\documentclass[twoside]{article} \usepackage{amsmath,amsfonts,amssymb} \pagestyle{myheadings} \markboth{\hfil Existence and multiplicity results \hfil EJDE--2002/25} {EJDE--2002/25\hfil D. G. Costa, Y. Guo, \& M. Ramos \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 25, pp. 1--15. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence and multiplicity results for nonlinear elliptic problems in $\mathbb{R}^N$ with an indefinite functional % \thanks{ {\em Mathematics Subject Classifications:} 35J25, 35J20, 58E05. \hfil\break\indent {\em Key words:} Superlinear elliptic problems, Morse index, minimax methods. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted June 23, 2001. Published March 4, 2002.} } \date{} % \author{David G. Costa, Yuxia Guo, \& Miguel Ramos} \maketitle \begin{abstract} We prove the existence of a nontrivial solution for the nonlinear elliptic problem $-\Delta u=\lambda h(x)u + a(x)g(u)$ in $\mathbb{R}^N$, where $g$ is superlinear near zero and near infinity, $a(x)$ changes sign, $\lambda$ is positive, and $h(x)\geqslant 0$ is a weight function. For $g$ odd, we prove the existence of an infinite number of solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} We consider the elliptic problem $$-\Delta u = \lambda h(x) u + a(x) g(u), \quad u\in {\cal D}^{1,2}(\mathbb{R}^N) \label{Plambda}$$ where $N\geqslant 3$, $\lambda >0$ is a real parameter, $g\in C^1(\mathbb{R};\mathbb{R})$, $a\in C^1(\mathbb{R}^N)$ changes sign, and $$0\leqslant h(x)\in L^{N/2}\cap L^{\infty}\cap C^1(\mathbb{R}^N), \quad h\not\equiv 0.\label{H1}$$ Here, we denote by ${\cal D}^{1,2}(\mathbb{R}^N)$ the closure of ${\cal D}(\mathbb{R}^N)$ with respect to the norm $(\int_{\mathbb{R}^N}|\nabla u|^2)^{1/2}$. The corresponding problem over a {\em bounded} domain $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$ has been considered by several authors in recent years; we refer the reader to \cite{AdP,AT1,AT2,BCN,DR,RTT} and references therein. However, not so much seems to be known in such indefinite situation where the domain is unbounded. Here we consider the setting introduced in \cite{CT} and assume the following: \begin{gather} a(x)\in C^{1}(\mathbb{R}^N) \quad \mbox{with}\quad \limsup_{|x|\to \infty}a(x)<0,\label{A1}\\ \nabla a(x)\neq 0,\quad \forall x: a(x)=0.\label{A2} \end{gather} We explicitly observe that $a(x)$ may be unbounded. Concerning the function $g$, we assume that $g$ is {\em superlinear}, {\em subcritical} and a further {\em sign condition}. More precisely, we assume $g\in C^1(\mathbb{R},\mathbb{R})$ and that for some positive constants $\ell$ and $C$, it holds \begin{gather} g(0)=g'(0)=0,\label{G1} \\ \lim_{|s|\to\infty}\frac{g'(s)}{|s|^{p-2}}=\ell, \quad \mbox{with}\quad 20$(see also Remark 3.2 at the end of section 3). To state Theorem \ref{thm1}, let us first recall the well-known fact that (\ref{H1}) implies the existence of a sequence$\mu_i(h)$of eigenvalues of the linear problem $$-\Delta u=\mu h(x)u, \quad u\in {\cal D}^{1,2}(\mathbb{R}^N)$$ such that$0 < \mu_1(h) < \mu_2(h)\leqslant \mu_3(h)\leqslant \ldots \to \infty$. This is due to the fact that the map$u\mapsto \int hu^2$is compact in${\cal D}^{1,2}(\mathbb{R}^N)$. \begin{theorem} \label{thm1} Consider problem {\rm (\ref{Plambda})}, suppose that$a(x)$changes sign and that {\rm (\ref{H1})-(\ref{G4})} hold. If$\mu_k(h)<\lambda < \mu_{k+1}(h)$for some$k\geqslant 1$then {\rm (\ref{Plambda})} has a nonzero solution. \end{theorem} We point out that the existence of {\em positive} solutions for (\ref{Plambda}) was proved in \cite{CT} for$\lambda < \Lambda$, with$\Lambda$lying in a right neighborhood of the {\em first} eigenvalue$\mu_1(h)$. Since {\em positive} solutions cannot exist in general if$\lambda > \Lambda$(see \cite{CT}), it is natural to ask whether other (possibly sign-changing) solutions exist for any$\lambda>0$. Theorem \ref{thm1} answers this question in the affirmative whenever$\lambda$is not one of the eigenvalues$\mu_j(h)$. It should also be noted that, in contrast with the assumptions made in \cite{CT} (see also the references therein), we assume that the function$a(x)$satisfies condition (\ref{A2}) instead of having a thick" zero set (see Lemma 1.2 in \cite{CT}) and, for Theorem \ref{thm1}, we do not assume that$g(s)$behaves like a superlinear power at zero. Condition (\ref{A2}) was introduced in \cite{BCN} and later used in \cite{DR,RTT}. Typically, Theorem \ref{thm1} applies to nonlinearities such as, for example,$g(s)=|s|^{p-2}s+\theta(s)|s|^{q-2}s$where$2 0,\quad \forall s\in \mathbb{R},\; s\neq 0, \label{G3'}\\ \lim_{s\to 0}\frac{sg(s)}{|s|^q}=\ell_0\in (0,\infty)\quad \mbox{ for some } q>0 \label{G4'} \end{gather} and prove the following. \begin{theorem} \label{thm2} Consider