\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \markboth{Double resonant problems non-quadratic at infinity} {Marcelo F. Furtado \& Elves A. B. Silva} \begin{document} \setcounter{page}{155} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent USA-Chile Workshop on Nonlinear Analysis, \newline Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 155--171.\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Double resonant problems which are locally non-quadratic at infinity % \thanks{ {\em Mathematics Subject Classifications:} 35J65, 58E05. \hfil\break\indent {\em Key words:} Resonance, non-quadratic at infinity, Morse index. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Published January 8, 2001. \hfil\break\indent The first author was supported by CAPES/Brazil. \hfil\break\indent The second author was partially supported by CNPq/Brazil. }} \def\intom#1{ \int_{\Omega} #1 \, dx} \def\norm#1{ \left\Vert #1 \right\Vert } \date{} \author{Marcelo F. Furtado \& Elves A. B. Silva} \maketitle \begin{abstract} We establish the existence of a nontrivial solution for a double resonant elliptic problem under a local non-quadraticity condition at infinity and pointwise limits. We also study the existence of a nonzero solution when there is resonance at the first eigenvalue. The first result is obtained as an application of an abstract theorem that establishes the existence of a nontrivial critical point for functionals of class $C^2$ on real Hilbert spaces. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} The main goal of this article is to establish sufficient conditions for the existence of a nontrivial solution for a double resonant semi-linear elliptic problem under a local non-quadraticity condition and pointwise limits. We also study the existence and the multiplicity of solutions when there is resonance at the first eigenvalue. To achieve our main objective, we prove a generalization of a critical point theorem due to Lazer-Solimini \cite{LSo}. Their theorem establishes the existence of a nontrivial critical point for a functional of class $C^2$ defined on a real Hilbert space, under the hypotheses of Rabinowitz's saddle point theorem \cite{Rab2}. As it is well known, minimax theorems and related results are based on the existence of a linking structure and on deformation lemmas \cite{AmbRab,BBF,Rab,MWillem,Struwe,Chang,Sil4}. In general, to be able to derive such deformation results, it is supposed that the functional satisfies a compactness condition. In this article, we assume the (SCe) condition introduced by Silva in \cite{Sil} and defined below (see Definition \ref{sce}). Denoting by $m(I,u)$ [$\overline{m}(I,u)$] the Morse index [augmented Morse index] of the functional $I \in C^2(E,\mathbb{R})$ at the point $u$, we prove the following result: \vspace{0.5cm} \begin{theorem} Let $E=V \oplus W$ be a real Hilbert space with $V$ finite dimensional and $W=V^{\bot}$. Suppose $I \in C^2(E,\mathbb{R})$ satisfies {\em (SCe)} and \begin{itemize} \item[$(I_1)$] there exists $\beta \in \mathbb{R}$ such that $I(v) \leq \beta$, for all $v$ in $V$, \item[$(I_2)$] there exists $\gamma \in \mathbb{R}$ such that $I(w) \geq \gamma$, for all $w$ in $W$, \item[$(I_3)$] the origin is a critical point of $I$, $I''(0)$ is a Fredholm operator and either $\dim V < m(I,0)$ or $\overline{m}(I,0) < \dim V$. \end{itemize} Then $I$ possesses a nonzero critical point. \label{intro_th1} \end{theorem} The above result is a generalization of the Theorem 1.1 in \cite{LSo} since the condition $(I_1)$ does not imply the anti-coercivity of $I$ on the subspace $V$. Theorem \ref{intro_th1} also complements a recent result by Perera-Schechter \cite{PerSch}. Note that in \cite{PerSch} it is assumed the Palais-Smale compactness condition (PS) which is stronger than condition (SCe) and may not be true under the hypotheses of our application. As in \cite{LSo}, our proof of Theorem \ref{intro_th1} is based on the infinite dimensional Morse theory. To compensate the lack of anticoercivity of $I$ on $V$, we use a deformation result, due to Silva \cite{Sil} (see also \cite{SilTese,Sil2}), that sends $V \cap \partial B_R(0)$, for $R>0$ sufficiently large, below the level surface $\gamma$ of $I$, preserving the linking between $V \cap \partial B_R(0)$ and the subspace $W$. As observed above, Theorem \ref{intro_th1} is motivated by the semilinear elliptic problem, \begin{equation} - \Delta u = f(x,u) \mbox{ in } \Omega, \quad u=0 \mbox{ on } \partial\Omega, \label{P} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N\geq 1$) and the nonlinearity $f \in C^1(\overline{\Omega}\times \mathbb{R},\mathbb{R})$ satisfies $f(x,0) \equiv 0$ and the subcritical growth condition: \begin{enumerate} \item[$(f_1)$\ ] there are constants $a_1,a_2 >0$ such that $$|f_s(x,s)| \leq a_1|s|^{\sigma-2}+ a_2,$$ for all $x \in \Omega$, $s\in \mathbb{R}$ where $\sigma>2$ ($2<\sigma < 2N(N-2)^{-1}$ if $N\geq 3$). \end{enumerate} Standard arguments show that the associated functional $I:H_0^1(\Omega) \to \mathbb{R}$ given by \begin{equation} I(u) = \frac{1}{2} \int_{\Omega} |\nabla u|^2 \, dx - \int_{\Omega} F(x,u) \, dx, \label{definition_functional} \end{equation} with $F(x,s) = \int_0^s f(x,t) \, dt$, is of class $C^2$ and that the classical solutions of (\ref{P}) are the critical points of $I$. We also note that $u \equiv 0$ is a (trivial) solution for the problem (\ref{P}). Thus, our first objective is to establish the existence of a nonzero critical point for $I$. Denoting by $\lambda_k$ the $k^{th}$ eigenvalue of $-\Delta$ on $\Omega$ with zero boundary conditions and considering the limits \begin{equation} L(x) = \liminf_{|s| \to \infty} \frac{2F(x,s)}{s^2} \quad\mbox{and}\quad K(x) = \limsup_{|s| \to \infty} \frac{2F(x,s)}{s^2}, \label{int1} \end{equation} taken in the pointwise sense, we suppose \begin{enumerate} \item[$(F_1)$] $\lambda_j \leq L(x) \leq K(x) \leq \lambda_{j+1}$, a.e. $x \in \Omega$. Furthermore, if $L(x) \equiv \lambda_j$, there exists $D_+ \in L^1(\Omega)$ such that \begin{equation} F(x,s) \geq \frac{\lambda_j}{2}s^2 + D_+(x), \quad\forall\, s\in\mathbb{R}, \mbox{ a.e. } x \in \Omega, \label{eq_f1} \end{equation} \end{enumerate} and \begin{enumerate} \item[$(F_2)$] there exist $q>1$ ($q\geq \frac{N}{2}$, if $N\geq3$), $A \in L^q(\Omega)$ and $B \in L^1(\Omega)$ such that $$ |F(x,s)| \leq A(x)s^2 + B(x), \quad\forall\, s\in\mathbb{R}, \mbox{ a.e } x\in \Omega. $$ \end{enumerate} We note that $(F_1)$ characterizes (\ref{P}) as a double resonant problem (see \cite{DMag,DMag2,COl,FigBer,Gonc}) and that the conditions $(F_1)$ and $(F_2)$ are related to the geometry of the saddle point theorem. The necessary Morse index estimates for the functional $I$ at $u \equiv 0$ are provided by a local condition for $f(x,s)$ at the origin: \begin{enumerate} \item[$(f_2)$\ ] $f_s(x,0) \not\leq \lambda_j$ or $\lambda_{j+1} \not\leq f_s(x,0)$. \end{enumerate} Here, we write $f_s(x,0) \not \leq (\not\geq) \lambda_k$ to indicate that $f_s(x,0) \leq (\geq) \lambda_k$ with strict inequality holding on a set of positive measure. In this article, we also consider a local non-quadraticity condition at infinity on the primitive $F$. More specifically, setting $H(x,s)=f(x,s)s-2F(x,s)$, we suppose \begin{enumerate} \item[$(NQ)_+$\hspace{-0.52cm} ] \hspace{0.30cm} there exist $\Omega_0 \subset \Omega$ with positive measure and $C_+ \in L^1(\Omega)$ such that $ \hspace{0.52cm} \begin{array}{rll} \mbox{(i)} & \lim\limits_{|s|\to \infty} H(x,s) = \infty, & \mbox{a.e. } x\in \Omega_0, \vspace{0.1cm} \\ \mbox{(ii)} & H(x,s) \geq C_+(x), & \forall\thinspace s\in\mathbb{R}, \mbox{a.e. } x\in \Omega. \end{array} $ \end{enumerate} We note that condition $(NQ)_+$ with $\Omega_0=\Omega$ has been assumed in the works \cite{DMag,Gonc} (see also \cite{Sch1,Sch2}). In this article we show that the hypothesis $(NQ)_+$, for $\Omega_0$ sufficiently large, and weaker versions of $(F_1)$ and $(F_2)$ provide the necessary compactness for the functional $I$. Now, denoting by $\mu (U)$ the Lebesgue measure of a measurable set $U \subset \mathbb{R}^N$, we state our application of Theorem \ref{intro_th1}: \begin{theorem} There exists $0<\alpha<\mu (\Omega)$ such that, if $f \in C^1(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ satisfies $f(x,0) \equiv 0$, $(f_1)$, $(f_2)$, $(F_1)$, $(F_2)$ and $(NQ)_+$ with $\mu(\Omega_0)>\alpha$, then the problem (\ref{P}) possesses a nonzero solution. \label{intro_th2} \end{theorem} It is worthwhile mentioning that in \cite{DMag,COl,FigBer,Gonc} the authors establish the existence of solution for (\ref{P}) (without supposing $f(x,0)\equiv 0$). Actually, applying a version of the saddle point theorem due to Silva \cite{Sil} and the technical results proved in this article, we are able to study (\ref{P}) in this setting when $f \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})$ and satisfies \begin{enumerate} \item[($\hat{f_1}$) ] there are constants $a_3,a_4 >0$ such that $$|f(x,s)| \leq a_3|s|^{\sigma-1}+ a_4,$$ for all $x \in \Omega$, $s\in \mathbb{R}$ where $\sigma>1$ ($1 < \sigma < 2N(N-2)^{-1}$ if $N\geq 3$). \end{enumerate} Noting that on this case the functional $I$ is of class $C^1$, and that the critical points of $I$ are weak solutions of (\ref{P}), we obtain \begin{theorem} There exists $0<\alpha<\mu(\Omega)$ such that, if $f \in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ satisfies $(\hat{f_1})$, $(F_1)$, $(F_2)$ and $(NQ)_+$ with $\mu(\Omega_0)>\alpha$, then the problem (\ref{P}) possesses a weak solution. \label{intro_th3} \end{theorem} In this work, we also present versions of Theorems \ref{intro_th2} and \ref{intro_th3} under conditions that are dual to $(NQ)_+$ and $(F_1)$ (see Theorems \ref{intro_th2'} and \ref{intro_th3'}). In our final result, we study the existence of a nonzero solution for the problem (\ref{P}) under a resonant condition at the first eigenvalue. On this case, we may replace the condition $(F_2)$ by \begin{enumerate} \item[$(\hat{F_2})$] there exist $q>1$ ($q\geq \frac{N}{2}$ if $N\geq3$), $A \in L^q(\Omega)$ and $B \in L^1(\Omega)$ such that $$F(x,s) \leq A(x)s^2 + B(x) , \quad\forall\, s\in\mathbb{R}, \mbox{ a.e } x\in \Omega,$$ \end{enumerate} and consider the following local condition for the primitive $F$ \begin{enumerate} \item[$(F_3)$] there exists $r_1>0$ such that $$F(x,s) \geq 0 , \quad\forall\ 00$ such that \begin{equation} F(x,s) \geq \frac{\lambda_1}{2}s^2, \quad\forall\, 00$, there exist $\varepsilon \in (0,\overline{\varepsilon})$, $R>0$ and a continuous function $\eta:[0,1] \times E \to E$ such that \begin{enumerate} \item[$(\eta_1)$] $\eta(t,u)=u$, for every $u \in W$, \item[$(\eta_2)$] $\eta(t,\cdot):E \to E$ is a homeomorfism, for every $t \in [0,1]$, \item[$(\eta_3)$] $\eta(1,u) \in I^{\gamma-\varepsilon}$, for every $u \in V \cap \partial B_R(0)$, \item[$(\eta_4)$] $I(\eta(t,u)) \leq I(u)$, for every $u \in E$, $t \in [0,1]$. \end{enumerate} \label{lmdef_elvis} \end{lemma} Given two topological spaces $A$ and $X$, with $A$ a subspace of $X$, let $H_n(X,A)$ denote the $n^{th}$ relative singular homology group with coefficient group equal to the real numbers. The next result is based on \cite{LSo}. \begin{proposition} Let $E=V \oplus W$ be a real Hilbert space with $V$ finite dimensional and $W=V^{\bot}$. Suppose $I \in C^2(E,\mathbb{R})$ satisfies $(I_1)$, $(I_2)$, {\em (SCe)} and has only a finite number of critical points, all of which are nondegenerate. Then $I$ possesses a critical point $u$ such that $m(I,u)=\dim V$. \label{pc_indmorse_k} \end{proposition} \paragraph{Proof.} Let $k=\dim V$. When $k=0$, the condition $(I_2)$ implies that $I$ is bounded from below and consenquently, by (SCe), the infimum of $I$ is attained at a point $u \in E$ (see \cite{BBF,Rab}). Since $u$ is nondegenerate, $m(I,u)=0$. Now, let us consider $k \geq 1$. Arguing by contradiction, we suppose that $I$ has no critical point with Morse index equals to $k$. First, we choose $d_0 < \gamma$ such that $I$ has no critical points on $I^{d_0}$. Noting that $I$ satisfies $(I_2)$ with $\gamma$ replaced by $d_0$, we may apply Lemma \ref{lmdef_elvis} to obtain $\varepsilon \in (0,1)$, $R>0$ and $\eta \in C([0,1]\times E,E)$ satisfying $(\eta_1), \, (\eta_2), \, (\eta_4)$ and \begin{equation} \tilde{S}=\eta(1,S) \subset I^{d_0-\varepsilon} \subset I^{d_0} \subset E\setminus W, \label{ab_11} \end{equation} where $S = V \cap \partial B_R(0)$. Moreover, by $(\eta_1)$ and $(\eta_2)$, we have that $\eta(1,E\setminus W)=E\setminus W$. Thus, considering the inclusions $i:\tilde{S} \to E\setminus W$ and $l:S \to E\setminus W$, we obtain the following commutative diagram of homomorphisms $$ \begin{array}{ccc} H_*(S) & \stackrel{l_*}{\longrightarrow} & H_*(E\setminus W) \\ \psi_{1*} \downarrow & & \downarrow \psi_{2*} \\ H_*(\tilde{S}) & \stackrel{i_*}\longrightarrow & H_*(E\setminus W), \end{array} $$ where $\psi_1 = \eta(1,\cdot)|_{S}$ and $\psi_2 = \eta(1,\cdot)|_{E\setminus W}$. Invoking $(\eta_2)$ one more time, we have that $\psi_{1*}$ and $\psi_{2*}$ are isomorphisms. Therefore, since $S$ is a strong deformation retract of $E \setminus W$, we conclude that $$ i_* : H_*(\tilde{S}) \to H_*(E\setminus W) $$ is an isomorphism. Now, let $j: \tilde{S} \to I^{d_0}$ and $h:I^{d_0} \to H \setminus W$ be the inclusion maps. Since, by (\ref{ab_11}), $i = h \circ j$, we have that $i_* = h_* \circ j_*$. Hence, $h_*$ is surjective and $$ \dim H_{k-1}(I^{d_0}) \geq \dim H_{k-1}(E\setminus W) = \dim H_{k-1}(S). $$ Therefore, \begin{equation} \dim H_{k-1}(I^{d_0}) \geq \left\{ \begin{array}{ll} 1, & \mbox{ if } k > 1, \\ 2, & \mbox{ if } k = 1. \end{array} \right. \label{contr1} \end{equation} Now, consider $$ c_1 < c_2 < \cdots < c_m, $$ the possible critical levels of $I$, and take $d_1, \ldots, d_m$ real numbers such that \mbox{$d_{j-1} < c_j < d_j$}, for $j=1, \ldots, m$. If $I$ does not possess critical values, set $d_1 = d_m > d_0$. Recalling that $\dim H_k(I^{d_j},I^{d_{j-1}})$ is equal to the number of critical points of $I$ with Morse index $k$ at the critical level $c_j$ (see \cite{MWillem}), we have \begin{equation} \dim H_k(I^{d_j},I^{d_{j-1}}) = 0,\quad\forall\, j=1,\ldots,m. \label{dim0_1} \end{equation} The exactness of the sequence $$ \cdots {\longrightarrow} H_k \left( I^{d_{j-1}}, I^{d_{0}} \right) {\longrightarrow} H_k \left( I^{d_{j}}, I^{d_{0}} \right) {\longrightarrow} H_k \left( I^{d_{j}}, I^{d_{j-1}} \right) {\longrightarrow} \cdots $$ implies that $$ \dim H_k \left( I^{d_{j}}, I^{d_{0}} \right) \leq \dim H_k \left( I^{d_{j}}, I^{d_{j-1}} \right) + \dim H_k \left( I^{d_{j-1}}, I^{d_{0}} \right). $$ Consequently, by the above inequality and (\ref{dim0_1}), we obtain \begin{equation} \dim H_k(I^{d_m},I^{d_0}) \leq ~\sum_{j=1}^m \dim H_k(I^{d_j},I^{d_{j-1}}) = 0. \label{ab_12} \end{equation} Considering the exact sequence of the pair $(I^{d_m},I^{d_0})$, we conclude that $$ \dim H_{k-1}(I^{d_0}) \leq \dim H_{k}(I^{d_m},I^{d_0}) + \dim H_{k-1}(I^{d_m}). $$ Since $I^{d_m}$ is a strong deformation retract of $E$ (see Remark \ref{rem_b}), by (\ref{ab_12}) and the above expression, we have $$ \dim H_{k-1}(I^{d_0}) \leq \dim H_{k-1}(E) = \left\{ \begin{array}{ll} 0, & \mbox{ se } k > 1, \\ 1, & \mbox{ se } k = 1, \end{array} \right. $$ which contradicts (\ref{contr1}) and concludes the proof of Proposition \ref{pc_indmorse_k}.\hfill$\diamondsuit$\smallskip Proposition \ref{pc_indmorse_k} is a key ingredient for the proof of Theorem \ref{intro_th1}. We also need the following version of a result by Marino-Prodi \cite{MProdi} (see also \cite{Sol}). \begin{lemma} Suppose $I \in C^2(E,\mathbb{R})$ satisfies {\em (SCe)}, $u_0$ is an isolated critical point of $I$ and $I''(u_0)$ is a Fredholm operator. Then, given $\varepsilon>0$, there exists $J \in C^2(E,\mathbb{R})$ such that $J$ satisfies {\em (SCe)}, $I(u)=J(u)$ for $\norm{u-u_0} \geq \varepsilon$, $J$ has only a finite number of critical points all of which are nondegenerate in the open ball $B_{\varepsilon}(u_0)$, and $\norm{I^{(j)}(u)-J^{(j)}(u)} < \varepsilon$, for $j=0,1,2$ and $u \in E$. \label{marino} \end{lemma} \begin{remark}\rm In \cite{MProdi}, the authors consider the above lemma with the hypothesis that $I$ satisfies the Palais-Smale condition. Noting that for bounded sequences the (SCe) condition is equivalent to the Palais-Smale condition and that the functionals $I$ and $J$ are equals on the complement of $B_{\varepsilon}(u_0)$, we see easily that the result in \cite{MProdi} holds with the weaker condition (SCe). \end{remark} We are ready to prove our abstract theorem. The following argument is due to Lazer--Solimini \cite{LSo} and it will be presented for the sake of completness. \paragraph{Proof of Theorem \ref{intro_th1}. } Without loss of generality we may assume that 0 is an isolated critical point. Since $I''(0)$ is a Fredholm operator, we can use the spectral theory to obtain a constant $b>0$ and an orthogonal decomposition $E=E^0 \oplus E^+ \oplus E^-$, where $E^0$ is the kernel of $I''(0)$, $E^+$ and $E^-$ are closed and invariants under $I''(0)$, and \begin{eqnarray} \langle I''(0)u,u \rangle &\geq& b \norm{u}^2,\quad\forall\, u \in E^+, \label{th1_11} \\ \langle I''(0)u,u \rangle &\leq& -b \norm{u}^2,\quad\forall\, u \in E^-. \label{th1_11'} \end{eqnarray} Now, let $J$ be the functional given by Lemma \ref{marino}, with $\varepsilon>0$ such that $0 < \varepsilon < \frac{b}{3}$ and $\norm{I''(u)-I''(0)} < \frac{b}{3}$, if $\norm{u} < \varepsilon$. It is clear that $ J$ also satisfies $(I_1)$ and $(I_2)$ for appropriated constants. To prove Theorem \ref{intro_th1} it suffices to show that $J$ possesses a critical point on the complement of $B_{\varepsilon}(0)$. Suppose, by contradiction, that this is not the case. Then, by Proposition \ref{pc_indmorse_k}, there exists a critical point $\overline{u}$ of $J$ such that $m(J,\overline{u})=\dim V$. By our choice of $\varepsilon$ and Lema \ref{marino}, we obtain $$ \norm{J''(\overline{u})-I''(0)} \leq \norm{J''(\overline{u})-I''(\overline{u})} + \norm{I''(\overline{u})-I''(0)} < \frac{2b}{3}. $$ Hence, by (\ref{th1_11}) and (\ref{th1_11'}), we get $$ \displaylines{ \langle J''(\overline{u})u,u \rangle \geq \frac{b}{3}\norm{u}^2,\quad\forall\, u \in E^+, \cr \langle J''(\overline{u})u,u \rangle \leq -\frac{b}{3}\norm{u}^2,\quad\forall\, u \in E^-. }$$ Consequently, $$ m(I,0)=\dim E^- \leq m(J,\overline{u}) \leq \dim \left( E^0 \oplus E^- \right)= \overline{m}(I,0). $$ The last expression and the hypotheses of Theorem \ref{intro_th1} show that $m(J,\overline{u}) \neq \dim V$. This concludes the proof of Theorem \ref{intro_th1}. \hfill $\diamondsuit$\smallskip By the above argument and repeated applications of Lemma \ref{marino}, we get \begin{proposition} Let $E=V \oplus W$ be a real Hilbert space with $V$ finite dimensional and $W=V^{\bot}$. Suppose $I \in C^2(E,\mathbb{R})$ satisfies {\em (SCe)}, $(I_1)$, $(I_2)$ and \begin{itemize} \item[$(\hat{I_3})$] $I$ possesses a finite number of critical points $\{ u_j \}_{j=1}^m$ such that $I''(u_j)$ is a Fredholm operator and either $\dim V < m(I,u_j)$ or $\overline{m}(I,u_j) < \dim V$, for $j=1,\ldots,m$. \end{itemize} Then there exists a critical point $u$ of $I$ with $u \neq u_j, \, j=1,\ldots,m.$ \end{proposition} Finally, we state the version of the saddle point theorem, due to Silva \cite{Sil} (see also \cite{SilTese,Sil2}), that will be used in the proof of Theorem \ref{intro_th3}. \begin{theorem} Let $E=V \oplus W$ be a real Hilbert space with $V$ finite dimensional and $W=V^{\bot}$. Suppose $I \in C^1(E,\mathbb{R})$ satisfies $(I_1)$, $(I_2)$ and {\em (SCe)}$_b$ for every $b \geq \gamma$. Then $I$ possesses a critical value $b \in [\gamma,\beta]$. \label{elves_th} \end{theorem} \section{Proof of Theorem \ref{intro_th2}} We denote by $|\cdot|_p$ the $L^p(\Omega)$-norm ($1 \leq p \leq \infty$) and by $\norm{\cdot}$ the norm in $E=H_0^1(\Omega)$ induced by the inner product $$ \langle u,v\rangle = \intom{\nabla u \cdot \nabla v},\quad\forall\, u, \, v \in H_0^1(\Omega). $$ As mentioned previously, the condition $(f_1)$ implies that the functional $I$ given by (\ref{definition_functional}) is of class $C^2$, and the classical solutions of (\ref{P}) are the critical points of $I$. A standard argument \cite{Rab,Chang} shows that for $\varphi, \psi \in H$, \begin{equation} I''(0)(\varphi,\psi) = \intom{\nabla\varphi \cdot \nabla\psi} - \intom{f_s(x,0)\varphi\psi}\,. \label{derivseg} \end{equation} We denote by $E_k$ the subspace of $E$ spanned by $\{ \phi_1, \ldots, \phi_k \}$, where $\{ \phi_1, \phi_2, \ldots \}$ is the orthonormal basis of $E$ formed by the eigenfunctions associated to the eigenvalues $0 < \lambda_1 < \lambda_2 \leq \ldots \leq \lambda_n \leq \ldots$ of $- \Delta$ on $\Omega$ with zero boundary conditions. Given $N \geq 3$, consider $2^* = \frac{2N}{N-2}$ and set $$ S = S(N) = \inf\limits_{u \in E, \, u \not\equiv 0} \frac{\intom{|\nabla u|^2}}{\left(\intom{|u|^{2^*}}\right) ^{2/2^*}} > 0, $$ the best constant for the Sobolev embedding $H_0^1(\Omega) \hookrightarrow L^{2^*}(\Omega)$. Note that $S$ depends only on the dimension $N$ \cite{Struwe}. We start with a technical result \begin{lemma} Suppose $f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R})$ satisfies $K(x) \leq \lambda_{j+1}$. Then, given $\varepsilon>0$, there exists $\tilde{\Omega} \subset \Omega$ and a constant $M_1=M_1(\varepsilon)>0$ such that $\mu(\Omega \setminus \tilde{\Omega}) < \varepsilon$ and \begin{equation} F(x,s) \leq \frac{1}{2} ( \varepsilon + \lambda_{j+1} )s^2 + M_1, \quad\forall\, s \in \mathbb{R}, \, \mbox{a.e. } x \in \tilde{\Omega}. \label{ap_31} \end{equation} \label{egorov} \end{lemma} \paragraph{Proof.