\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 124, pp. 1--22.\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 2003 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2003/124\hfil Resonance and Strong Resonance] {Resonance and strong resonance for semilinear elliptic equations in $\mathbb{R}^N$} \author[Gabriel L\'{o}pez G. \& Adolfo J. Rumbos\hfil EJDE-2003/124\hfilneg] {Gabriel L\'{o}pez Garza \& Adolfo J. Rumbos} % in alphabetical order \address{Gabriel L\'{o}pez Garza \hfill\break Dept. of Math., Claremont Graduate University\\ Claremont California 91711, USA} \email{Gabriel.Lopez@cgu.edu} \address{Adolfo J. Rumbos \hfill\break Department of Mathematics, Pomona College\\ Claremont, California 91711, USA} \email{arumbos@pomona.edu} \date{} \thanks{Submitted June 3, 2003. Published December 16, 2003.} \thanks{G. L\'{o}pez was supported by CONACYT M\'{e}xico} \subjclass[2000]{35J20} \keywords{Resonance, strong resonance, concentration-compactness} \begin{abstract} We prove the existence of weak solutions for the semilinear elliptic problem $$ -\Delta u=\lambda hu+ag(u)+f,\quad u\in \mathcal{D}^{1,2}({\mathbb{R}^N}), $$ where $\lambda \in \mathbb{R}$, $f\in L^{2N/(N+2)}$, $g:\mathbb{R} \to \mathbb{R}$ is a continuous bounded function, and $h \in L^{N/2}\cap L^{\alpha}$, $\alpha>N/2$. We assume that $a \in L^{2N/(N+2)}\cap L^{\infty}$ in the case of resonance and that $a \in L^1 \cap L^{\infty}$ and $f\equiv 0$ for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem using the concentration-compactness lemma of Lions. Then we prove the existence of weak solutions by applying the saddle point theorem of Rabinowitz for the cases of non-resonance and resonance, and a linking theorem of Silva in the case of strong resonance. The main theorems in this paper constitute an extension to $\mathbb{R}^N$ of previous results in bounded domains by Ahmad, Lazer, and Paul \cite{ALP}, for the case of resonance, and by Silva \cite{EL} in the strong resonance case. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lema}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \newtheorem{coro}[theorem]{Corollary} \section{Introduction} Let $\mathcal{D}^{1,2}$ be the completion of $C^{\infty}_{c}({\mathbb{R}^N})$ with respect to the norm $$ \|u\|=\Big(\int |\nabla u|^2\Big)^{1/2}. $$ It is known that $\mathcal{D}^{1,2}$ is a Hilbert space with inner product $\langle u,v\rangle =\int \nabla u\cdot \nabla v$. It is also known that $\mathcal{D}^{1,2}$ is embedded in $L^{2^*}(\mathbb{R}^N)$ (cf. \cite{B}). In fact, \begin{equation} |u|_{L^{2^*}}^{2^*}\leqslant C^*\|u\|^2 , \label{embed} \end{equation} where $2^*=2N/(N-2)$ and $C^*$ is a constant depending on $N$. In this paper we study the existence of solutions to the boundary--value problem \begin{equation} \label{Plf} \begin{gathered} -\Delta u = \lambda h(x) u + a(x)g(u)+f(x),\quad x\in{\mathbb{R}^N},\\ u\in\mathcal{D}^{1,2}, \end{gathered} \end{equation} where $\lambda \in {\mathbb{R}}$, $f\in L^{2N/(N+2)}$, $g:\mathbb{R} \to \mathbb{R}$ is a continuous, bounded function, $h \in L^{N/2}\cap L^{\alpha}$, for $\alpha>N/2$, and $a \in L^{\infty}$. \subsection*{Definition} For a bounded nonlinearity $g$, problem \eqref{Plf} is said to be at resonance if $\lambda$ is an eigenvalue of the boundary-value problem \begin{equation} \begin{gathered} -\Delta u=\lambda h(x)u,\quad x\in{\mathbb{R}^N},\\ u\in \mathcal{D}^{1,2}. \end{gathered}\label{a1} \end{equation} If, in addition, $g(s)\to 0$ as $|s|\to\infty$, problem \eqref{Plf} is said to be strongly resonant. If $\lambda$ is not an eigenvalue of (\ref{a1}), then \eqref{Plf} is said to be a non-resonance problem. \smallskip It is well known that, for $h\in L^{N/2}({\mathbb{R}^N})\cap L^{\alpha}({\mathbb{R}^N})$, $\alpha>N/2$, and $h>0$ a.e., problem (\ref{a1}) possesses a sequence $\{\lambda_{j}\}$ of eigenvalues satisfying $0<\lambda_1<\lambda_2\leqslant\lambda_3\dots$, with $\lambda_{j}\to\infty$ as $j\to\infty$, and the corresponding family of eigenfunctions, $\{\varphi_n\}$, forms a complete orthonormal system for $\mathcal{D}^{1,2}$. Furthermore, $\varphi_1$ can be chosen to be positive a.e. in $\mathbb{R}^N$. The goal of this paper is to extend the solvability of a family of elliptic problems on bounded domains to the whole space ${\mathbb{R}^N}$, $N\geqslant3$. In particular, we study the existence of weak solutions for problem \eqref{Plf} with $a \in L^{2N/(N+2)}\cap L^{\infty}$, for the case of resonance, and with $a \in L^1\cap L^{\infty}$ and $f=0$, for the case of strong resonance. We prove the existence of weak solutions of \eqref{Plf} using variational methods; i.e., solutions of \eqref{Plf} are realized as critical points of the functional $$ J_{\lambda}(u)=\frac{1}{2}\int |\nabla u|^2-\frac{\lambda}{2}\int hu^2 -\int aG(u)-\int fu, $$ where $G(s)=\int ^{s}_{0}g(t)dt$, $s\in{\mathbb{R}}$. Our results are obtained using the saddle point theorem by Rabinowitz \cite{R} and a linking theorem in \cite{EL}, in conjunction with the concentration-compactness lemma of Lions \cite{L1}. The solvability of \eqref{Plf} in the resonance case can be obtained by imposing conditions on either $g$ or $G(s)$. We prove the following existence results: \begin{theorem} \label{Thm1} Let $g\in C({\mathbb{R},\mathbb{R}})$ be bounded and $a$ be an element of $L^{2N/(N+2)}\cap L^{\infty}$. If $\lambda\in (\lambda_1,\lambda_2)$, where $\lambda_1$ and $\lambda_2$ are the first two eigenvalues of (\ref{a1}), then problem \eqref{Plf} has at least one solution for any $f\in L^{2N/(N+2)}$. \end{theorem} \begin{theorem} \label{Thm2} Let $g\in C({\mathbb{R},\mathbb{R}})$ be bounded and $a \in L^{2N/(N+2)}\cap L^{\infty}$. If \begin{equation} \lim_{|t|\to\infty}\Big\{\int a(x)G(t\varphi_1)+t\int f(x)\varphi_1\Big\}=+\infty, \label{alp} \end{equation} then Problem \eqref{Plf} with $\lambda=\lambda_1$ has a weak solution. \end{theorem} \begin{theorem} \label{Thm3} Suppose that $g:{\mathbb{R}}\to {\mathbb{R}}$ is continuous and satisfies $\lim_{|s|\to\infty}g(s)=0$, and that $a\in L^{1}\cap L^{\infty}$. Let \begin{equation} \Lambda :=\liminf_{\|u\|\to\infty,\,u\in L_n} \int a(x)G(u)dx, \label{MyStrongRes} \end{equation} where $L_n:=\mathop{\rm span}\{\varphi_i:\lambda_{i}=\lambda_{n}\}$. Then, if $\Lambda\in{\mathbb{R}}$ and \begin{equation} a(x)G(s)\leqslant a(x)|a|^{-1}_{L^{1}}\Lambda\quad \mbox{for every $s\in \mathbb{R}$ and a.e. $x\in\mathbb{R}^N$}, \label{StrongRes2} \end{equation} problem \eqref{Plf} with $\lambda=\lambda_{n}$ possesses at least one solution. \end{theorem} The non-resonance result of Theorem \ref{Thm1} can be proved in the more general case in which $\lambda$ lies between two consecutive eigenvalues $\lambda_k<\lambda_{k+1}$ of problem (\ref{a1}). Similarly, the resonance result of Theorem \ref{Thm2} also holds for higher eigenvalues $\lambda_k<\lambda_{k+1},\;k>1$. In this case the solvability condition \eqref{alp} has to be modified appropriately. Problems at resonance have been of interest to researchers ever since the pioneering work of Landesman and Lazer \cite{LL} in 1970 for second order elliptic operators in bounded domains. The literature on resonance problems in bounded domains is quite vast; of particular interest to this paper are the works of Ahmad, Lazer and Paul \cite{ALP} in 1976 and of Rabinowitz in 1978, in which critical point methods are applied. Theorem \ref{Thm2} is an extension to $\mathbb{R}^N$ of the Ahmad, Lazer and Paul result. There is also an extensive literature on strongly resonant problems in bounded domains. Theorem \ref{Thm3} is an extension to $\mathbb{R}^N$ of a result of Silva in \cite{EL}. Resonance problems on unbounded domains, and in particular in $\mathbb{R}^N$, have been studied recently by Costa and Tehrani \cite{CO-T1} and by Jeanjean \cite{J} for the operator $-\Delta+K$ for $K$ positive, and by Stuart and Zhou \cite{SZ} for radially symmetric solutions for asymptotically linear problems in $\mathbb{R}^N$. In all these references variational methods were used. Previously, Metzen \cite{M} had used the method of approximated domains to obtain existence for non-resonant problems in unbounded domains, and Hetzer and Landesman \cite{HL} for resonant problems for a class of operators which includes the Schr\"{o}dinger operator. The main difficulty in proving Theorems \ref{Thm1}, \ref{Thm2} and \ref{Thm3} arises in showing that some kind of compactness occurs, the so called Palais-Smale condition $(PS)_c$, when using the variational approach. Even in bounded domains, to prove that the $(PS)_c$ condition holds is a very delicate issue. As an example, in bounded domains $\Omega\subset\mathbb{R}^ N$, it has been proved \cite{W} that for certain functionals the $(PS)_c$ condition does not hold at the constant $c=(1/N) S^{N/2}$, where $$ S=\inf_{\phi\in H^{1}_{0}(\Omega),\,|\phi|_{L^{2^*}}=1} \int |\nabla\phi|^2. $$ The lack of compactness for problems in unbounded domains has been overcome by different approaches; for instance, approximation by bounded domains mentioned above, the use of Sobolev spaces of symmetric functions which possess compact embedding properties, or the use of weighted Sobolev spaces (see \cite{CO-T} and references therein). From an heuristic point of view it seems that for each problem the $(PS)_c$ condition requires a specific and particular approach. In this paper we apply the concentration-compactness method of Lions \cite{L1}, which basically consists of proving the existence of a set where compactness is available by using the restrictions imposed on the $(PS)_c$ sequences by the energy functional associated with the problem \eqref{Plf}. \section{Variational Setting for Non-Resonance Problems} We study the existence of solutions for semilinear elliptic equations in $\mathbb{R}^N$ ($N\geqslant 3$) of the form $$ -\Delta u = \lambda h(x) u + a(x)g(u)+f(x), $$ where $\lambda \in {\mathbb{R}}$, $f\in L^{2N/(N+2)}$, $g:{\mathbb{R}} \to {\mathbb{R}}$ is a continuous function, $|g(s)|\leqslant M$ for all $s\in{\mathbb{R}}$, $h \in L^{N/2}\cap L^{\alpha}$, $\alpha>N/2$ and $a \in L^{2N/(N+2)}\cap L^{\infty}$. In particular, we consider the boundary-value problem \eqref{Plf} which is is a non-linear perturbation of the linear eigenvalue problem (\ref{a1}). It can be shown that if $h\in L^{N/2}(\mathbb{R}^N)\cap L^\alpha (\mathbb{R}^N)$, for $\alpha>\frac{N}{2}$, and $h> 0$ a.e., then Problem (\ref{a1}) has an increasing sequence of eigenvalues $0<\lambda_1\leqslant\lambda_2,\dots$ with $\lambda_j\to \infty$ as $j\to\infty$ and a corresponding sequence of eigenfunctions, $\{\varphi_j\}$, which forms a complete orthonormal system for $\mathcal{D}^{1,2}$. This is a consequence of the following result which is easily derived from \cite[Lemma 2.1]{CI-G}. \begin{lema} \label{Proposition 2.2} If $h \in L^{N/2}(\mathbf R \it^N)\cap L^\alpha (\mathbb{R}^N)$ for $\alpha>\frac{N}{2}$, then $$ -\Delta w=hu \quad \mbox{in }\mathcal{D}^{1,2}(\mathbb{R}^N) $$ has a weak solution in $\mathcal{D}^{1,2}$ for every $u\in \mathcal{D}^{1,2}(\mathbb{R} \it^N)$. Moreover the operator $T_h:\mathcal{D}^{1,2} ({\mathbb{R}^N})\to \mathcal{D}^{1,2}(\mathbb{R}^N)$, defined by $T_h(u)=T(u)=w$, is compact. \end{lema} \begin{coro} \label{Corollary 2.3} Let $h\in L^{N/2}\cap L^\alpha$ for $\alpha>N/2$ and define $F:\mathcal{D}^{1,2}\to {\mathbb{R}}$ by $F(u):=\int hu^2$, then $F$ is weakly continuous; that is, if $u_n\to u$ weakly in $\mathcal{D}^{1,2}$, then $F(u_n)\to F(u)$. \end{coro} Moreover, the condition $h\in L^{\alpha}$ for $\alpha >N/2$ can also be used to show, as a consequence of the weak Harnack inequality \cite[Theorem 8.20]{G-T} that $\varphi_1>0$ a.e. in $\mathbb{R}^N$, $\lambda_1$ is simple, and the zero-set of the eigenfunctions $\varphi_j$, $j\geqslant 1$, has Lebesgue measure zero. This last property is known as {\it unique continuation\/} \cite{AR}. Solutions of \eqref{Plf} happen to be critical points of the functional \begin{equation} J_{\lambda}(u)=\frac{1}{2}\int|\nabla u|^2-\frac{\lambda}{2}\int h(x)u^2 -\int a(x) G(u)-\int f(x)u \label{J} \end{equation} for $u\in\mathcal{D}^{1,2}$, where $J_{\lambda}\in\mathcal{C}^1(\mathcal{D}^{1,2},{\mathbb{R}^N})$ has Fr\'{e}chet derivative $$ J'_{\lambda}(u)v= \int \nabla u \nabla v-\lambda \int h(x)uv- \int a(x)g(u)v -\int f(x)v $$ for all $u,v\in\mathcal{D}^{1,2}$. This is a straightforward consequence of the definition of Fr\'{e}chet derivative and the conditions on $a$, $f$, $g$, and $h$. We will use the following version of the concentration-compactness lemma of Lions \cite{L1}. \begin{lema}[Lions Concentration-Compactness Lemma] \label{lions} Let $(\rho _{n})_{n\geqslant 1}$ be a sequence in $L^1(\mathbb{R}^N)$ satisfying: $\rho_n\geqslant 0$ in $\mathbb{R}^N$ and $\int \rho_n dx =\sigma$, where $\sigma>0$ is fixed. Then there exists a subsequence $(\rho_{n_{k}})_{k\geqslant 1}$ satisfying one of the three possibilities: \begin{itemize} \item[(i)] (Compactness) There exists $y_k\in \mathbb{R}^N$ such that $\rho_{n_{k}}(\cdot+y_k)$ is tight; that is, for all $\varepsilon >0$ there exists $R<\infty$ such that $\int_{y_{k}+B_{R}}\rho_{n_{k}}(x)\geqslant \sigma - \varepsilon$ for all $k$. \item[(ii)] (Vanishing) $\lim_{k\to \infty} \sup_{y \in \mathbb{R}^N}\int_{y+B_{R}}\rho_{n_{k}}(x)dx=0$ for all $R<\infty$. \item[(iii)] (Dichotomy) There exists $\alpha \in (0,\sigma)$ such that, for all $\varepsilon>0$, there