problem {\rm (\ref{Plambda})}, suppose that $a(x)$ changes sign, $g$ is odd and that {\rm (\ref{A1})-(\ref{G2})} and {\rm (\ref{H1'})-(\ref{G4'})} hold. Then, for any $\lambda \in \mathbb{R}$, problem {\rm (\ref{Plambda})} has infinitely many solutions. \end{theorem} We observe that similar conclusions for {\em bounded} domains $\Omega$ were obtained in \cite{AT1} for odd $g$ and in \cite{B,T1} for perturbations of an odd function $g$, under a different set of assumptions (again, the thick zero set" assumption on $a(x)$ was used by those authors). Concerning (\ref{H1'}), we mention that the assumption on the positivity can be replaced by the weaker assumption that $h>0$ a.e. over the bounded open set $\{ x: a(x)>0\}$ (see Lemma \ref{epsilon-estimate} in section 2). The proofs of Theorems \ref{thm1} and \ref{thm2} are given in sections 3 and 4, respectively. In order to prove our results, we have to face the lack of compactness due to the unboundedness of the domain and the fact that $a(x)$ changes sign. We overcome this by constructing an appropriate sequence of solutions of the equation in (\ref{Plambda}), lying in $H_0^1(\Omega_n)$, where $\Omega_n=B_{R_n}(0)\subset \mathbb{R}^N$ with $R_n\to \infty$; the estimates on the Morse indices of these solutions \cite{DR} insure the boundedness of the sequence and allows us to take its limit in the space ${\cal D}^{1,2}(\mathbb{R}^N)$ (weak limit, in case of Theorem \ref{thm1}; strong limit, in case of Theorem 2). Roughly speaking, our argument relies on the fact that the Morse index estimates provide Palais-Smale sequences for (\ref{Plambda}) with the additional property that the sequence is bounded in $L^{\infty}(\mathbb{R}^N)$; together with a version of Brezis-Lieb lemma proved in section 2, this yields compactness for the problem. Regarding the multiplicity result, it will follow from the observation that, under the assumptions of Theorem 2, these Palais-Smale sequences can be constructed at arbitrarily large levels of energy. We mention that, although our results in section 2 suggest that perhaps one could work directly in a convenient Banach subspace of the Hilbert space ${\cal D}^{1,2}(\mathbb{R}^N)$, we prefer to use the approximated sequence of Hilbert spaces $H_0^1(\Omega_n)$. This is mainly because, in the latter case, Morse index estimates in Hilbert spaces and their connections with blow-up techniques (see \cite{DR}) can be directly applied to our problem without the need of additional theoretical developments. \section{Preliminary results} We recall that we denote by ${\cal D}^{1,2}(\mathbb{R}^N)$ the closure of ${\cal D}(\mathbb{R}^N)$ with respect to the norm $\|u\|=(\int_{\mathbb{R}^N}|\nabla u|^2)^{1/2}$. The notation $\|u\|_r$ ($1\leqslant r\leqslant \infty$) stands for the norm in $L^r$ spaces and the Sobolev continuous immersion of ${\cal D}^{1,2}(\mathbb{R}^N)$ into $L^{2^{\star}}(\mathbb{R}^N)$ will be repeatedly used (see \cite{W}). As mentioned in the Introduction, we prove Theorems 1 and 2 through an approximation argument in bounded open balls of $\mathbb{R}^N$. Under assumption (\ref{H1}), we denote by $(\mu_i(h))_{i\in \mathbb{N}}$ and $(\mu_i^R(h))_{i\in \mathbb{N}}$ (for each $R>0$) the sequence of eigenvalues of the problems \begin{gather*} -\Delta u =\mu h(x)u, \quad u\in {\cal D}^{1,2}(\mathbb{R}^N) \quad \mbox{and }\\ -\Delta u =\mu^R h(x)u, \quad u\in H_0^1(B_R(0)). \end{gather*} \begin{lemma}\label{eigenvalues} Given $k\in \mathbb{N}$ and $\varepsilon >0$ there exists $R_0>0$ such that $$|\mu_k(h)-\mu_k^R(h)| < \varepsilon,\quad \forall R\geqslant R_0.$$ \end{lemma} \paragraph{Proof.} We first recall that the theory of compact symmetric operators on Hilbert spaces implies the following variational characterization of $\mu_k(h)$: $$\mu_k(h)= \min_{{\rm dim}X=k}\max_{u\in X,u\neq 0} \frac{\int_{\mathbb{R}^N}|\nabla u|^2}{\int_{\mathbb{R}^N}hu^2} = \max_{u\in X_k,u\neq 0} \frac{\int_{\mathbb{R}^N}|\nabla u|^2}{\int_{\mathbb{R}^N}hu^2}\,,$$ where we denote by $X_k$ the eigenspace associated with the first $k$ eigenvalues and $X$ runs through the $k$-dimensional subspaces of ${\cal D}^{1,2}(\mathbb{R}^N)$. A similar formula holds for $\mu_k^R(h)$ where, now, $X$ is a subspace of $H_0^1(B_R(0))$. Since $H_0^1(B_R(0))\subset {\cal D}^{1,2}(\mathbb{R}^N)$, this shows in particular that $\mu_k(h) \leqslant \mu_k^R(h)$. On the other hand, let $X_k$ be spanned by $\{ \varphi_1,\ldots,\varphi_k\}$ and denote by $\Psi_R$ a smooth function $\Psi_R\in {\cal D}(\mathbb{R}^N)$ such that $0\leqslant \Psi_R \leqslant 1$, $\Psi_R=1$ in $B_R(0)$, $\Psi_R=0$ in $\mathbb{R}^N\setminus B_{2R}(0)$ and $\|\nabla \Psi_R\|_{\infty}\leqslant CR^{-1}$ for every $R>0$. By unique continuation, the space spanned by $\{ \Psi_R\varphi_1,\ldots,\Psi_R\varphi_k\}$ has dimension $k$. Therefore, in view of the above variational characterizations, the lemma will be proved if we show that, for every large $R$, $$\label{eigenvalues-1} \frac{\int|\nabla( u\Psi_R)|^2}{\int h(u\Psi_R)^2} \leqslant \varepsilon + \frac{\int|\nabla u|^2}{\int h u^2}, \quad \forall u \in X_k.