} Effectively, given $\varepsilon>0$, we apply the Egorov's Theorem for the sequence $$ G_n(x) = \sup \left \{ \frac{2F(x,s)}{s^2} ~|~ |s| \geq n \right\}, $$ to obtain $\tilde{\Omega} \subset \Omega$ such that $\mu(\Omega \setminus \tilde{\Omega}) < \varepsilon$ and $$ \limsup_{|s| \to \infty} \frac{2F(x,s)}{s^2} \leq \lambda_{j+1},\quad \mbox{unif. for a.e. } x \in \tilde{\Omega}. $$ The above expression provide $r=r(\varepsilon)>0$ such that \begin{equation} F(x,s) \leq \frac{1}{2} ( \varepsilon + \lambda_{j+1} )s^2, \quad\forall\, |s| \geq r, \, \mbox{a.e. } x \in \tilde{\Omega}. \label{eq_egorov} \end{equation} Now, by the continuity of $F(x,s)$ on $\overline{\Omega} \times [-r,r]$, $$ F(x,s) \leq M_1,\quad\forall\, |s| \leq r, \, \mbox{a.e. } x \in \Omega. $$ The above expression and (\ref{eq_egorov}) prove the statement (\ref{ap_31}). \hfill $\diamondsuit$ \begin{remark}\rm If we suppose that for every $r>0$ there exists $B_r \in L^1(\Omega)$ such that $$ F(x,s) \leq B_r(x),\quad\forall\, |s| \leq r, \, \mbox{a.e. } x \in \Omega, $$ we obtain a version of the above result with $M_1 \in L^1(\Omega)$ without to assume the continuity of $f$. \end{remark} Now, we prove the compactness condition for the functional $I$. \begin{lemma} There exists $0 < \alpha < \mu(\Omega)$ such that, if $f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R})$ satisfies $(\hat{f_1})$, $(\hat{F_2})$, $K(x) \leq \lambda_{j+1}$ and $(NQ)_{+}$ with $\mu(\Omega_0) > \alpha$, then the functional $I$ satisfies {\em (SCe)}. \label{aplm1} \end{lemma} \paragraph{Proof.} Let $(u_n) \subset E$ be such that $I(u_n) \to c \in \mathbb{R}$ and $I'(u_n) \to 0$. If $(u_n)$ is bounded, hypothesis $(\hat{f_1})$ implies that $(u_n)$ possesses a convergent subsequence. Consequently, by Definition \ref{sce}, it suffices to verify that $\Vert I'(u_{n_j})\Vert \Vert u_{n_j} \Vert \to \infty$, for every sequence $(u_{n_j}) \subset (u_n)$ such that $\Vert u_{n_j} \Vert \to \infty$. Arguing by contradiction, we suppose that there exists a subsequence, which we will denote by $(u_n)$, such that $\norm{u_n} \to \infty$ and $\norm{I'(u_n)}\norm{u_n}$ is bounded. This assumption and the fact that $I(u_n) \to c$ provide $M \in \mathbb{R}$ such that \begin{equation} \liminf \intom{H(x,u_n)} = \liminf \left[ 2I(u_n)-I'(u_n)u_n \right] \leq M. \label{aplm1eq1} \end{equation} On the other hand, given $\varepsilon>0$, for $n$ sufficiently large, we have \begin{equation} \frac{1}{2} \norm{u_n}^2 \leq (\varepsilon +c) + \intom{F(x,u_n)}. \label{ap_32} \end{equation} Now, we apply Lemma \ref{egorov} to obtain $M_1>0$ and $\tilde{\Omega} \subset \Omega$ such that $\mu(\Omega \setminus \tilde{\Omega}) < \varepsilon$ and $$ F(x,u_n) \leq \frac{1}{2} ( \varepsilon + \lambda_{j+1} )u_n^2 + M_1, \quad\mbox{a.e. } x \in \tilde{\Omega}. $$ Considering $N\geq3$, we may use the above expression, (\ref{ap_32}), H\"older inequality, $(\hat{F_2})$ and the Sobolev Embeding Theorem, to obtain \begin{eqnarray*} \frac{1}{2} \norm{u_n}^2 &\leq& (c+\varepsilon) + \frac{1}{2}(\varepsilon+\lambda_{j+1})|u_n|_2^2 + M_1\mu(\Omega) \\ && + \int_{\Omega\setminus\tilde{\Omega}} \left[ Au_n^2 + B \right] \, dx \\ &\leq& \frac{1}{2}(\varepsilon+\lambda_{j+1})|u_n|_2^2 + S^{-1}|A|_{L^{\frac{N}{2}}(\Omega\setminus\tilde{\Omega})}\norm{u_n}^2 + \tilde{M}, \end{eqnarray*} where $\tilde{M}=\tilde{M}(\varepsilon)=c+\varepsilon+M_1\mu(\Omega)+|B|_1$, or equivalently \begin{equation} \left( \frac{1}{2} - S^{-1}|A|_{L^{\frac{N}{2}}(\Omega\setminus\tilde{\Omega})} \right) \norm{u_n}^2 \leq \frac{1}{2}(\varepsilon+\lambda_{j+1})|u_n|_2^2 + \tilde{M}. \label{aplm1eq3} \end{equation} Defining $\tilde{u}_n = \frac{u_n}{\norm{u_n}}$ we may suppose that $\tilde{u}_n \rightharpoonup \tilde{u}$ weakly in $E$, $\tilde{u}_n \to \tilde{u}$ strongly in $L^2(\Omega)$ and $\tilde{u}_n(x) \to \tilde{u}(x)$ for almost everywhere $x \in \Omega$. Thus, dividing (\ref{aplm1eq3}) by $\norm{u_n}^2$, taking $n \to \infty$, $\varepsilon \to 0$, we conclude that \begin{equation} 1 \leq \lambda_{j+1}|\tilde{u}|_2^2. \label{aplm1eq4} \end{equation} At this point, we claim that there exists a measurable set $\Omega_1 \subset \Omega_0$ with positive measure such that \begin{equation} \tilde{u}(x) \neq 0,\quad \mbox{ a.e. } x \in \Omega_1. \label{omega1} \end{equation} Supposing the claim, we use $(NQ)_+$ and Fatou's lemma to conclude that $$ \liminf \intom{ H(x,u_n) } \geq \intom{ \liminf H(x,u_n) } = \infty, $$ which contradicts (\ref{aplm1eq1}). To prove the claim we set $$ \alpha = \mu(\Omega) - \left( \frac{S}{\lambda_{j+1}} \right)^{N/2} > 0, $$ and suppose, by contradiction, that the claim is false. Then, by (\ref{aplm1eq4}), we have \begin{eqnarray*} 1 & \leq & \lambda_{j+1} \intom{|\tilde{u}(x)|^2} \leq \lambda_{j+1} \left[\int_{\Omega \setminus \Omega_0} |\tilde{u}(x)|^{2^*} \, dx \right]^{2/2^*} \mu(\Omega\setminus\Omega_0)^{2/N} \\ & \leq & \lambda_{j+1}S^{-1}\norm{\tilde{u}}^2 \mu(\Omega\setminus\Omega_0)^{2/N} \leq \lambda_{j+1}S^{-1}\mu(\Omega\setminus\Omega_0)^{2/N} < 1. \end{eqnarray*} For $N=1$ or $N=2$, we use the Sobolev embedding $H_0^1(\Omega) \hookrightarrow L^r(\Omega)$, $10$ and using $(NQ)_+$, we obtain \begin{equation} \frac{d}{ds}\left[ \frac{G(x,s)}{s^2} \right] = \frac{g(x,s)s-2G(x,s)}{s^3} \geq \frac{C_+(x)}{s^3}. \label{aplm2eq1} \end{equation} Integrating the above expression on $[s,t] \subset (0,\infty)$, we get $$ \frac{G(x,s)}{s^2} \leq \frac{G(x,t)}{t^2} - \frac{C_+(x)}{2} \left[ \frac{1}{s^2} - \frac{1}{t^2} \right]. $$ Since $K(x) \leq \lambda_{j+1}$, we have that $\limsup_{t \to \infty} t^{-2}G(x,t) \leq 0$ for almost everywhere $x \in \Omega$, and therefore $$ G(x,s) \leq -\frac{C_+(x)}{2},\quad \forall\, s>0, \, \mbox{a.e. } x\in \Omega. $$ The proof for $s < 0$ is similar. \hfill $\diamondsuit$\smallskip The above lemma is used to verify the condition $(I_2)$ of Theorem \ref{intro_th1}. The next result is a version of Proposition 2 in \cite{COl} (see also \cite{Maw}). It provides estimates on subsets of $\Omega$ and it will be used to verify the hypothesis $(I_1)$ of Theorem \ref{intro_th1}. \begin{lemma} Suppose $(F_1)$ and $L(x) \not\geq \lambda_j$. Then there exist $\delta_1, \varepsilon>0$ such that, for every subset $\tilde{\Omega} \subset \Omega$ with $\mu(\Omega \setminus \tilde{\Omega}) < \varepsilon$, we have $$ \norm{u}^2 - \int_{\tilde{\Omega}} L(x)u^2 \, dx \leq -\delta_1 \norm{u}^2,\quad\forall \, u \in E_j. \label{ap_310} $$ \label{aplm3'} \end{lemma} \paragraph{Proof.} By Proposition 2 in \cite{COl}, the lemma holds for $\tilde{\Omega}=\Omega$ and $\hat{\delta}_1>0$. Let $u \in E_j$, $\tilde{\Omega} \subset \Omega$. Supposing $N \geq 3$, using H\"older inequality and the Sobolev Embeding Theorem, we obtain \begin{eqnarray*} \norm{u}^2 - \int_{\tilde{\Omega}} L(x)u^2 \, dx &=& \norm{u}^2 - \intom{L(x)u^2} + \int_{\Omega \setminus \tilde{\Omega}} L(x)u^2 \, dx \\ & \leq & -\hat{\delta}_1\norm{u}^2 + |L|_{\infty}S^{-1}\mu(\Omega\setminus\tilde{\Omega})^\frac{2}{N} \norm{u}^2 \\ &=& \norm{u}^2 \left( -\hat{\delta}_1 + |L|_{\infty}S^{-1} \mu(\Omega\setminus\tilde{\Omega})^\frac{2}{N} \right). \end{eqnarray*} Hence, the lemma holds for $\delta_1 = \hat{\delta}_1/2$ and $\varepsilon>0$ sufficiently small. The argument for the cases $N=1$ or $N=2$ is similar. \hfill $\diamondsuit$ \paragraph{Proof of Theorem \ref{intro_th2}} Without loss of generality, we may suppose that the origin is an isolated critical point of $I$. Taking $V=E_k$ and $W=V^{\perp}$, by Lemma \ref{aplm2}, we have $$ I(w) = \frac{1}{2} \big( \norm{w}^2 - \intom{K(x)w^2} \big) - \intom{ \big[ F(x,w) - \frac{K(x)}{2}w^2 \big] } \geq -\frac{|C_+|_1}{2}\,, $$ for every $w \in W$. Consenquently, $I$ satisfies $(I_2)$. To verify $(I_1)$, we first suppose that $L(x) \equiv \lambda_j$. Then, using (\ref{eq_f1}), we have $$ I(v) \leq \frac{1}{2} \left( \norm{v}^2 - \lambda_j|v|_2^2 \right) + |D_+|_1 \leq |D_+|_1,\quad\forall\, v \in V. $$ Thus, $I$ satisfies $(I_1)$ on this case. Now, let us consider $L(x) \not\geq \lambda_j$. Consider $\delta_1, \varepsilon > 0$ given by Lemma \ref{aplm3'}. Using the same argument employed in the proof of Lemma \ref{egorov}, we obtain $M_1=M_1(\varepsilon)>0$ such that $$ 2F(x,v) \geq (L(x)-\varepsilon) v^2 - M_1,\quad \forall\, v\in V, \, \mbox{a.e. } x \in \tilde{\Omega}, $$ where $\tilde{\Omega} \subset \Omega$ satisfies $\mu(\Omega\setminus\tilde{\Omega}) < \varepsilon$. Taking $N\geq3$, the above expression, Lemma \ref{aplm3'}, $(F_2)$, H\"older inequality and the Sobolev Embeding Theorem imply \begin{eqnarray*} 2I(v) & \leq & \norm{v}^2 - \int_{\tilde{\Omega}} L(x)v^2 \, dx + \varepsilon|v|^2_2 + M_1\mu(\Omega) + 2 \int_{\Omega\setminus\tilde{\Omega}} \left[ Av^2 + B \right] \, dx \\ & \leq & \left( -\delta_1 + \frac{\varepsilon}{\lambda_1} + 2S^{-1}|A|_{L^\frac{N}{2}(\Omega\setminus\tilde{\Omega})}\right)\norm{v}^2 + M_1\mu(\Omega)+2|B|_1, \end{eqnarray*} for every $v \in V$. Considering $\varepsilon$ smaller if necessary, we conclude that $I(v) \to -\infty$, when $\norm{v} \to \infty$. Thus, $(I_1)$ also holds on this case. The proof for $N=1$ and $N=2$ is similar. Since $f$ satifies $(f_1)$, $I''(0)$ is a Fredholm operator. To establish the Morse Index estimates, we suppose first that $f_s(x,0) \not\geq \lambda_{j+1}$. Using (\ref{derivseg}) and the same argument employed on the proof of the Lemma \ref{aplm3'}, we obtain $\delta_1>0$ such that $$ I''(0)(u,u) = \norm{u}^2 - \intom{f_s(x,0)u^2} \leq -\delta_1 \norm{u}^2,\quad\forall\, u \in E_{j+1}. $$ Consequently, $m(I,0) \geq j+1 > \dim V$. An analogous argument shows that, $\overline{m}(I,0) \leq j-1 < \dim V$, whenever $f_s(x,0) \not\leq \lambda_j$. Hence, $I$ satisfies $(I_3)$. Now, we invoke Lemma \ref{aplm1} and Theorem \ref{intro_th1} to obtain a nonzero solution for (\ref{P}). \hfill $\diamondsuit$\smallskip We also observe that the problem (\ref{P}) can be considered under conditions that are dual to $(NQ)_+$ and $(F_1)$. More specifically, assuming \begin{enumerate} \item[$(\hat{F_1})$] $\lambda_j \leq L(x) \leq K(x) \leq \lambda_{j+1}, \mbox{ a.e. } x \in \Omega$. Furthermore, if $K(x)\equiv \lambda_{j+1}$, there exist $D_- \in L^1(\Omega)$ such that $$ F(x,s) \leq \frac{\lambda_{j+1}}{2}s^2 + D_-(x), \quad\forall\, s\in\mathbb{R}, \mbox{ a.e. } x \in \Omega,$$ \end{enumerate} and \begin{enumerate} \item[$(NQ)_-$\hspace{-0.52cm} ] \hspace{0.30cm} there exist $\Omega_0 \subset \Omega$ with positive measure and $C_- \in L^1(\Omega)$ such that $ \hspace{0.52cm} \begin{array}{rll} \mbox{(i)}& \lim\limits_{|s|\to \infty} H(x,s) = -\infty, & \mbox{a.e. } x\in \Omega_0, \vspace{0.1cm} \\ \mbox{(ii)} & H(x,s) \leq C_-(x), & \forall\thinspace s\in\mathbb{R}, \mbox{a.e. } x\in \Omega, \end{array} $ \end{enumerate} an argument similar to the one employed in the proof of Theorem \ref{intro_th2} provides the following theorem. \begin{theorem} There exists $0<\alpha<\mu(\Omega)$ such that, if $f \in C^1(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ satisfies $f(x,0) \equiv 0$, $(f_1)$, $(f_2)$, $(\hat{F_1})$, $(F_2)$ and $(NQ)_-$ with $\mu(\Omega_0)>\alpha$, then the problem (\ref{P}) possesses a nonzero solution. \label{intro_th2'} \end{theorem} \section{Proofs of Theorems \ref{intro_th3} and \ref{intro_th4}} We start by observing that, under hypothesis $(\hat{f_1})$, the functional $I$ given by (\ref{definition_functional}) is of class $C^1$ in $E=H_0^1(\Omega)$ and that the weak solutions of (\ref{P}) are the critical points of $I$. \paragraph{Proof of Theorem \ref{intro_th3}} The same arguments employed in the proof of Theorem \ref{intro_th2} show that $I$ satisfies $(I_1)$ and $(I_2)$. Thus, we may invoke Lemma \ref{aplm1} and Theorem \ref{elves_th} to derive the existence of a critical point for $I$. \hfill $\diamondsuit$\smallskip Considering the dual conditions $(NQ)_-$ and $(\hat{F_1})$, we may apply the above argument to prove \begin{theorem} There exist $0<\alpha<\mu(\Omega)$ such that, if $f \in C(\overline{\Omega}\times\mathbb{R},\mathbb{R})$ satisfies $(\hat{f_1})$, $(\hat{F_1})$, $(F_2)$ and $(NQ)_-$ with $\mu(\Omega_0)>\alpha$, then the problem (\ref{P}) possesses a weak solution. \label{intro_th3'} \end{theorem} Finally, let us establish the multiplicity of solutions for the resonance at the first eigenvalue. \paragraph{Proof of Theorem \ref{intro_th4}} Using $(\hat{f_1})$, $(\hat{F_2})$, $K(x) \leq \lambda_1$ and Remark \ref{sce_lambda1}, we conclude that $I$ satisfies (SCe). The Lemma \ref{aplm2} and the argument used in the proof of Theorem \ref{intro_th2} imply that $I$ is bounded from bellow. Thus, by (SCe), the infimun of $I$ is attained at a critical point $u \in E$. Hence, to prove Theorem \ref{intro_th4}, it suffices to verify that $I(t\phi_1) \leq 0$, for some $t>0$, where $\phi_1$ is the first eigenfunction of $-\Delta$ on $\Omega$ with zero boundary conditions. Suppose first that $L_0(x) \equiv \lambda_1$. The regularity of $\phi_1$ implies that, for $t>0$ sufficiently small, we have $00$, we may use the definition of $L_0(x)$ and the same argument of the proof of Lemma \ref{egorov}, to obtain $\tilde{\Omega} \subset \Omega$ such that $\mu(\Omega \setminus \tilde{\Omega}) < \varepsilon$ and \begin{equation} F(x,s) \geq \frac{1}{2} ( L_0(x) - \varepsilon) s^2,\quad00$. Furthermore, we may suppose that $\varepsilon$ and $r$ are so that \begin{equation} \int_{\tilde{\Omega}} (L_0(x)-\lambda_1) \phi_1^2 \, dx \geq \alpha>0, \label{eq_11_2} \end{equation} and $r0$ sufficiently small and for almost everywhere $x\in \Omega$. 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Matem\'atica, Universidade de Bras\'{\i}lia \\ 70910 Brasilia, DF Brazil \\ e-mail: furtado@mat.unb.br \smallskip \noindent{\sc Elves A. B. Silva }\\ Dpto. Matem\'atica, Universidade de Bras\'{\i}lia \\ 70910 Brasilia, DF Brazil \\ e-mail: elves@mat.unb.br \end{document}