exist $k_o \geqslant 1$, a sequence $\{y_n\}\subset {\mathbb{R}^N}$, a number $R>0$ and a sequence $\{R_n\}\subset {\mathbb{R}}_{+}$, with $RN/2$, and $a \in L^{2N/(N+2)}\cap L^{\infty}$. Then, if every $(PS)_c$ sequence for $J_{\lambda}$ is bounded, $J_{\lambda}$ satisfies the $(PS)_c$ condition. \end{prop} \begin{proof} Let $(u_n)$ be a $(PS)_c$ sequence for $J_{\lambda}$. Thus, by assumption, $(u_n )$ is bounded. Without loss of generality, we may assume that $\|u_n\|^2=\int|\nabla u_n|^{2}>0$ for all $n$. Define $$ \rho_n:=|\nabla u_n|^2\quad\mbox{for all } n. $$ Thus $(\rho_n)$ is a sequence in $L^1(\mathbb{R}^N)$ satisfying (passing to a subsequence if necessary) $\int \rho_n \to \tau >0$ as $n\to\infty$. Defining $$ \rho'_n=\frac{\rho_n}{\int \rho_n} \quad\mbox{for all } n, $$ we have $\int\rho'_n=1>0$ for all $n$. Hence, using $(\rho'_n )$ for $(\rho_n )$, we may assume that $(\rho_n )$ satisfies the hypotheses of the Lions Concentration-Compactness Lemma \ref{lions} with $\sigma=1$. \smallskip \noindent \textbf{(A)} Claim: Vanishing does not hold. Let $B_R(y)=\{x\in {\mathbb{R}^N}:|x-y|0$, there exists $R'>0$ such that $$ \Big(\int_{[B_{R'}(0)]^{c}}|\nabla\phi|^2\Big)^{1/2} <\frac{\varepsilon}{2\sup_{k}\|u_{n_{k}}\|} . $$ On the other hand, by the Cauchy-Schwartz inequality, \[ \Big|\int \nabla u_{n_{k}}\cdot\nabla\phi\Big| \leqslant \|u_{n_{k}}\|\Big(\int_{[B_{R'}(0)]^{c}}|\nabla\phi|^2\Big)^{1/2} +\|\phi\|\Big(\int_{B_{R'}(0)}|\nabla u_{n_{k}}|^2\Big)^{1/2}. \] Moreover, since vanishing in Lemma \ref{lions} implies the existence of a subsequence $(u_{n_k} )$ and $k_0$ such that $$ \int _{B_{R'}(0)}|\nabla u_{n_{k}}|^{2} <\big(\frac{\varepsilon}{2\|\phi\|}\big)^2\quad\mbox{if } k\geqslant k_0, $$ it follows that $\int \nabla u_{n_{k}}\cdot\nabla\phi<\varepsilon$ for all $k\geqslant k_o $. Since $\varphi\in\mathcal{D}^{1,2}$ was arbitrary, $u_{n_{k}}\to 0$ weakly in $\mathcal{D}^{1,2}$. Now, using the assumption that $\|J'_{\lambda}(u_{n_{k}})\|\to 0\;\mbox{as}\;k\to\infty$, we have \begin{equation} \int |\nabla u_{n_{k}}|^2 -\int ag(u_{n_{k}})u_{n_{k}} =o(1) \quad\mbox{as } k\to\infty, \label{myeq1} \end{equation} since the map $u\mapsto\int hu^2$ is weakly continuous by Corollary \ref{Corollary 2.3}, and $u\mapsto \int fu$ is in $(\mathcal{D}^{1,2})^*$. On the other hand, since $a\in L^{2N/(N+2)}$, given $\varepsilon>0$, there exists $R_*$ such that \begin{equation} \Big(\int_{[B_{R_{*}}(0)]^{c}}|a|^{2N/(N+2)}\Big)^\frac{N+2}{2N} <\frac{\varepsilon}{2M\sup_{k}\|u_{n_{k}}\|}. \label{myeq2} \end{equation} Moreover, vanishing implies that there exists $k_1$ such that for $k>k_1$, \begin{equation} \int_{B_{R_{*}}(0)}|\nabla u_{n_{k}}|^2< \Big(\frac{\varepsilon}{2M|a|_{L^{\frac{2N}{N+2}}}}\Big)^{2\cdot2^*}. \label{myeq3} \end{equation} Thus, applying H\"{o}lder's inequality and the estimates in (\ref{myeq2}) and (\ref{myeq3}), we conclude $$ \Big|\int a(x)g(u_{n_{k}})u_{n_{k}}\Big| <\varepsilon \quad\mbox{for all } k\geqslant k_1 . $$ Hence, since $\varepsilon$ was arbitrary, it follows from (\ref{myeq1}) that $\lim_{k\to\infty}\int|\nabla u_{n_{k}}|^2=0$, which contradicts $\int |\nabla u_{n_{k}}|^{2}=\sigma>0$ for all $k$. \smallskip \noindent \textbf{(B)} Claim: Dichotomy does not hold. If dichotomy occurs, then there exists $\alpha\in (0,\sigma)$ such that, given $\varepsilon>0$, we can chose $R>0$ with $$ \lim_{k\to\infty} \sup_{y\in{\mathbb{R}}^{\rm N}} \int_{B_{\frac{R}{2}}(y)}|\nabla u_{n_{k}}|^2>\alpha-\varepsilon. $$ Moreover, there exists $k_o\geqslant 1$ such that, for $k\geqslant k_o$, $$ \alpha -\varepsilon<\sup_{y\in{\mathbb{R}^N}}\int_{B_{\frac{R}{2}}(y)} |\nabla u_{n_{k}}|^2<\alpha+\varepsilon; $$ thus, for each $k\geqslant k_o$, there exists $y_k\in {\mathbb{R}^N}$ such that \begin{equation} \alpha -\varepsilon<\int_{B_{\frac{R}{2}}(y_k)}|\nabla u_{n_{k}}|^2 <\alpha+\varepsilon. \label{1} \end{equation} Furthermore, from Property \eqref{D} in Lemma \ref{lions}, there exists an increasing sequence $(R_k )$, with $R_1\geqslant R$ and $R_k \to \infty$ as $k\to\infty$, such that \begin{equation} \sigma-\alpha-\epsilon\leqslant\int_{\{ B_{3R_{k}}(y_k)\}^c} |\nabla u_{n_{k}}|^2\leqslant\sigma-\alpha+\epsilon \quad\mbox{for all } k\geqslant k_o. \label{2} \end{equation} Consequently, \begin{equation} \int_{\frac{R}{2}<|x-y_{k}|<3R_{k}}|\nabla u_{n_{k}}|^2<2\varepsilon \quad\mbox{for all } k\geqslant k_o . \label{3} \end{equation} Note that (\ref{3}) implies \begin{equation} \int_{R<|x-y_{k}|\leqslant 2R_{k}}|u_{n_{k}}|^{2^*}\leqslant \theta(\varepsilon) \quad\mbox{for all }\ k\geqslant k_o, \label{4} \end{equation} where $\theta\to 0$ as $\varepsilon\to 0$. To see why (\ref{4}) holds, take $\eta_k\in C^{\infty}_{0}(\mathbb{R}^N)$ such that $\eta_k(x)=0$ if $|x|\leqslant R/2$ or $|x|\geqslant 3R_k$, and $\eta_k (x)=1$ if $R\leqslant|x|\leqslant 2R_k$. By Sobolev's inequality (\ref{embed}) we have that \begin{align*} \Big(\int |\eta_k u_k|^{2^*}\Big)^{1/2^*} &\leqslant C\Big(\int|\nabla (\eta_k u_k)|^2\Big)^{1/2} \\ &\leqslant C\Big(\int |\nabla\eta_k|^2 u^{2}_{k} +2\int u_k \eta_k\nabla\eta_k\cdot\nabla u_k+\int \eta^{2}_{k} |\nabla u_{k}|^2\Big)^{1/2}\\ &\leqslant C\Big(C_1\int_{\Omega}|u_{k}|^2+C_2\int_{\Omega}|u_k| +C_3\int_{\frac{R}{2}\leqslant |x-y_k|\leqslant 3R_k}|\nabla u_k|^2\Big)^{1/2}. \end{align*} Where we have written $u_k$ for $u_{n_{k}}$, and $\Omega=\{x:\nabla\eta_k\not=0\}=\{R/2\leqslant|x-y_k|\leqslant R\}\cup\{2R_k \leqslant|x-y_k|\leqslant 3R_k\}$. Clearly, $\Omega\subset\{R/2\leqslant|x-y_k|\leqslant 3R_k\}$. By (\ref{3}) in conjunction with Sobolev's inequality, we also obtain that $\int_{\Omega}|u_{k}|^2 < C\varepsilon $ and $\int_{\Omega}|u_k|0$, there exists $R'>0$ such that \begin{equation} \Big(\int_{[B_{R'}(0)]^{c}}|\nabla\phi|^2\Big)^{1/2} <\frac{\varepsilon}{\sup_{k}\|u_{k}\|}. \label{my47} \end{equation} Since $\{y_k\}$ is bounded in $\mathbb{R}^N$, there exists $y^*$ such that $y_k\to y^*$ (taking a subsequence if necessary). Choose $n_0 \geqslant k_o$ such that $B_{R'}(0)\subset B_{R_{n_{0}}}(y^*)$; this is possible since $R_k\to\infty$. Then, for $k>n_0$, by (\ref{8}), \begin{align*} \int \nabla u^{\ddag}_k \cdot\nabla\phi &=\int_{[B_{R_{n_{0}}}(y^*)]^{c}}\nabla u^{\ddag}_k\cdot \nabla\phi\\ &\leqslant \|u^{\ddag}_{k}\|\Big(\int_{[B_{R_{n_{0}}}(y^*)]^{c}}|\nabla\phi|^2 \Big)^{1/2}\\ &\leqslant \|u^{\ddag}_{k}\|\Big(\int_{[B_{R'}(0)]^{c}}|\nabla \phi|^2\Big)^{1/2}<\varepsilon, \end{align*} by the Cauchy-Schwarz inequality, the fact that $[B_{R_{n_{0}}}(y^*)]^{c}\subset [B_{R'}(0)]^{c}$ and (\ref{my47}). Therefore, since $\phi\in \mathcal{D}^{1,2}$ and $\varepsilon >0$ were arbitrary, it follows that $u^{\ddag}_k\to0$ weakly in $\mathcal{D}^{1,2}$. Thus, $\int hu^{\ddag}_k u_k\to 0$ as $k\to \infty$ as stated in (\ref{*}). On the other hand, since $a\in L^{2N/(N+2)}$, given $\varepsilon>0$, there exists $R''$ such that $$ \Big(\int_{[B_{R''}(0)]^{c}}|a|^{2N/(N+2)}\Big)^\frac{N+2}{2N} <\frac{\varepsilon}{M\sup_{k}\|u_k\|}. $$ Thus, by (\ref{8}) and similar arguments to those used above, there exist $n_0 \geqslant k_o$ such that $B_{R''}(0)\subset B_{R_{n_{0}}}(y^*)$ and \begin{align*} \big|\int a(x)g(u_k)u^{\ddag}_k\big| &\leqslant M \int a(x)|u^{\ddag}_k|\\ &\leqslant M\int_{[B_{R_{n_{0}}}(y^*)]^{c}}a|u^{\ddag}_k|\\ &\leqslant M\|u^{\ddag}_{k}\| \Big(\int_{[B_{R_{n_{0}}}(y^*)]^{c}}|a|^{2N/(N+2)}\Big)^{(N+2)/(2N)}\\ &\leqslant \varepsilon,\quad \mbox{if } k>n_0. \end{align*} Consequently, \begin{equation} \int ag(u_k)u^{\ddag}_{k}=o(1)\quad\mbox{as } k\to\infty . \label{**} \end{equation} Now, \begin{align*} &\Big| \int\nabla u_k \cdot\nabla u^{\ddag}_k-\int|\nabla u^{\ddag}_k|^2\Big|\\ &=\Big|\int_{R_{k}\leqslant|x-y_k|\leqslant 2R_{k}}\nabla(\phi_k u_k)\cdot \nabla u_k\Big|\\ &\leqslant \int_{R_{k}\leqslant|x-y_k|\leqslant 2R_{k}}|u_k||\nabla u_k\cdot \nabla\phi_k|+\int_{R_{k}\leqslant|x-y_k|\leqslant 2R_{k}}|\phi_k||\nabla u_k|^2 \\ &\leqslant C_1\int_{R_{k}\leqslant|x-y_k|\leqslant 2R_{k}}|u_k|+ C_2\int_{R_{k}\leqslant|x-y_k|\leqslant 2R_{k}}|\nabla u_k|^2. \end{align*} Hence, by (\ref{4}) and a similar argument used to obtain (\ref{4}) applied to the first integral, as well as the Sobolev embedding Theorem, \begin{equation} \int\nabla u_k\cdot\nabla u^{\ddag}_k=\int|\nabla u^{\ddag}_k|^2+\theta_2(\varepsilon) \quad\mbox{for } k\geqslant k_o , \label{***} \end{equation} where $ \theta_2\to 0$ as $\varepsilon\to 0$. Since $(u_k)$ is a $(PS)_c$ sequence, as $k\to\infty$, $$ \langle J'_{\lambda}(u_k),u^{\ddag}_k\rangle=\int \nabla u_k\cdot\nabla u^{\ddag}_k-\lambda\int h u_k u^{\ddag}_k-\int ag(u_k)u^{\ddag}_k -\int f u^{\ddag}_k=o(1)\,. $$ It then follows from (\ref{*}), (\ref{**}) and (\ref{***}) together with the fact that $\int fu^{\ddag}_{k}\to 0$ as $k\to\infty$, that $$ \int|\nabla u^{\ddag}_k|^2=o(1)\quad \mbox{as } k\to\infty . $$ Therefore, from (\ref{2}), $$ \sigma-\alpha-\varepsilon\leqslant\int_{|x-y_k| \geqslant 3R_k}|\nabla u^{\ddag}_k|^2 \leqslant\int |\nabla u^{\ddag}_k|^2=o(1)\quad\mbox{as }k\to\infty , $$ with $\sigma-\alpha-\varepsilon>0$ for $\varepsilon$ small, which is a contradiction. Consequently, $(y_k)$ cannot be bounded and dichotomy does not hold in this case. \smallskip \noindent (ii) Assume now that $(y_k)$ is unbounded. For this case we use $u^{\dag}_k$ to get a contradiction. First, we show that $u^{\dag}_k\stackrel{k}{\to}0$ weakly in $\mathcal{D}^{1,2}$. Let $\phi$ be any function in $\mathcal{D}^{1,2}$. Given $\varepsilon>0$, there exists $R'>0$ such that $\int_{[B_{R'}(0)]^{c}}\nabla \phi<\varepsilon/\sup_k \|u_k\|$. Since $\{y_k\}$ is not bounded, there exists $n_0$ such that $|y_{n_{0}}|>R'+2R$, where $R$ is as in (\ref{7}). We then have that $B_{2R}(y_{n_{0}})\subset [B_{R'}(0)]^{c}$, so in view of (\ref{7}), \begin{align*} \int \nabla u^{\dag}_{k}\cdot\nabla\phi&= \int_{[B_{2R}(y_{n_{0}})]} \nabla u^{\dag}_{k}\cdot\nabla\phi\\ &\leqslant \|u^{\dag}_{k}\|\int_{[B_{2R}(y_{n_{0}})]}\nabla\phi\\ &\leqslant \|u^{\dag}_{k}\|\int_{[B_{R'}(0)]^{c}}\nabla\phi\\ &\leqslant \varepsilon,\quad\mbox{if } k>n_0. \end{align*} Since $\phi\in \mathcal{D}^{1,2}$ and $\varepsilon>0$ are arbitrary, we conclude that $u^{\dag}_k\stackrel{k}{\to}0$ weakly in $\mathcal{D}^{1,2}$. From the assumption that $(u_m )$ is a bounded $(PS)_c$ sequence, we have \begin{equation} \int \nabla u_k\cdot\nabla u^{\dag}_k-\lambda\int h u_k u^{\dag}_k -\int ag(u_k)u^{\dag}_k-\int fu^{\dag}_k=o(1)\quad \mbox{as } k\to\infty. \label{rev1} \end{equation} Observe that \begin{align*} \Big|\int h u^{\dag}_k(u^{\dag}_k-u_k)\Big| &\leqslant \int |h||u^{\dag}_k||u^{\dag}_k-u_k|\\ &\leqslant \int_{R\leqslant|x-y_{k}|\leqslant 2R}|h||\zeta_{k}||\zeta_{k}-1||u_k|^2\\ &\leqslant C|h|_{\frac{N}{2}} \Big(\int_{R\leqslant |x-y_k|\leqslant 2R}|u_k|^{2^*}\Big)^{2/2^*}, \end{align*} by H\"{o}lder's inequality. So, by (\ref{4}), since $R0$, since $a\in L^{2N/(N+2)}$, there exists $R'''>0$ such that \begin{equation} \Big(\int_{[B_{R'''}(0)]^c}|a|^{2N/(N+2)}\Big)^\frac{N+2}{2N} <\frac{\varepsilon}{M\sup_{k}\|u_k\|}. \label{my48} \end{equation} Since $\{y_k\}$ is unbounded, we take $n_0$ such that $|y_{n_{0}}|>R'''+2R$. Then $B_{2R}(y_{n_{0}})\subset[B_{R'''}(0)]^{c}$; thus, by (\ref{7}) and (\ref{my48}), \begin{align*} \Big|\int ag(u_k)u^{\dag}_{k}\Big| &\leqslant M\int_{B_{2R}(y_{n_{0}})}|a||g(u_k)u^{\dag}_{k}|\\ &\leqslant M\|u_k\|\Big(\int _{[B_{R'''}(0)]^c}|a|^{2N/(N+2)}\Big) ^{(N+2)/(2N)}\\ &< \varepsilon\quad\mbox{for all } k, \end{align*} from which (\ref{rev3}) follows. Finally, \begin{equation} \int\nabla u_k\cdot\nabla u^{\dag}_k=\int|\nabla u^{\dag}_k|^2 +\theta_4(\varepsilon),\label{rev4} \end{equation} where $\theta_4\to 0$ as $\varepsilon\to 0$. This follows from (\ref{4}), the estimate \[ \Big| \int\nabla u_k \cdot\nabla u^{\dag}_k-\int|\nabla u^{\dag}_k|^2\Big| \leqslant C_1\int_{R\leqslant|x-y_k|\leqslant 2R}|u_k|+C_2\int_{R\leqslant|x-y_k| \leqslant 2R}|\nabla u_k|^2 , \] and an argument similar to that used to obtain (\ref{4}), by observing that $\{R\leqslant|x-y_k|\leqslant 2R\}\subset\{R\leqslant |x-y_k|\leqslant 2R_k\}$ and applying the Sobolev's embedding Theorem. Thus, using (\ref{rev2}), (\ref{rev3}) and (\ref{rev4}) in (\ref{rev1}), and recalling (\ref{1}), we obtain $$ 0<\alpha-\varepsilon\leqslant\int_{B_\frac{R}{2}(y_k)}|\nabla u^{\dag}_k|^2 \leqslant\int |\nabla u^{\dag}_k|^2=o(1)\quad\mbox{as }k\to\infty, $$ which is a contradiction. Since vanishing and dichotomy in Lemma \ref{lions} do not hold, necessarily compactness holds; i.e., there exists $\{ y_n\} \subset\mathbb{R}^N $ such that for all $\varepsilon$ there exists $R>0$ such that \begin{equation} \int_{B_{R}(y_{n})}|\nabla u_n|^2\geqslant\sigma-\varepsilon \quad\mbox{for all }n. \label{newcomp} \end{equation} Now, it follows from (\ref{newcomp}) and the fact that $\int |\nabla u_n|^2 = \sigma$ for all $n$ that \begin{equation} \int_{[B_R(y_n)]^c}|\nabla u_n|^2<\varepsilon\quad\mbox{for all } n. \label{9} \end{equation} \smallskip \noindent \textbf{Claim:} $\{y_n\}$ is bounded. If $\{y_n\}$ is not bounded, then $ u_n\to 0$ weakly in $\mathcal{D}^{1,2}$ as $n\to\infty$. To see why this is the case, take $\phi\in \mathcal{D}^{1,2}$ and let $\varepsilon>0$. There exists $R'>0$ such that \begin{equation} \Big(\int_{[B_{R'}(0)]^{c}}|\nabla \phi|^2\Big)^{1/2}< \varepsilon/(2\sup_{n}\|u_n\|). \label{revision1} \end{equation} Since $\{y_n\}$ is not bounded, we may