$$ Except otherwise indicated, all integrals are taken over the whole space $\mathbb{R}^N$. On the other hand, (\ref{eigenvalues-1}) will follow once we show that $$\label{eigenvalues-2} \frac{\int|\nabla( u\Psi_R)|^2}{\int h(u\Psi_R)^2} - \frac{\int|\nabla u|^2}{\int h u^2}\to 0 \quad \mbox{ as } \quad R\to \infty,$$ uniformly for $u\in X_k$, $\int|\nabla u|^2 =1$. In order to prove (\ref{eigenvalues-2}), let $R_n$ be any sequence such that $R_n \to \infty$, denote $\Psi_n=\Psi_{R_n}$ and let $u_n\in X_k$ be any sequence such that $\int|\nabla u_n|^2 =1$. Since $X_k$ is finite dimensional, $$\varepsilon_n:=\int_{\mathbb{R}^N\setminus B_{R_n}(0)} |u_n|^{2^{\star}}\to 0.$$ Similarly, $$\int |\nabla u_n|^2 \Psi_n^2 \to 1 \quad \mbox{ and } \quad \liminf_{n\to \infty} (\int hu_n^2\; \int h u_n^2\Psi_n^2)>0.$$ Recalling that $\int|\nabla u_n|^2 =1$, it remains to prove that $$\label{eigenvalues-3} \int|\nabla( u_n\Psi_n)|^2\; \int h u_n^2 - \int h u_n^2\Psi_n^2 \to 0.$$ Observing that, by H\"older inequality, $$\int hu_n^2(1-\Psi_n^2)\leqslant \int_{\mathbb{R}^N\setminus B_{R_n}(0)}hu_n^2 \leqslant \|h\|_{N/2} \;\varepsilon_n^{2/2^{\star}}\to 0,$$ the expression in (\ref{eigenvalues-3}) can be written as $$(\int|\nabla( u_n\Psi_n)|^2-1)\; \int h u_n^2 +\mbox{o}(1),$$ where $\mbox{o}(1)\to 0$ as $n\to \infty$. Finally, we observe that \begin{align*} |\int |\nabla (u_n\Psi_n)|^2-1|\leqslant & \mbox{o}(1) + \int u_n^2 |\nabla \Psi_n|^2 + 2 \int |u_n|\,|\Psi_n|\, |\nabla u_n|\,|\nabla \Psi_n|\\ \leqslant & \mbox{o}(1) + CR_n^{-2} \int_{\mathbb{R}^N\setminus B_{R_n}(0)}u_n^2\\ &+ 2 (R_n^{-2} \int_{\mathbb{R}^N\setminus B_{R_n}(0)}u_n^2)^{1/2}\; (\int|\nabla u_n|^2)^{1/2}\\ \leqslant & \mbox{o}(1) + C' (\varepsilon_n^{2/2^{\star}} + \varepsilon_n^{1/2^{\star}})\to 0 \end{align*} and the lemma follows.\hfill$\Box$\smallskip For any $R>0$ and $k\in \mathbb{N}$ we denote by $X_{k,R}$ the closure in $H_0^1(B_R(0))$ of the eigenspaces associated with the eigenvalues $\mu_i^R(h)$ for $i\geqslant k+1$. \begin{lemma}\label{epsilon-estimate} Assume {\rm (\ref{H1'})} and let $p\in [1,2^{\star})$. Given $\varepsilon >0$ and $k_1\in \mathbb{N}$ there exist $R_0 >0$ and $k\in \mathbb{N}$, $k\geqslant k_1$, such that $$\int_{B_1(0)}|u|^{p} \leqslant \varepsilon \Big( \int_{B_R(0)} |\nabla u|^2 \Big)^{p/2} \quad \forall R\geqslant R_0 \, \forall u \in X_{k,R}.$$ \end{lemma} \paragraph{Proof.} Assuming the contrary, there exist sequences $k_n\to \infty$, $R_n \to \infty$, $u_n\in X_{k_n,R_n}$ such that $\int_{B_{R_n}(0)} |\nabla u_n|^2 =1$ and $\int_{B_1(0)}|u_n|^{p} \geqslant \varepsilon$. Up to a subsequence, we may assume that $(u_n)$ converges weakly in ${\cal D}^{1,2}(\mathbb{R}^N)$ to some function $u$ and that $u_n \to u$ strongly in $L^{p}(B_1(0))$. In particular, $\int_{B_1(0)}|u|^{p}\geqslant \varepsilon$. Since $u_n\in X_{k_n,R_n}$, it follows from the definition of $\mu_{k_n}^{R_n}(h)$ that $$\label{definition-of-eigenvalues} 1=\int_{B_{R_n}(0)} |\nabla u_n|^2\geqslant \mu_{k_n}^{R_n}(h) \int_{B_{R_n}(0)} hu_n^2.$$ On the other hand, as observed at the beginning of the proof of Lemma \ref{eigenvalues}, for every $n$ we have that $$\label{monotonicity-of-eigenvalues} \mu_{k_n}^{R_n}(h)\geqslant \mu_{k_n}(h)\to \infty.$$ Combining this with (\ref{definition-of-eigenvalues}) yields that $$\int_{B_1(0)} hu^2 = \lim_{n\to \infty} \int_{B_1(0)} hu_n^2\leqslant \lim_{n\to \infty} \int_{B_{R_n}(0)} hu_n^2=0.$$ Since $h>0$ in $B_1(0)$, this implies $u=0$ in $B_1(0)$, contradicting the fact that $\int_{B_1(0)}|u|^{p}\geqslant \varepsilon$.\hfill$\Box$\smallskip We end this section with a version of the well-known Brezis-Lieb lemma which is suitable for our purposes. We first recall the following. \begin{lemma}[Brezis-Lieb lemma] Let $H: \mathbb{R}\to\mathbb{R}$ be continuous, $H\geqslant 0$ and satisfy $$\label{a-kind-of-convexity} \forall \varepsilon \; \exists C_{\varepsilon}:\; |H(s+t)-H(s)|\leqslant \varepsilon|H(s)|+C_{\varepsilon}|H(t)|,\quad \forall s,t\in \mathbb{R}.$$ For a given sequence $(w_n)$ of measurable functions in $\mathbb{R}^N$, suppose $w_n \to w$ a.e., $\sup_n\int H(w_n) <\infty$ and $\int H(w) <\infty$. Then $\sup_n\int H(w_n-w) <\infty$ and $$\int |H(w_n)-H(w)-H(w_n-w)|\to 0 \quad \mbox{ as } \; n\to \infty.$$ \end{lemma} \paragraph{Proof.