. H. Miyagaki, Multiple nontrivial solutions of semilinear strongly resonant elliptic equations, {\it Nonlin. Anal. T.M.A.} {\bf 19} (1992), 43--52. \bibitem{hk} P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with indefinite weight function, {\it Comm. Partial Differential Equations} {\bf 5} (1980), 999--1030. \bibitem{ll} E. A. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary-value problems at resonance, {\it J.~Math. Mech.} {\bf 19} (1976), 609--623. \bibitem{tfma} T. F. Ma, A note on the existence of two nontrivial solutions of a resonance problem, {\it Portugal. Math.} {\bf 51} (1994), 517--523. \bibitem{mp} S. Minakshisundaran and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemann manifolds, {\it Canad. J.~Math.} {\bf 1} (1949), 242--256. \bibitem{p} A. Pleijel, On the eigenvalues and eigenfunctions of elastic plates, {\it Comm. Pure Appl. Math.} {\bf 3} (1950), 1--10. \bibitem{r1} P. Rabinowitz, {\it Minimax Methods in Critical Point Theory with Applications to Differential Equations}, CBMS Regional Conf. Ser.in Math., Amer. Math. Soc., Providence, RI, vol. 65, 1986. \bibitem{r2} M. Ramos and C. Rebelo, A unified approach to min-max critical points theorems, {\it Portugal. Math.} {\bf 51} (1994), 489--516. \bibitem{sw} A. Szulkin and M. Willem, Eigenvalue problems with indefinite weight, {\it Studia Math.} {\bf 135} (1999), 191--201. \end{thebibliography} \end{document}