assume that $|y_n |\to\infty$ as $n\to\infty$, and so there exists $n_0$ such that $|y_{n_{0}}|>R'+R_0$, where we choose $R_o >0$, whose existence is guaranteed by part (i) of Lemma \ref{lions} (see also (\ref{9})), such that \begin{equation} \Big(\int_{[B_{R_{0}}(y_{n_{0}})]^{c}} |\nabla u_n|^2\Big)^{1/2}<\varepsilon/(2\|\phi\|). \label{revision2} \end{equation} Also, $B_{R_0}(y_{n_{0}})\subset [B_{R'}(0)]^{c}$. Thus, \begin{align*} \Big|\int \nabla u_n\cdot\nabla\phi\Big| &\leqslant \|u_n\|\Big(\int_{B_{R_0}(y_{n_{0}})}|\nabla\phi|^2\Big)^{1/2} +\|\phi\|\Big(\int_{[B_{R_0}(y_{n_{0}})]^{c}}|\nabla u_n|^2\Big)^{1/2}\\ &\leqslant \|u_n\|\Big(\int_{[B_{R'}(0)]^{c}}|\nabla\phi|^2\Big)^{1/2} +\|\phi\|\Big(\int_{[B_{R_0}(y_{n_{0}})]^{c}}|\nabla u_n|^2\Big)^{1/2}, \end{align*} so that, by (\ref{revision1}) and (\ref{revision2}), $$ \big|\int \nabla u_n \cdot \nabla\phi\big|<\varepsilon\quad\mbox{for all }n>n_0 . $$ Since $\phi\in \mathcal{D}^{1,2}$ was arbitrary, we conclude that $ u_n\to 0$ weakly in $\mathcal{D}^{1,2}$ as stated. Consequently, using the assumption that $(u_n )$ is a bounded $(PS)_c$ sequence, we obtain \begin{equation} \int|\nabla u_n|^2-\int ag(u_n)u_n =o(1)\quad\mbox{as }n\to\infty, \label{comp1} \end{equation} since $u\to \int hu^2$ is weakly continuous by Corollary \ref{Corollary 2.3}, and $u\mapsto \int fu$ is also weakly continuous. Moreover, \begin{equation} \int ag(u_n)u_n=\int_{B_{R}(y_n)}ag(u_n)u_n+\int_{[B_{R}(y_n)]^{c}} ag(u_n)u_n. \label{comp2} \end{equation} Since $\{y_n\}$ is not bounded, it follows that $$ \Big(\int_{B_{R}(y_n)}|a|^{2N/(N+2)}\Big)^\frac{N+2}{2N}\to 0 \quad\mbox{as } n\to\infty. $$ Therefore, \begin{equation} \Big|\int_{B_{R}(y_n)}ag(u_n)u_n\Big|\leqslant C \Big(\int_{B_{R}(y_n)}|a|^{2N/(N+2)}\Big)\to 0 \quad\mbox{as } n\to\infty. \label{comp3} \end{equation} In addition, by (\ref{embed}) and (\ref{9}), $$ \int_{[B_{R}(y_n)]^{c}}|u_n|^{2N/(N-2)}0$ such that $B_R (y_n )\subset B_{R^*} (0)$ for all $n=1,2,3,\ldots$. We may also choose $R^*$ large enough so that \begin{equation} \int_{B_{R^*} (0)^c} |a|^{\frac{2N}{N+2}} < \varepsilon^{\frac{2N}{N+2}}. \label{aestimate} \end{equation} Put $\Omega = B_{R^*}(0)$ and note that $\Omega$ satisfies the hypotheses of the Rellich-Kondrachov Theorem \cite[Theorem 7.26, p. 171]{G-T}. Given that $({u}_n)$ is a bounded sequence, there exists a subsequence, $(u_{n_k } )$, such that ${u}_{n_k} \to u $ weakly in $\mathcal{D}^{1,2}$ as $k\to\infty$. Moreover, given $1\leqslant t<2^*$, we may assume that ${u}_{n_k} \to u$ strongly in $L^t(\Omega)$ as $k\to\infty$, since $\Omega$ is bounded, passing to a subsequence if necessary, by the Rellich-Kondrachov Theorem. Observe that, since $B_R (y_n )\subset \Omega$ for all $n=1,2,3,\ldots$, then $\Omega^c \subset B_R (y_n )^c $ for all $n$. It then follows from (\ref{9}) that \begin{equation} \int_{\Omega^c}|\nabla u_n|^2<\varepsilon\ \mbox{ for all } n. \label{new9} \end{equation} We want to show that \begin{equation} \int |\nabla (u_{n_k } -u)|^2\to 0\quad\mbox{as } k\to\infty. \label{comp4} \end{equation} We have \begin{align*} \int |\nabla (u_{n_k}-u)|^2 &=\int \nabla (u_{n_k}-u)\cdot\nabla(u_{n_k}-u)\\ &=\int \nabla u_{n_k}\cdot\nabla(u_{n_k}-u)-\int \nabla u\cdot\nabla(u_{n_k}-u), \end{align*} where $\int \nabla u\cdot\nabla(u_{n_k}-u)\to 0$ as $k\to\infty$, by the definition of weak convergence in $\mathcal{D}^{1,2}$. Consequently, (\ref{comp4}) will follow if we can prove that \begin{equation} \lim_{k\to\infty} \int \nabla u_{n_k}\cdot\nabla(u_{n_k}-u) = 0. \label{comp5} \end{equation} Now from the fact that $(u_n )$ is a bounded $(PS)_c$ sequence it follows that \begin{equation} \Big|\int \nabla u_{n_k} \cdot\nabla (u-u_{n_k})-\lambda\int hu_{n_k} (u-u_{n_k})- \int ag(u_{n_k})(u-u_{n_k})\Big|=o(1) \label{comp6} \end{equation} as $k\to\infty$, since $\int f(u-u_n)\to 0$ as $n\to\infty$. We estimate the second integral on the left--hand side of (\ref{comp6}) as follows: $$ \Big|\int hu_{n_k} (u-u_{n_k})\Big|\leqslant \Big|\int_{\Omega} hu_{n_k}(u-u_{n_k})\Big| + \Big|\int_{{\Omega}^c} hu_{n_k}(u-u_{n_k})\Big|, $$ where, by H\"older's inequality, $$ \Big|\int_{\Omega} hu_{n_k}(u-u_{n_k})\Big| \leqslant |h|_{_{L^\alpha}} \Big(\int_{\Omega} |u_{n_k}|^{2N/(N-2)} \Big) ^{(N-2)/(2N)}\Big(\int_{\Omega} |u-u_{n_k}|^s\Big)^{1/s}, $$ where $\frac{1}{s}=\frac{1}{2} + \frac{2}{N}-\frac{1}{\alpha}$, so that $s<2^*$. Hence, by the Rellich-Kondrachov Theorem, we may assume that $u_{n_k}\to u$ strongly in $L^s(\Omega)$. Consequently, \begin{equation} \int_{\Omega} hu_{n_k}(u-u_{n_k})=o(1)\quad\mbox{as }k\to\infty. \label{comp7} \end{equation} On the other hand, by H\"older's inequality and the assumption that $(u_n )$ is bounded, $$ \Big|\int_{\Omega^c} hu_{n_k}(u-u_{n_k})\Big| \leqslant C|h|_{_{L^{\frac{N}{2}}}} \Big(\int_{\Omega^c} |u_{n_k}|^{2N/(N-2)} \Big) ^{(N-2)/(2N)} . $$ Thus, by (\ref{embed}) and (\ref{new9}), $$ \Big|\int_{\Omega^c} hu_{n_k}(u-u_{n_k})\Big| \leqslant C\varepsilon\quad\mbox{for all } k. $$ Therefore, it follows from (\ref{comp7}) that \begin{equation} \limsup_{k\to\infty} \left| \int hu_{n_k}(u-u_{n_k}) \right| \leqslant C\varepsilon . \label{comp8} \end{equation} Similarly, for the third integral on the left-hand side of (\ref{comp6}), \begin{equation} \Big|\int_{\Omega} ag(u_{n_k})(u-u_{n_k})\Big| \leqslant C\int_{\Omega}|u-u_{n_k}|\to 0\quad\mbox{as } k\to\infty, \label{comp9} \end{equation} since $a\in L^\infty$ and $u_{n_k}\to u$ strongly in $L^1(\Omega)$. To estimate the integral over $\Omega^c$ use H\"older's inequality together with the assumptions that $(u_n )$ and $g$ are bounded to obtain $$ \Big|\int_{\Omega^c } ag(u_{n_k})(u-u_{n_k})\Big|\leqslant C\Big( \int_{\Omega^c } |a|^{\frac{2N}{N+2}}\Big)^{(N+2)/(2N)}. $$ It then follows from (\ref{aestimate}) that $$ \Big|\int_{\Omega^c } ag(u_{n_k})(u-u_{n_k})\Big|\leqslant C\varepsilon. $$ Consequently, by (\ref{comp9}), \begin{equation} \limsup_{k\to\infty} \Big|\int ag(u_{n_k})(u-u_{n_k})\Big|\leqslant C\varepsilon . \label{comp10} \end{equation} Hence, since $\varepsilon$ is arbitrary, (\ref{comp5}) follows from (\ref{comp6}), (\ref{comp8}) and (\ref{comp10}), and (\ref{comp5}) in turn implies (\ref{comp4}); that is, $$ \|u_{n_k}-u\|^2 = \int |\nabla(u_{n_k}-u)|^2=o(1)\quad\mbox{as }k\to\infty. $$ i.e. $u_{n_k}\to u$ strongly in $\mathcal{D}^{1,2}$. \end{proof} Now, with $J_{\lambda}$ satisfying the $(PS)_c$ condition, once we can show that every $(PS)_{c}$ sequence is bounded, we are able to prove some existence results for problem \eqref{Plf}. Existence will be obtained as a consequence of the following saddle point theorem of Rabinowitz. \begin{theorem}[Saddle Point Theorem \cite{R}] \label{sadd} Let $E=V\oplus X $, where $E$ is a real Banach space and $V\not= \{0\}$ and is finite dimensional. Suppose $I\in \mathcal{C}^1(E,{\mathbb{R}})$ satisfies the $(PS)$ condition, \begin{itemize} \item[($I_1$)] there is a constant $\alpha$ and a bounded neighborhood $D$ of 0 in $V$ such that $I\big|_{\partial D}\leqslant\alpha$, and \item[($I_2$)] there is a constant $\beta>\alpha$ such that $I|_{X}\geqslant \beta$. \end{itemize} Then, $I$ possesses a critical value $c\geqslant\beta$. Moreover $c$ can be characterized as $$c=\inf_{h\in\Gamma}\max_{u\in\overline{D}} I(h(u)),$$ where $\Gamma=\{h\in\mathcal{C}(\overline{D},E):h=\mbox{id on }\partial D\}$. \end{theorem} First, we consider the case when $\lambda$ in the problem \eqref{Plf} is not an eigenvalue of the eigenvalue problem (\ref{a1}): %eqn12 \begin{gather*} -\Delta u=\lambda h(x)u\quad \mbox{in } \mathbb{R}^N,\;h>0\;a.e. ,\\ u\in \mathcal{D}^{1,2} . \end{gather*} We are now in a position to prove Theorem \ref{Thm1}. % \label{Proposition 3.3} \begin{proof}[Proof of Theorem \ref{Thm1}] Suppose $(u_n)$ is a $(PS)_c$ sequence. First, we show that $(u_n)$ is bounded in $\mathcal{D}^{1,2}$. We argue by contradiction. Assume $(\int|\nabla u_n|^2)^{1/2} =t_n \to \infty$ and define $v_n = u_n/t_n$; then, $(\int|\nabla v_n|^2)^{1/2} =1$ for all $n$. So we have, passing to a subsequence if necessary, that $v_n \rightharpoonup v$ weakly in $\mathcal{D}^{1,2}(\mathbb{R}^N)$, since $(v_n)$ is bounded. Now we claim that $v(x)\equiv 0$. As a consequence of the assumption $\|J'_{\lambda}(u_n)\|\to 0$ as $n\to\infty$, we have \begin{align*} \langle J'_{\lambda}(u_n),\phi \rangle &= \int \nabla u_n\cdot \nabla \phi dx-\lambda \int h(x)u_n \phi dx\\ &\quad -\int a(x)g(u_n)\phi dx-\int f(x)\phi=o(1)\|\phi\| \end{align*} as $n\to\infty$, for all $\phi \in \mathcal{D}^{1,2}$. Dividing the previous equation by $t_n=|\nabla u_n |_2$ we obtain \begin{equation} \int \nabla v_n \cdot\nabla \phi- \lambda \int h(x)v_n \phi-\int a(x) \frac{g(u_n)}{t_n}\phi -\int \frac{f(x)\phi}{t_{n}}=o(1) \label{13} \end{equation} as $n\to\infty$. Given that $g$ is bounded and that $\phi \in \mathcal{D}^{1,2}$ implies $\phi \in L^{2N/(N-2)}$, we obtain from (\ref{13}) \begin{eqnarray} \int \nabla v_n \cdot\nabla\phi -\lambda\int hv_n \phi-\frac{C}{t_n}=o(1) \quad\mbox{as }n\to\infty, \label{14} \end{eqnarray} by H\"{o}lder's inequality. Thus, letting $n\to\infty$ in (\ref{14}), $$ \int \nabla v \cdot\nabla\phi-\lambda \int hv\phi=0. $$ Since $\lambda$ is not an eigenvalue of problem (\ref{a1}), we conclude that $v=0$ a.e. in $\mathbb{R}^N$. Substituting $\phi=v_n$ in (\ref{14}) we obtain \begin{equation} \int |\nabla v_n|^2-\lambda\int hv^{2}_{n}-\frac{C}{t_n}=o(1) \quad\mbox{as }n\to\infty, \label{my50} \end{equation} where we have used the fact that $\int h v^{2}_{n}\to 0$ and $\int fv_n\to 0$ as $n\to\infty$ since $v_n \to v = 0$ weakly in $\mathcal{D}^{1,2}$ (see Corollary \ref{Corollary 2.3}). Moreover, given that $\int |\nabla v_{n}|^2=\|v_n\|^2=1$ we obtain from (\ref{my50}) that $$ 1-\frac{C}{t_n}=o(1)\quad\mbox{as }n\to\infty, $$ which leads clearly to a contradiction as $t_n\to \infty$. Therefore, $u_n$ is bounded in $\mathcal{D}^{1,2}$. Thus, every $(PS)_c$ sequence is bounded. Hence, by Proposition \ref{Proposition 3.2}, $J_{\lambda}$ satisfies the $(PS)_c$ condition. Next we prove that $J_{\lambda}$ satisfies the hypotheses of the Saddle Point Theorem \ref{sadd}. Let $\varphi_1$ be an eigenfunction corresponding to $\lambda_1$. Recall that $\|\varphi_1\|=1$. Consider \begin{align*} J_{\lambda}(t\varphi_1) &=\frac{1}{2}t^2 \int|\nabla\varphi_1|^2 -\frac{\lambda}{2}t^2\int h(x)\varphi^{2}_{1}-\int a(x)G(t\varphi_1) -t\int f(x)\varphi_1\\ &=\frac{1}{2}t^2\big(1-\frac{\lambda}{\lambda_1}\big) -\int a(x)G(t\varphi_1)-t\int f(x)\varphi_1. \end{align*} Recall that $|G(s)|\leqslant C|s|$ for all $s\in{\mathbb{R}}$. Therefore, \begin{align*} J_{\lambda}(t\varphi_1) &\leqslant-\frac{1}{2}\big(\frac{\lambda}{\lambda_1}-1\big)t^2 +C|t|\int a(x)|\varphi_1|\\ &\leqslant -\frac{1}{2}\big(\frac{\lambda}{\lambda_1}-1\big)t^2+C_1|t| \big(|a|_{L^{2N/(N+2)}}+|f|_{L^{2N/(N+2)}}\big)|\varphi_1|_{L^{2N/(N-2)}}. \end{align*} Let $V=\mathop{\rm span}\{\varphi_1\}$; it then follows from the last inequality that $$ \lim_{\|v\|\to\infty,\;v\in V}J_{\lambda}(v)=-\infty. $$ Finally, let $X=V^{\perp}=\{w\in\mathcal{D}^{1,2}:\langle w,\varphi_1\rangle=0\}$. Then $\lambda_2\int hw^2\leqslant\int|\nabla w|^2$ for all $w\in X$ and $$ J_{\lambda}(w)\geqslant\frac{1}{2}\big(1-\frac{\lambda}{\lambda_2}\big) \|w\|^2-C_1\big(|a|_{L^{2N/(N+2)}}+|f|_{L^{2N/(N+2)}}\big)\|w\| $$ for any $w\in X$. Therefore, $J_{\lambda}(w)\to+\infty$ as $\|w\|\to\infty$ in $X$. Consequently, $(I_1 )$ and $(I_2 )$ in the Saddle Point Theorem \ref{sadd} hold, and so $J_{\lambda}$ has a critical point, which establishes Theorem \ref{Thm1}. \end{proof} \noindent\textbf{Remark.} This argument can be extended to the case $\lambda_k<\lambda<\lambda_{k+1}$ where $\lambda_k$ and $\lambda_{k+1}$ are consecutive eigenvalues of problem (\ref{a1}). \section{A Resonance Problem} In this section we consider the problem \begin{equation} \label{Pl1} \begin{gathered} -\Delta u = \lambda_1 h(x) u + a(x)g(u)+f(x),\\ u\in\mathcal{D}^{1,2}, \end{gathered} \end{equation} where $\lambda_1$ is the first eigenvalue of (\ref{a1}) over $\mathbb{R}^N$. We can solve problem \eqref{Pl1} if we impose a condition similar to one used by Ahmad, Lazer and Paul in \cite{ALP} on $G(u)$ and $f$; that is condition \eqref{alp} in the statement of Theorem \ref{Thm2}. \begin{proof}[Proof of Theorem \ref{Thm2}] % \ref{Propsion 3.3} We first show that $J_{\lambda_{1}}$ satisfies the $(PS)_{c}$ condition for any $c\in\mathbb{R}$, and then we verify that $J_{\lambda_{1}}$ satisfies the conditions of the saddle point theorem of Rabinowitz (cf. Theorem \ref{sadd}). Let $(u_m)$ be a $(PS)_c$ sequence for the functional $J_{\lambda_{1}}$ defined in (\ref{J}) for $\lambda=\lambda_1$. We claim that $(u_m)$ is bounded. Write $u_m=v_m+w_m$, where $v_n\in span\{\varphi_1\}=V$ and $w_m\in V^{\perp}=X$ for each $m\in\mathbb{N}$. First we show that $(w_m)$ is bounded in $\mathcal{D}^{1,2}$. Since $\|J'_{\lambda_{1}}(u_m)\|\stackrel{m}{\to} 0$, there exists $m_0\in \mathbb{N}$ such that if $m\geqslant m_0$, then $$ \Big|\int\nabla u_m\cdot\nabla v-\lambda_1\int hu_m v-\int ag(u_m)v-\int fv\Big| \leqslant\|v\| \quad\mbox{for all } v\in \mathcal{D}^{1,2} . $$ In particular, if $v=w_m$, we have $$ \Big|\int|\nabla w_m|^2-\lambda_1\int hw^{2}_{m}\Big|\leqslant \|w_m\| +\Big|\int ag(u_m)w_m\Big|+\Big|\int fw_m\Big|\quad\mbox{ for } m\geqslant m_o . $$ Given that $\lambda_2\int hv^2\leqslant\|v\|^2$ for all $v\in X$, we obtain $$ \big(1-\frac{\lambda_1}{\lambda_2}\big)\|w_m\|^2 \leqslant\|w_m\|+C\big(|a|_{L^{2N/(N+2)}}+|f|_{L^{2N/(N+2)}}\big)\|w_m\| \quad\mbox{ for } m\geqslant m_o, $$ from which it follows that $(w_m)$ is bounded in $\mathcal{D}^{1,2}$. Next we show that $(v_m)$ is bounded. Observe that $J_{\lambda_1}(u_m)\stackrel{m}{\to}c$ implies that $J_{\lambda_1}(u_m)$ is bounded; say $|J_{\lambda_1}(u_m)|\leqslant C_1$ for all $m$, where \begin{align*} J_{\lambda_1}(u_m)&= \frac{1}{2}\int|\nabla u_m|^2-\frac{\lambda_1}{2}\int hu^{2}_{m}-\int aG(u_m)\\ &= \frac{1}{2}\int |\nabla w_m|^2-\frac{\lambda_1}{2}\int hw^{2}_{m}\\ &\quad-\int a\left[G(v_m+w_m)-G(v_m)\right]-\int aG(v_m)-\int fv_m . \end{align*} Note that $|G(v_m+w_m)-G(v_m)|\leqslant M|w_m|$. Hence, $$ \Big|\int a[G(v_m+w_m)-G(v_m)]\Big|\leqslant M\int a|w_m|\leqslant C_3\|w_m\| . $$ So we obtain \begin{align*} \Big|\int aG(v_m)+\int fv_m\Big| &\leqslant |J(u_m)|+C_2\|w_m\|^2+ C_3\|w_m\|\\ &\leqslant C_1+C_2\|w_m\|^2+C_3\|w_m\|. \end{align*} Given that $(w_m)$ is bounded, we have $$ \Big|\int a(x) G(v_m)+\int fv_m\Big|\leqslant C\quad \mbox{for all}\ m . $$ Therefore, if \eqref{alp} holds, then $(v_m)$ is bounded in $\mathcal{D}^{1,2}$, otherwise $\int aG(v_m)+\int fv_m$ would approach infinity as $m\to\infty$, by \eqref{alp}. We therefore conclude that $(u_m )$ is bounded, and so by Theorem \ref{Thm1} we have that $J_{\lambda_1}$ satisfies the $(PS)_c$ condition. To show that the other hypotheses of the Saddle Point Theorem \ref{sadd} are satisfied, we proceed as in the proof of Theorem \ref{Thm1}. If $u\in X$, we have $u=\sum^{\infty}_{j=2}a_j\varphi_j$, hence $$ \int|\nabla u|^2-\lambda_1\int hu^2=\sum^{2}_{j=2}a^{2}_{j} \big(1-\frac{\lambda_{1}}{\lambda_{j}}\big)\geqslant \big(1-\frac{\lambda_{1}}{\lambda_{2}}\big)\|u\|^2. $$ Moreover, since $|g(s)|\leqslant M$ for all $s\in {\mathbb{R}}$, we have that, for all $u\in \mathcal{D}^{1,2}$, $$ \Big|\int aG(u)\Big|\leqslant M\int |a||u| \leqslant M|a|_{L^{2N/(N+2)}}|u|_{L^{2N/(N-2)}}\leqslant C\|u\|. $$ Therefore $J_{\lambda}$ is bounded from below on $X$; i.e. ($I_2$) in the Saddle Point Theorem \ref{sadd} holds. Finally, if $v\in V$, we have $$ J_{\lambda_{1}}(v)=-\int aG(v)-\int fv. $$ But $\int aG(v)+\int fv\to\infty$ as $\|v\|\to\infty$ by \eqref{alp} and, therefore, ($I_1$) in the Saddle Point Theorem (\ref{sadd}) also holds. Hence, $J_{\lambda_1}$ has a critical point and the theorem follows. \end{proof} \noindent\textbf{Remark.} The existence result in Theorem \ref{Thm2} can be extended to the problem \eqref{Plf} with $\lambda=\lambda_{n}$ for $n>1$ by modifying condition \eqref{alp} appropriately. In fact, suppose the eigenspace corresponding to $\lambda_n$ is $E_{\lambda_{n}}=\mathop{\rm span}\{\varphi_{n_{1}},\varphi_{n_{2}},\dots, \varphi_{n_{k}}\}$, then \eqref{alp} is replaced by $$ \lim_{t^{2}_{1}+\cdots+t^{2}_{k}\to\infty}\int a(x)G(t_{1}\varphi_{n_{1}} +\dots+t_{k}\varphi_{n_{k}})+\int f(t_{1}\varphi_{n_{1}}+\dots +t_{k}\varphi_{n_{k}})=\infty. $$ \textbf{Remark.} Suppose $\lim_{s\to\infty} g(s)=g_{\infty}$ and $\lim_{s\to-\infty} g(s)=g_{-\infty}$ exist. Then, if $g_{\infty}>0$ and $g_{-\infty}<0$, $G(s)=\int^{s}_{0}g(t)dt\to\infty$ as $|s|\to\infty$. Consequently, by L' H\^{o}spital's rule, the Lebesgue dominated convergence theorem and the fact that $\varphi_1>0$ a.e. in $\mathbb{R}^N$ we have that $$ \lim_{|t|\to\infty}\frac{1}{t}\int a(x)G(t\varphi_1) =\lim_{|t|\to\infty}\int ag(t\varphi_1)\varphi_1 = \begin{cases} g_{\infty}\int a\varphi_1 &\mbox{as }t\to\infty,\\ g_{-\infty}\int a\varphi_1 &\mbox{as }t\to-\infty . \end{cases} $$ Thus, the condition \eqref{alp} in the resonance Theorem \ref{Thm2} holds if \[ g_{\infty}\int a\varphi_1+\int f\varphi_1>0\quad \mbox{and}\quad g_{-\infty}\int a\varphi_1+\int f\varphi_1<0 , \] or \begin{equation} \label{LL} g_{-\infty}\int a\varphi_1<-\int f\varphi_10$ and $\alpha>\gamma$ such that $I(u)\geqslant\alpha$ for every $u\in X_2$ with $\|u\|_{E}\geqslant r_1$. \end{itemize} If $I$ satisfies the $(PS)_c$ condition for every $c>\gamma$ and every $(PS)_{c}$ sequence is bounded, then $I$ possesses a critical value $b\geqslant\gamma$. \end{theorem} \begin{proof}[Proof of Theorem \ref{Thm3}] Observe that since $ L^{1}\cap L^{\infty}\subset L^q$ for any $q\in (1,\infty)$, Proposition \ref{Proposition 3.2} applies to the functional $ J_{\lambda_{n}} $ given in equation (\ref{J}) with $f=0$. Define the subspaces $E_k:=\mathop{\rm span}\{\varphi_1,\dots,\varphi_k\}$ and $L_k:=\mathop{\rm span}\{\varphi_i:\lambda_{i}=\lambda_{k}\}$ for every $k\in \mathbb{N}$; also, set $E_0=\{0\}$. We show first that $J_{\lambda_{n}}(u)$ satisfies the $(PS)_c$ condition for every $c\in (-\Lambda,\infty)$. By Proposition \ref{Proposition 3.2} it is enough to show that if $c\in (-\Lambda,\infty)$ and $(u_m)$ is a $(PS)_c$ sequence, then $(u_m)$ is bounded. Let $(u_m)$ be a $(PS)_{c}$ sequence for $J_{\lambda}$. Assume by contradiction that $(u_m)$ is not bounded. Write $u_m=u^{+}_{m}+u^{0}_{m}+u^{-}_{m}$, where $u^{+}_{m}\in (E_n)^{\perp}$, $u^{0}_{m}\in L_n$, and $u^{-}_{m}\in E_{n-1}$. Since $\|J'_{\lambda_{n}}(u_m )\|\to 0$ as $m\to\infty$, it follows that there exists $m_o \in \mathbb{N}$ for which \begin{equation} \Big|\int\nabla u_m\nabla u^{+}_{m}-\lambda_n\int hu_mu^{+}_{m} -\int ag(u_m)u^{+}_{m}\Big| \leqslant \|u^{+}_{m}\| \quad\mbox{for } m\geqslant m_o . \label{my51} \end{equation} On the other hand, since $g$ is bounded, $$ \Big|\int ag(u_m)u^{+}_{m}\Big|\leqslant M\int |a||u^{+}_{m}| \leqslant M|a|_{L^{2N/(N+2)}}\Big(\int|u^{+}_{m}|^\frac{N-2}{2N}\Big)^{2N/(N-2)} \leqslant C\|u^{+}_{m}\| $$ for all $m$. Consequently, it follows from (\ref{my51}) that there exists $C_1>0$ such that $$ \frac{\lambda_{n+1}-\lambda_{n}}{\lambda_{n+1}}\|u^{+}_{m}\|^2 \leqslant C_1\|u^{+}_{m}\| \quad\mbox{ for }\ m\geqslant m_o . $$ Therefore, $(u^{+}_{m})$ is bounded. For $(u^{-}_{m})$, since $u^{-}_{m}\in E_{n-1}$ implies $\lambda_{n-1}\geqslant\frac{\int|\nabla u^{-}_{m}|^2}{\int h(u^{-}_{m})^2}$, by similar calculations as for $u^{+}_{m}$, we obtain that there exists $C_2>0$ such that $$ \Big|\frac{\lambda_{n-1}-\lambda_n}{\lambda_{n-1}}\Big|\|u^{-}_{m}\|^2 \leqslant C_2\|u^{-}_{m}\| \quad\mbox{for}\ m\geqslant m_o . $$ Consequently, $(u^{-}_m)$ is also bounded. Moreover, we will show shortly that \begin{equation} \|u^{\pm}_{m}\|\to 0\quad\quad\mbox{as }m\to\infty. \label{cero} \end{equation} This will follow from the fact that $g(s)\to 0$ as $|s|\to\infty$. In fact, from $$ \Big|\frac{\lambda_{n+1}-\lambda_n}{\lambda_{n+1}}\Big|\|u^{+}_{m}\|^2 -\Big|\int ag(u_m)u^{+}_{m}\Big| \leqslant \Big|\langle J'_{\lambda_{n}}(u_m),u^{+}_{m}\rangle\Big| $$ and H\"{o}lder's inequality, we obtain \begin{equation} \Big|\frac{\lambda_{n+1}-\lambda_n}{\lambda_{n+1}}\Big|\|u^{+}_{m}\| \leqslant \|J'_{\lambda_{n}}(u_m)\| +C\Big(\int |a|^{\frac{2N}{N+2}}|g(u_m)|^{\frac{2N}{N+2}}\Big)^{(N+2)/(2N)}. \label{r1} \end{equation} Since $\|J'_{\lambda_{n}}(u_m)\|\to 0$ as $m\to\infty$, condition (\ref{cero}), for $u_m^{+}$, follows from (\ref{r1}) once we show that \begin{equation} \lim_{m\to\infty}\int |a|^{\frac{2N}{N+2}}|g(u_m)|^{\frac{2N}{N+2}}=0. \label{r2} \end{equation} Define $v_m=u_{m}^{0}/\|u_{m}^{0}\|$ for all $m$. Then, since $L_n$ is finite dimensional, we may assume, passing to a subsequence if necessary, that there exists $v\in L_n$ such that $\|v\|=1$ and \begin{equation} v_m(x)\to v(x)\quad\mbox{a.e. as } m\to\infty. \label{r3} \end{equation} For a given $\varepsilon>0$, find $R>0$ such that \begin{equation} \int _{[B_{R}(0)]^{c}} |a|^{2N/(N+2)}<\frac{\varepsilon}{M^{2N/(N+2)}}. \label{r4} \end{equation} Since $\|u_{m}^{+}\|$ and $\|u_{m}^{-}\|$ are bounded, we may assume, as a consequence of the Rellich-Kondrachov Theorem \cite[Theorem 7.26]{G-T}, passing to subsequences if necessary, that there exist functions $w^{\pm}\in H^{1}(B_{R}(0))$ such that $u_{m}^{\pm}(x)\to w^{\pm}$ a.e. in $B_{R}(0)$ as $m\to \infty$ \cite[p. 58]{B}. It then follows from $$ u_m(x)=\|u_{m}^{0}\|\Big(\frac{u_{m}^{-}(x)}{\|u_{m}^{0}\|}+v_m(x) +\frac{u_{m}^{+}(x)}{\|u_{m}^{0}\|}\Big), $$ (\ref{r3}), and the unique continuation property of the eigenfunctions that $$ |u_m(x)|\to\infty\quad\mbox{a.e. in }B_{R}(0)\quad\mbox{as }m\to\infty, $$ since $\|u_{m}^{0}\|\to\infty$ as $m\to\infty$. Therefore, by the Lebesgue dominated convergence theorem and the fact that $g(s)\to 0$ as $s\to\infty$, $$ \lim_{m\to\infty} \int_{B_{R}(0)} |a|^{2N/(N+2)}|g(u_m)|^{2N/(N+2)}=0 . $$ Hence, in view of (\ref{r4}), $$ \limsup_{m\to\infty} \int |a|^{2N/(N+2)}|g(u_m)|^{2N/(N+2)}\leqslant \varepsilon , $$ from which (\ref{r2}) follows. Consequently, (\ref{cero}) is established for $(u_m^{+})$. Similar calculations lead to the analogous result for $(u_m^{-})$. Now, from $$ G(u_m)-G(u^{0}_{m})=\int^{1}_{0}g(u^{0}_{m}+t(u^{+}_{m}+u^{-}_{m}))(u^{+}_{m} +u^{-}_{m})dt\,, $$ we obtain \begin{align*} \Big|\int aG(u_m)-\int aG(u^{0}_{m})\Big| &\leqslant \Big|\int\int^{1}_{0}ag(u^{0}_{m}+t(u^{+}_{m}+u^{-}_{m}))(u^{+}_{m} +u^{-}_{m})\Big|\\ &\leqslant \int^{1}_{0}\int |a||g(u^{0}_{m}+t(u^{+}_{m}+u^{-}_{m}))(u^{+}_{m} +u^{-}_{m})|dtdx\\ &\leqslant M|a|_{\frac{2N}{N+2}}\Big(\int |u^{+}_{m} +u^{-}_{m}|^\frac{N-2}{2N}dx\Big)^{2N/(N-2)}\\ &\leqslant C\|u^{+}_{m}+u^{-}_{m}\| . \end{align*} It then follows from (\ref{cero}) and the above inequality that $$ \int aG(u_m)=\int aG(u^{0}_{m})+o(1)\;\mbox{as}\;m\to\infty . $$ Thus, \[ \liminf_{m\to\infty}\int aG(u_m)\geqslant \liminf_{m\to\infty}\int aG(u^{0}_{m}) \geqslant \liminf_{\stackrel{\|u\|\to\infty}{u\in L_n}}\int aG(u)=\Lambda . \] Hence, \begin{equation} \liminf_{m\to\infty}\int aG(u_m)\geqslant\Lambda. \label{15} \end{equation} On the other hand, by $J_{\lambda_{n}}(u_m)\to c$ and (\ref{cero}) we have \begin{align*} c&= \lim_{m\to\infty}J_{\lambda_{n}}(u_m)=\lim_{m\to\infty} \Big\{\int|\nabla u_m|^2-\lambda_n\int hu^{2}_{m}-\int aG(u_m)\Big\}\\ &= \lim_{m\to\infty} \Big\{\int |\nabla u^{+}_{m}|-\lambda_n\int h(u^{+}_{m})^2 +\int |\nabla u^{-}_{m}|^2-\lambda_n\int h|u^{-}_{m}|^2-\int a G(u_m)\Big\}\\ &= -\lim_{m\to\infty}\int aG(u_m) . \end{align*} By hypothesis, $c>-\Lambda$, thus $\lim_{m\to\infty}\int aG(u_m)<\Lambda $, which contradicts (\ref{15}). Therefore, $(u_m)$ must be bounded if $c\in (-\Lambda,\infty)$. To show the existence of a weak solution we use the Linking Theorem, Theorem \ref{link} (see \cite{EL}). Define $(E_n)^{\perp}:=X_2$, then $\lambda_{n+1}\leqslant\frac{\int|\nabla u|^2}{\int hu^2}$ for all $u\in X_2$. So, given any $u\in X_2$, it follows from (\ref{StrongRes2}) that \begin{align*} J_{\lambda_{n}}(u)&= \frac{1}{2}\int|\nabla u|^2-\frac{1}{2}\lambda_n\int hu^2-\int aG(u)\\ &\geqslant \frac{1}{2}\big( 1-\frac{\lambda_{n}}{\lambda_{n+1}}\big) \|u\|^2 - \int\frac{a}{|a|_{L^{1}}}\Lambda\\ &\geqslant -\Lambda; \end{align*} i.e., $J_{\lambda_{n}}(u)\geqslant\gamma\in {\mathbb{R}}$ for all $u\in X_2$ with $\gamma:=-\Lambda$. So condition $(I_1)$ in Theorem \ref{link} holds. On the other hand, from $$ J_{\lambda_{n}}(u) \geqslant \frac{1}{2}\big(1-\frac{\lambda_{n}}{\lambda_{n+1}}\big) \|u\|^2+\gamma , $$ it follows that $J_{\lambda_{n}}(u)\to\infty$ as $\|u\|\to\infty$ (since $\lambda_{n+1}>\lambda_n$), and therefore ($I_2$) in Theorem \ref{link} also holds. Now, define $X_1:=E_n$. If $n>1$, we may write $u=u_1+u_0$ where $u_1\in E_{n-1}$ and $u_0\in L_n$. For $u\in E_{n-1}$ we know that $\lambda_{n-1}\geqslant \frac{\int |\nabla u|^2}{\int hu^2}$; thus, for $u\in X_1$, \begin{align*} J_{\lambda_{n}}(u)&= \frac{1}{2}\int |\nabla (u_0+u_1)|^2-\frac{\lambda_n}{2} \int h(u_0+u_1)^2-\int aG(u)\\ &= \frac{1}{2}\int |\nabla u_1|^2-\frac{\lambda_n}{2}\int hu^{2}_{1}-\int aG(u)\\ &\leqslant - \frac{1}{2}\big(\frac{\lambda_n}{\lambda_{n-1}}-1 \big)\|u_1\|^2 -\int(aG(u)-aG(u_0))-\int a G(u_0). \end{align*} From $G(u)-G(u_0)=\int^{1}_{0}g(u_0+tu_1)u_1dt$ we get that $|G(u)-G(u_0)|\leqslant M|u_1|$, so that the above inequality becomes \begin{equation} J_{\lambda_{n}}(u)\leqslant - \frac{1}{2}\big(\frac{\lambda_n}{\lambda_{n-1}}-1 \big)\|u_1\|^2 + C\|u_1\| -\int aG(u_0). \label{strong1} \end{equation} It then follows from (\ref{strong1}) and the condition (\ref{MyStrongRes}) that there exists a real constant $\beta$ such that $J_{\lambda_{n}}(u)\leqslant \beta$ for all $u\in X_1$ which is condition ($I_0$) in Theorem \ref{link}. For $n=1$ we have $$ J_{\lambda_{n}}(u) = -\int aG(u)\quad\mbox{for}\ u\in X_1, $$ which, by (\ref{MyStrongRes}), yields $\beta\in {\mathbb{R}}$ such that $J_{\lambda_{n}}(u)\leqslant\beta$ for all $u\in X_1$ i.e., ($I_0$) in Theorem \ref{link} holds. Therefore, $J_{\lambda_{n}}(u)$ has a critical value $b\geqslant\gamma$ and the theorem is established. \end{proof} As a consequence of Theorem \ref{Thm3}, we have the following statement. \begin{coro} \label{coro 3.6} Let $g:{\bf R}\to {\bf R}$ be a continuous function satisfying $\lim_{|s|\to\infty}g(s)=0$. Suppose that $a>0$ a.e., $$ G(s)\to\xi\in{\mathbb{R}}\quad\mbox{as }|s|\to\infty,\quad \mbox{and}\quad G(s)\leq\xi\quad\mbox{for every }s\in{\mathbb{R}}. $$ Then problem \eqref{Pl0} has a weak solution. \end{coro} This corollary follows from Theorem \ref{Thm3}, the unique continuation property of the eigenfunctions, and the Lebesgue dominated convergence theorem. \begin{thebibliography}{00} \bibitem{AR} {\sc N. 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