} An inspection of the proof as given for example in Lemma 1.32 of \cite{W} shows that condition (\ref{a-kind-of-convexity}) is sufficient to deduce the conclusion of the lemma.\hfill$\Box$\smallskip The next lemma provides a sufficient condition for (\ref{a-kind-of-convexity}) to hold. \begin{proposition}\label{getting-a-kind-of-convexity} Let $H: \mathbb{R} \to \mathbb{R}$ be continuous, $H(s)>0$ for all $s\neq 0$ and suppose that for some 0 0 \quad \mbox{ and } \quad \lim_{|s|\to \infty}\frac{H(s)}{|s|^p} =\ell_{\infty} >0. $$Then H satisfies condition {\rm (\ref{a-kind-of-convexity})}. \end{proposition} \paragraph{Proof.} Step 1. Given \varepsilon >0, fix 0 < \varepsilon_0 < R_0 and C_2/C_1<1+\varepsilon, C'_2/C'_1<1+\varepsilon, in such a way that \begin{gather*} C_1|s|^q\leqslant H(s) \leqslant C_2|s|^q, \quad \forall |s| \leqslant 2\varepsilon_0, C'_1|s|^p\leqslant H(s) \leqslant C'_2|s|^p, \quad \forall |s| \geqslant R_0. \end{gather*} Then condition (\ref{a-kind-of-convexity}) is trivially satisfied in case |t|\leqslant \varepsilon_0 and |s|\leqslant \varepsilon_0; and also in case |t|\geqslant R_0, |s|\geqslant R_0 and |t+s|\geqslant R_0. \noindent Step 2. Take \lambda>0 such that $$\label{tricky-estimate-1} H(t)\geqslant \lambda, \quad \forall t: \; \varepsilon_0 \leqslant |t|\leqslant R_0.$$ Next, choose \delta \in ]0,\varepsilon_0[ in such a way that \begin{gather}\label{tricky-estimate-2} |H(s+t)-H(s)| \leqslant \varepsilon\lambda, \quad \forall |s|\leqslant R_0,\; \forall |t|\leqslant \delta, \\ \label{tricky-estimate-3} |H(s+t)-H(s)| \leqslant \varepsilon H(s), \quad \forall |s|\geqslant R_0,\; \forall |t|\leqslant \delta. \end{gather} \noindent Step 3. We prove condition (\ref{a-kind-of-convexity}) in the case |t|\leqslant \delta. As mentioned in step 1 above, we may assume that |s|\geqslant \varepsilon_0. Thus, if |s|\geqslant R_0 the conclusion follows from (\ref{tricky-estimate-3}) while if |s|\leqslant R_0 the conclusion follows from (\ref{tricky-estimate-1}) and (\ref{tricky-estimate-2}). \noindent Step 4. It remains to check the case where |t|\geqslant \delta. First, we observe that there exists C_{\delta}>0 and C_3>0 such that $$\label{tricky-estimate-4} C_{\delta}|t|^p \leqslant H(t), \; \forall |t|\geqslant \delta \mbox{ and } \; |H(t)|=H(t)\leqslant C_3(1+|t|^p), \; \forall t\in \mathbb{R}.$$ Suppose then that |t|\geqslant \delta. In case |s|\leqslant R_0+|t| we deduce from (\ref{tricky-estimate-4}) that$$ |H(s+t)-H(s)| \leqslant H(s+t)+H(s) \leqslant C_4(1+|t|^p) \leqslant C_{\varepsilon}H(t),$$for some large constant C_{\varepsilon} >0. Finally, in case |s|\geqslant R_0 +|t| we have that |s|\geqslant R_0, |s+t|\geqslant R_0 and we can argue as in step 1, thanks to the first inequality in (\ref{tricky-estimate-4}).\hfill\Box \section{Proof of Theorem \ref{thm1}} Throughout this section we assume that the conditions in Theorem \ref{thm1} are satisfied. As a consequence of Lemma \ref{eigenvalues}, it follows from the assumptions of Theorem 1 that we can fix R so large that the constant \lambda appearing in (\ref{Plambda}) satisfies, for every large R>0, $$\label{nonresonance} \mu_k^R(h) < \lambda < \mu_{k+1}^R(h).$$ Take any sequence R_n\to \infty and let \Omega_n=B_{R_n}(0). We denote by I the functional $$\label{energy-functional} I(u)=\frac{1}{2}\int(|\nabla u|^2-\lambda h(x)u^2) - \int a(x)G(u).$$ Under assumptions (\ref{H1}), (\ref{A1}), (\ref{G1}), (\ref{G2}) and for each n\in \mathbb{N}, I is a C^2 functional over H_0^1(\Omega_n) and its critical points in H_0^1(\Omega_n) correspond to solutions of (\ref{Plambda}) lying in H_0^1(\Omega_n). For any such critical point u, we denote by m(u) the Morse index of u with respect to I, that is, the supremum of the dimensions of the linear subspaces of H_0^1(\Omega_n) on which the quadratic form D^2I(u) is negative definite. \begin{proposition}\label{bounded-domain} Under the assumptions of Theorem \ref{thm1}, for every n the equation in {\rm (\ref{Plambda})} has a nonzero solution u_n\in H_0^1(\Omega_n) and the following holds:\\[1mm] {\rm (i)} (u_n) is bounded in L^{\infty}(\Omega_n).\\[1mm] {\rm (ii)} Either m(u_n)\leqslant k-1 for every n or else \limsup_{n\to \infty}I(u_n) >0. \end{proposition} \paragraph{Proof.} Step 1. Since the proof is based on \cite{DR,RTT}, we shall be sketchy. At first we observe that the existence of nonzero solutions (u_n) follows straightforwardly from the main theorem in \cite{RTT}, which was proved by means of a truncation argument and the use of a critical point theorem in \cite{LL,LW}. We point out that, since a(x) changes sign, the truncation argument is needed in order to insure the Palais-Smale condition for the functional I over H_0^1(\Omega_n) as well as to obtain the required geometric condition on I. At this point, assumptions (\ref{A1}), (\ref{G3}) and (\ref{G4}) are not used; regarding the unique continuation property mentioned in Lemma 2 of \cite{RTT}, we also observe that the equation -\Delta u =\mu h(x) u can be written as Ku=u/\mu where Ku=(-\Delta)^{-1}(h(x)u) in H_0^1(\Omega_n), so that the proof of the quoted lemma remains unchanged. \noindent Step 2. By construction, the sequence of the Morse indices of these solutions is bounded (m(u_n)\leqslant k for every n, see \cite{DR,RTT}). Also, our regularity assumptions imply that u_n\in C(\overline{\Omega_n})\cap C^2(\Omega_n). Assume by contradiction that$$ M_n:=\|u_n\|_{\infty}=\max_{\Omega_n}u_n=u_n(x_n)\to +\infty$$for some x_n \in \Omega_n (the case where \|u_n\|_{\infty}=\max_{\Omega_n}(-u_n) is similar). Since \Delta u_n(x_n) \leqslant 0, the equation in (\ref{Plambda}) shows that$$ a^{-}(x_n)g(M_n) \leqslant CM_n + a^+(x_n)g(M_n),$$where we denoted a^+:=\max\{a,0\} and a^-:=\max\{-a,0\}. From assumption (\ref{A1}) we see that (x_n) is bounded. Thus, up to a subsequence, we can assume that x_n\to x_0 \in \mathbb{R}^N and a(x_0)\geqslant 0. At this point, the blow-up argument in section 3 of \cite{RTT} can be applied, leading to a contradiction. Indeed, since m(u_n)\leqslant k and \|u_n\|_{\infty} \to \infty, it is shown in \cite{RTT} that the sequence v_n(x)=u_n(\lambda_n x+x_n)/M_n, with \lambda_n=M_n^{(2-p)/2} or \lambda_n=M_n^{(2-p)/3} depending on whether a(x_0)>0 or a(x_0)=0, respectively, converges uniformly in compact sets to 0. Since, by definition, v_n(0)=1, this is impossible and therefore part (i) in Proposition \ref{bounded-domain} is proved. \noindent Step 3. Finally, as explained in Proposition 2 of \cite{DR}, each solution u_n can be chosen in such a way that either m(u_n)\leqslant k-1 or else, for some small r_n>0, $$\label{energy-level-estimate} I(u_n)\geqslant \inf\{ I(u): \; u\in X_{k,n},\;\|u\|=r_n\},$$ where we denote by X_{k,n} the closure of the eigenspaces associated with the eigenvalues \mu_i^{R_n}(h) for i\geqslant k+1. Actually, by the construction in \cite{DR}, (\ref{energy-level-estimate}) holds for a modified functional$$ \widetilde{I}(u)=\frac{1}{2}\int(|\nabla u|^2-\lambda h(x)u^2) - \int a^+(x)G(u)+ \int a^-(x)\widetilde{G}(u),where \widetilde{G} is a truncation of the function G which still satisfies (\ref{G3}). Thus, (\ref{energy-level-estimate}) should be written as $$\label{true-energy-level-estimate} I(u_n)\geqslant \inf\{ \widetilde{I}(u): \; u\in X_{k,n},\;\|u\|=r_n\}.$$ So, in order to prove (ii) in Proposition \ref{bounded-domain} it is enough to show that the right hand side of (\ref{true-energy-level-estimate}) can be bounded below by some positive constant which does not depend on n. Now, to show this, let u be any function in X_{k,n}. From (\ref{nonresonance}) we see that, for some constant \eta >0 independent of n, \begin{align*} \widetilde{I}(u) & \geqslant \eta \|u\|^2 - \int a^+G(u)+ \int a^-(x)\widetilde{G}(u) \\ & \geqslant \eta \|u\|^2 - \int a^+G(u) \end{align*} where we have used (\ref{G3}). To estimate the above integral term, we use the fact that (\ref{G1}) and (\ref{G2}) imply that |G(s)|\leqslant \varepsilon s^2 +C_{\varepsilon} |s|^{p} for any \varepsilon >0. As a consequence, and since a^+ has compact support (cf. (\ref{A1})), we obtain \int a^+u^2 \leqslant C(\int_{\{a>0\}} |u|^{2^{\star}})^{2/2^{\star}} \leqslant C(\int_{\Omega_n} |u|^{2^{\star}})^{2/2^{\star}} \leqslant C'\|u\|^2,$$and a similar estimate for \int a^{+}|u|^p. Therefore, $$\label{the-energy-is-bounded-below} \widetilde{I}(u) \geqslant \|u\|^2 \;(\eta - \varepsilon C'-C'_{\varepsilon}\|u\|^{p-2}),$$ where C' and C'_{\varepsilon} are independent of n. By choosing \varepsilon small we see that we can select r_n independently of n. This proves the claim and completes the proof of the proposition.\hfill\Box\smallskip Now we can complete the proof of Theorem \ref{thm1}. \paragraph{Proof of Theorem \ref{thm1} completed.} Let (u_n) be given by Proposition \ref{bounded-domain}. If we multiply the equation in (\ref{Plambda}) by u_n and integrate over \Omega_n we obtain $$\label{multiply-and-integrate} \int |\nabla u_n|^2+ \int a^-g(u_n)u_n = \lambda \int h u_n^2 + \int a^+ g(u_n)u_n.$$ Except othervise indicated, all integrals are taken over \mathbb{R}^N, by extending the solutions as zero outside \Omega_n. Since (\|u_n\|_{\infty}) is bounded, it follows from (\ref{multiply-and-integrate}) and (\ref{A1}) that $$\label{consequence-of-multiply-and-integrate} \int |\nabla u_n|^2+ \int g(u_n)u_n \leqslant \lambda \int h u_n^2 + C.$$ We claim that (\|u_n\|) is bounded. Indeed, suppose t_n:=\|u_n\|\to \infty and denote v_n:=u_n/t_n. Up to a subsequence, v_n\to v weakly in {\cal D}^{1,2}(\mathbb{R}^N) and a.\ e. Fix any function \varphi \in {\cal D}(\mathbb{R}^N). If we multiply the equation in (\ref{Plambda}) by \frac{au_n\varphi}{t_n^2} and integrate we see that $$\label{multiply-and-integrate-by-au} \int a^2\; \frac{g(u_n)u_n}{t_n^2}\; \varphi \leqslant C',$$ for some positive constant C' depending on \varphi. Since g is superlinear at infinity (cf. (\ref{G2})), we deduce from (\ref{multiply-and-integrate-by-au}) that a^2|v|\varphi=0. Since \varphi is arbitrary and since a vanishes on a set of measure zero, we obtain that v=0. Going back to (\ref{consequence-of-multiply-and-integrate}) and using the fact that the map u\mapsto \int hu^2 is compact in {\cal D}^{1,2}(\mathbb{R}^N), we conclude that \|v_n\|^2\to 0 as n\to \infty, which is a contradiction. This proves that (\|u_n\|) is bounded. Up to a subsequence, let u be a weak limit of (u_n) in {\cal D}^{1,2}(\mathbb{R}^N) such that u_n \to u a.\ e. Using test functions in (\ref{Plambda}) we see that u is a solution of (\ref{Plambda}). It remains to show that u \neq 0. Suppose by contradiction that (u_n) converges weakly to 0 in {\cal D}^{1,2}(\mathbb{R}^N). Since a^+ has compact support, it follows from (\ref{multiply-and-integrate}) and (\ref{G1})-(\ref{G2}) that$$ \int |\nabla u_n|^2+ \int a^-g(u_n)u_n \to 0 $$and subsequently, using (\ref{A1}), that $$\label{strongly-to-zero} \int |\nabla u_n|^2\to 0 \quad \mbox{ and } \quad \int |a|\;g(u_n)u_n \to 0.$$ Using (\ref{G3})-(\ref{G4}), this implies that $$\label{energy-to-zero} I(u_n)=\frac{1}{2}(\int |\nabla u_n|^2 -\lambda \int hu_n^2) - \int aG(u_n) \to 0$$ as n\to \infty. So, Proposition \ref{bounded-domain} (ii) implies that m(u_n)\leqslant k-1 for every n. On the other hand, we have seen in the proof of Lemma \ref{eigenvalues} (see (\ref{eigenvalues-1})) that, since \lambda >\mu_k(h), there exist a k-dimensional space X\subset H_0^1(B_R(0)) (for some large R>0) and a constant \eta >0 such that $$\label{large-Morse-index-1} I''(0)(v,v) = \int |\nabla v|^2-\lambda \int h v^2 \leqslant -2\eta \int |\nabla v|^2,\quad \forall v\in X.$$ Observe also that, by elliptic regularity, X\subset L^{\infty}(B_R(0)). Therefore, since u_n\to 0 a.e.\ and (\|u_n\|_{\infty}) is bounded, Lebesgue's dominated convergence theorem implies that $$\label{large-Morse-index-2} \int g'(u_n)v^2 \to 0, \quad \mbox{ uniformly in } v\in X: \; \int |\nabla v|^2=1.$$ Combining (\ref{large-Morse-index-1}) and (\ref{large-Morse-index-2}) we conclude that, for large n,$$ I''(u_n)(v,v) = \int |\nabla v|^2-\lambda \int h v^2 -\int a g'(u_n)v^2 \leqslant -\eta \int |\nabla v|^2,\quad \forall v\in X.$$By definition, this says that m(u_n)\geqslant k for large n, which contradicts the fact that m(u_n)\leqslant k-1 and concludes the proof of Theorem \ref{thm1}.\hfill\Box \begin{remark} \label{rmk3.2} \rm Under the assumptions of Theorem \ref{thm1}, suppose a(x) is bounded. In this case, it is sufficient to assume that the inequality in (\ref{G4}) holds for all |s|\leqslant \delta, for some \delta >0. Indeed, if a(x) is bounded the conclusion in (\ref{energy-to-zero}) is a consequence of the fact that \int |\nabla u_n|^2\to 0, \int g(u_n)u_n\to 0 (as follows from (\ref{strongly-to-zero})) and \begin{eqnarray*} \int |G(u_n)|&=&\int_{\{|u_n|\leqslant \delta\}} G(u_n)+\int_{\{|u_n|\geqslant \delta\}} G(u_n)\nonumber\\ &\leqslant& C\int g(u_n)u_n + C_{\delta}\int |u_n|^{2^{\star}}\to 0. \end{eqnarray*} \end{remark} \begin{remark} \label{rmk3.3}\rm In the same manner, one can also treat the case where the equation in (\ref{Plambda}) contains an extra term of the form -b(x)|u|^{r-2}u with b\geqslant 0, b bounded and r>2 small with respect to p. \end{remark} More generally, following \cite{DR}, the conclusion of Theorem \ref{thm1} holds for an equation of the form$$ -\Delta u = \lambda h(x) u + a(x) g(u)-b(x)f(u),$$where \lambda, h, a, g are as in Theorem \ref{thm1} and b\geqslant 0, b\in C^1\cap L^{\infty}(\mathbb{R}^N), f satisfies assumptions similar to (\ref{G1}), (\ref{G2}), (\ref{G4}) and, moreover,$$ |f'(s)| \leqslant C|s|^{r-2}\quad \mbox{and} \quad f(s)s-r\int_0^sf \leqslant Cs^2, \quad \forall |s|\geqslant 1,$$where C>0 and 20 we consider the modified problem $$-\Delta u = \lambda h(x) u + a^+(x)g(u) - a^-(x)g_j(u), \quad u\in H_0^1(B_R(0)),\label{PjR}$$ where a^{\pm}:=\max\{\pm a,0\}. The corresponding energy functional is {\em even} and is given by$$ I^j_R(u)=\frac{1}{2} \int_{B_R(0)} ( |\nabla u|^2 - \lambda h(x)u^2 ) - \int_{B_R(0)} a^+(x) G(u) + \int_{B_R(0)} a^-(x) G_j(u)$$for u \in H^1_0({B_R(0)}). For any \ell\in \mathbb{N} and R>0, we have the orthogonal sum$$ H^1_0({B_R(0)})= V_{\ell,R}\oplus X_{\ell,R},$$where V_{\ell,R} stands for the \ell-dimensional eigenspace associated with the first \ell eigenvalues \mu_i^R(h), i=1,\ldots,\ell. Finally, for any critical point u of I^j_R we denote by m^j_R(u) its Morse index. Recall that I^j_R satisfies the Palais-Smale condition over H^1_0({B_R(0)}) if every sequence (u_n)\subset H^1_0({B_R(0)}) such that (I^j_R(u_n)) is bounded and \|\nabla I^j_R(u_n)\| \to 0 has a convergent subsequence. The next lemma collects some facts that were proved in \cite[Prop.3]{RTT}. \begin{lemma}\label{results-from-DR} Assume {\rm (\ref{H1}), (\ref{A2})}, and {\rm (\ref{G2})}. Then \begin{enumerate} \item[(a)] For any j\in \mathbb{N} and R>0, the functional I^j_R satisfies the Palais-Smale condition over H^1_0({B_R(0)}). \item[(b)] For any j,\ell \in \mathbb{N} and R>0, I^j_R(u) \to -\infty as \|u\|\to \infty, u\in V_{\ell,R}. \item[(c)] For any \ell \in \mathbb{N} and R>0 there exist j_0\in \mathbb{N} and c>0 such that \|u\|_{\infty}\leqslant c for every j\geqslant j_0 and every critical point u of I^j_R such that m^j_R(u)\leqslant \ell. \end{enumerate} \end{lemma} Observe that (c) is an a-priori estimate in L^{\infty}(B_R(0)) for the solutions of (P)^{j}_{R} having bounded Morse index; for each fixed R, the estimate depends on R but {\em not} on j. Next, for every j,\ell\in \mathbb{N} and R>0 we let $$\label{definition-of-b^k} b^j_{\ell,R}=\inf \{ I^j_R(u): \; u\in X_{\ell-1,R},\; \|u\|=r_\ell\},$$ where r_\ell is a large constant to be chosen later. \begin{lemma}\label{estimate-from-below} Assume {\rm (\ref{H1'}), (\ref{A1}), (\ref{G1})--(\ref{G3})}, and let d\in \mathbb{R}. Then there exist \ell\in \mathbb{N} and R_0 >0 such that$$ b^j_{\ell,R} \geqslant d,\quad \forall j\in \mathbb{N}\;\; \forall R\geqslant R_0.$$\end{lemma} \paragraph{Proof.} The proof is similar to the one in (\ref{true-energy-level-estimate}) except that we now use Lemma \ref{epsilon-estimate}. Without loss of generality (cf. (\ref{A1})) we assume that \{ x: a(x)>0\} \subset B_1(0). At first we recall from (\ref{monotonicity-of-eigenvalues}) that we can fix \ell_1\in \mathbb{N} such that \mu_{\ell}^R(h) > \lambda for every R>0 and every \ell\geqslant \ell_1, so that, for some \eta >0,$$ \int_{B_R(0)} |\nabla u|^2 - \lambda\int_{B_R(0)} hu^2 \geqslant \eta \|u\|^2, \quad \forall \ell\geqslant \ell_1 \; \forall R>0\; \forall u\in X_{\ell-1,R}.$$Since, by (\ref{G2}), \int a^{+}G(u) \leqslant c\left( \int_{B_1(0)}|u|^p+1\right), we deduce that, for some positive constants c_1, c_2, $$\label{the-energy-is-uniformly-bounded-below} I^j_R(u) \geqslant c_1(c_2\|u\|^2 - \int_{B_1(0)}|u|^{p}-1),$$ for all j\in \mathbb{N}, R>0, \ell\geqslant \ell_1 and u\in X_{\ell-1,R}. Fix \epsilon >0 so small that $$\label{r-is-large-1} c_2-\varepsilon r^{p-2}=\frac{c_2}{2}$$ where r is given by $$\label{r-is-large-2} c_1c_2\frac{r^2}{2} -c_1\geqslant d.$$ For this value of \varepsilon, let \ell\geqslant \ell_1 and R_0 be given by Lemma \ref{epsilon-estimate}. Thanks to that lemma, we can rewrite (\ref{the-energy-is-uniformly-bounded-below}) as $$\label{the-energy-is-indeed-uniformly-bounded-below} I^j_R(u) \geqslant c_1 \|u\|^2(c_2 - \varepsilon \|u\|^{p-2})-c_1,\quad \forall j\in \mathbb{N}\; \forall R\geqslant R_0\; \forall u\in X_{\ell-1,R}.$$ From (\ref{r-is-large-1})--(\ref{the-energy-is-indeed-uniformly-bounded-below}) and the definition in (\ref{definition-of-b^k}) with r_{\ell}=r we conclude that$$ b^j_{\ell,R} \geqslant c_1 r^2 (c_2-\varepsilon r^{p-2})-c_1=c_1c_2\frac{r^2}{2}-c_1 \geqslant d$$for every j\in \mathbb{N} and R\geqslant R_0. \hfill\Box\smallskip In view of the above lemmas we can now prove the existence of a suitable sequence of approximating solutions to our original problem (\ref{Plambda}). \begin{proposition}\label{approximated-solutions} Assume {\rm (\ref{H1'}), (\ref{A1}), (\ref{A2}), (\ref{G1})--(\ref{G3})}, and let d\in \mathbb{R}. Then there exist R_0 >0 and C >0 such that for every R\geqslant R_0 the problem $$-\Delta u = \lambda h(x) u + a(x) g(u), \quad u\in H_0^1(B_R(0))\label{PlambdaR}$$ has a solution u_R satisfying I(u_R) \geqslant d and \|u_R\|_{\infty} \leqslant C. \end{proposition} \paragraph{Proof.} We recall from section 3 that I is the energy functional associated with (\ref{PlambdaR}) (cf. (\ref{energy-functional})). Let R_0 and \ell be given by Lemma \ref{estimate-from-below} and fix any R\geqslant R_0. For every j\in \mathbb{N} we consider the modified problem (P)^{j}_{R}. In view of Lemma \ref{results-from-DR} (b), for any j\in \mathbb{N} we can choose \rho_{\ell}^{j} > r_\ell (r_\ell as given in (\ref{definition-of-b^k})) in such a way that$$ a_{\ell,R}^{j}:=\sup\{ I^j_R(u)\, : \, u\in V_{\ell,R} \, : \, \|u\|=\rho_{\ell}^{j}\} \leqslant d-1.$$Denote \begin{gather*} D_{\ell,R}=\{ u \in V_{\ell,R}: \, \|u\|\leqslant\rho_{\ell}^{j}\},\\ \Gamma_{\ell,R}=\{ \gamma\in C(D_{\ell,R};H_0^1(B_R(0))\, : \, \gamma \mbox{ is odd and } \gamma|_{\partial D_{\ell,R}}=\mbox { identity}\},\\ c_{\ell,R}^{j}=\inf_{\gamma \in \Gamma_{\ell,R}}\sup_{u\in D_{\ell,R}} I^j_R(\gamma(u)). \end{gather*} Since a_{\ell,R}^{j} < b_{\ell,R}^{j} and since I^j_R satisfies the Palais-Smale condition (see Lemma \ref{results-from-DR} (a)), it is known that c_{\ell,R}^{j} is a critical value for I^j_R such that c_{\ell,R}^{j} \geqslant b_{\ell,R}^{j} (see e.g. \cite[Th. 5.2]{S}, \cite[Th. 3.6]{W}). In other words, there exists u_{R}^{j} such that$$ \nabla I^j_R (u_{R}^{j})=0 \quad \mbox{ and } \quad I^j_R(u_{R}^{j})=c_{\ell,R}^{j}\geqslant d.$$On the other hand, since V_{\ell,R} has dimension \ell, we can choose u_R^j in such a way that its Morse index is not greater than \ell; this follows readily from the arguments in e.g. \cite{LS,RS}. From Lemma \ref{results-from-DR} (c) we conclude that \|u_R^j\|_{\infty} is bounded independently of j. As a consequence, for j large we have that u_R^j is a solution of problem (\ref{PlambdaR}). At this point, for every R\geqslant R_0 we have constructed a solution u_R of (\ref{PlambdaR}) such that I(u_R)\geqslant d. Moreover, the Morse indices m(u_R) are bounded above by some fixed number \ell. To finish the proof of Proposition \ref{approximated-solutions} it remains to show that \|u_R\|_{\infty} is bounded independently of R. Since a^{+} has compact support, this follows as in step 2 of the proof of Proposition \ref{bounded-domain}.\hfill\Box \paragraph{Proof of Theorem 2 completed.} Step 1. Fix any d \in \mathbb{R} and take any sequence R_n\to \infty. Let (u_n) be the corresponding solutions of (P)_{\lambda,R_n} given by Proposition \ref{approximated-solutions}. As in the proof of Theorem \ref{thm1}, up to a subsequence, (u_n) has a weak limit u in {\cal D}^{1,2}(\mathbb{R}^N) such that u_n\to u a.\ e.\ and the sequence (\int |a|\;g(u_n)u_n) is bounded. Clearly, u is a solution of (\ref{Plambda}) and u\in L^{\infty}(\mathbb{R}^N). \noindent Step 2. By multiplying the equation in (\ref{Plambda}) by u\Psi_R where \Psi_R is as in the proof of Lemma \ref{eigenvalues} and by letting R\to \infty we readily see that$$ \int a^-g(u)u<\infty,so that \int |a|g(u)u is also finite. Since (\ref{G2}), (\ref{G3'}), and (\ref{G4'}) imply that |G(s)|=G(s)\leqslant Cg(s)s for all s\in \mathbb{R}, we see that \int |a|G(u) is finite and that (\int |a|G(u_n)) is bounded. In particular, I(u) and I'(u)u are finite numbers. \noindent Step 3. Denote v_n=u_n-u. In view of assumptions (\ref{G2}), (\ref{G3'}), (\ref{G4'}), and of Proposition \ref{getting-a-kind-of-convexity} (with respect to the measures a^{\pm}dx), we see that $$\label{Brezis-Lieb} \int a^{\pm}(H(u_n) - H(u)-H(v_n))\to 0 \quad \mbox{ as } \; n\to \infty,$$ where H is any of the functions H(s)=G(s) or H(s)=sg(s). As a consequence, \begin{align*} 0=I'(u_n)u_n & = I'(u)u + I'(v_n)v_n +\mbox{o}(1)\\ & = I'(v_n)v_n +\mbox{o}(1)\\ &= \int |\nabla v_n|^2 - \lambda \int h(x)v_n^2 - \int a(x)g(v_n)v_n +\mbox{o}(1). \end{align*} Since v_n\to 0 weakly in {\cal D}^{1,2}(\mathbb{R}^N) and strongly in L^{p}_{{\rm loc}}(\mathbb{R}^N) and since a^+ has compact support, this implies that $$\label{strongly-to-zero-2} \int |\nabla v_n|^2\to 0, \quad \int |a| g(v_n)v_n \to 0 \quad \mbox{ and } \quad \int |a| G(v_n)\to 0.$$ It follows from (\ref{strongly-to-zero-2}) and (\ref{Brezis-Lieb}) with H(s)=G(s) that u_n\to u strongly in {\cal D}^{1,2}(\mathbb{R}^N) and I(u)=\lim_{n\to \infty} I(u_n) \geqslant d.$Since$d$is any real number, this clearly yields infinitely many solutions for problem (\ref{Plambda}). \paragraph{Acknowledgments.} D. G. Costa thankfully aknowledges the hospitality and support of {\sc cmaf-ul} during his visit to the University of Lisbon. M. Ramos was partially supported by {\sc fct} and Y. 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Partial Differential Equations} {\bf 21} (1996), 541--557. \bibitem{T2} H. {\sc Tehrani}, Infinitely many solutions for an indefinite semilinear elliptic problem in$\mathbb{R}^N\$, {\it Advances in Differential Equations} {\bf 5} (2001), 1571--1596. \bibitem{W} M. {\sc Willem}, Minimax Theorems", Progr. Nonlinear Differential Equations Appl., Vol. 24, Birkh\"auser, Boston, 1996. \end{thebibliography} \noindent\textsc{David G. Costa}\\ Dept. of Math. Sciences, University of Nevada, \\ Las Vegas, NV 89154-4020, USA\\ e-mail: costa@unlv.edu \smallskip \noindent\textsc{Yuxia Guo}\\ Chinese Academy of Sciences \\ Beijing 100080, China\\ e-mail: yxguo@lsc02.iss.ac.cn \smallskip \noindent\textsc{Miguel Ramos }\\ CMAF, Univ. de Lisboa \\ Av. Prof. Gama Pinto, 1649-003 Lisboa, Portugal \\ e-mail: mramos@lmc.fc.ul.pt \end{document}