\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Existence of positive solutions \hfil EJDE--2001/11} {EJDE--2001/11 \hfil C. O. Alves \& O. H. Miyagaki \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 11, pp. 1--12. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence of positive solutions to a superlinear elliptic problem % \thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J10, 35A15. \hfil\break\indent {\em Key words:} Superlinear, Mountain Pass, Schrodinger equation, elliptic equation. \hfil\break\indent Partially supported by CNPq - Brazil and PRONEX-MCT \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted November 11, 2000. Published January 24, 2001.} } \date{} % \author{ C. O. Alves \& O. H. Miyagaki \\ \quad \\ {\em Dedicated to Professor J. V. Goncalves }} \maketitle \begin{abstract} We study the existence of positive solutions to the semilinear elliptic problem $$- \epsilon^2 \Delta u + V(z) u = f(u)$$ in $\mathbb{R}^N$ ($N\geq 2$), where the function $f$ has superlinear growth at infinity without any restriction from aboveon its growth. \end{abstract} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} We are concerned with the existence of positive solutions to the semilinear elliptic problem $$\label{Peps} - \epsilon^2 \Delta u + V(z) u = f(u), \quad \mbox{in } \mathbb{R}^N\ (N\geq 2),$$ where $\epsilon$ is a positive parameter, $V: \mathbb{R}^N \to [0,+ \infty)$ and $f:[0,+ \infty) \to [0, + \infty)$ are non-negative continuous functions. We study here the superlinear problem, that is, when the nonlinearity $f$ satisfies the conditions \begin{description} \item{\bf F1:} $\lim_{t \to \infty} \frac{f(t)}{t} = + \infty$. \item{\bf F2:} The Ambrosetti-Rabinowitz growth condition: There exists $\theta >2$ such that $$0 \leq \theta F(t) =\theta \int_{0}^{t} f(s)\, ds \leq t f(t), \quad t \in \mathbb{R}.$$ \end{description} % There are many papers that study (\ref{Peps}) under several assumptions on the potential $V$ and on the growth of $f$. It is well known that solvability of (\ref{Peps}) depends on the rate of growth of $f$ at infinity and that the cases $N\geq 3$ and $N=2$ are strikingly different. We can divide these studies in three cases as defined below, where we use the convention $$2^* := \frac{2N}{N-2}\,.$$ \paragraph{Subcritical growth:} $\displaystyle \lim_{t \to + \infty} \frac{|f(t)|}{|t|^{2^*}}= 0$, if $N \geq 3$; and $\displaystyle \lim_{t \to + \infty} \frac{|f(t)|}{\exp(\alpha t^2)}= 0$, for all $\alpha$, if $N=2$. \paragraph{Critical growth:} $\displaystyle \lim_{t \to + \infty} \frac{|f(t)|}{|t|^{2^*}}= L$ with $L>0$, if $N \geq 3$; and for $N=2$, there exists $\alpha_0 > 0$ such that $$\lim_{t \to + \infty} \frac{|f(t)|}{\exp(\alpha t^2)}= 0 \quad \forall \alpha > \alpha_0, \quad \lim_{t \to + \infty} \frac{|f(t)|}{\exp(\alpha t^2)}= +\infty\quad \forall \alpha < \alpha_0\,.$$ \paragraph{Supercritical growth:} $\displaystyle \lim_{t \to + \infty} \frac{|f(t)|}{|t|^{2^*}}= + \infty$, if $N \geq 3$; and $\displaystyle \lim_{t \to + \infty} \frac{|f(t)|}{\exp(\alpha t^2)}= +\infty$ for all $\alpha$, if $N =2$. \medskip We begin by recalling some results for subcritical growth case. For $(N\geq3)$, Rabinowitz \cite{R1} has found a solution with minimal energy for all small $\epsilon$, when $$\liminf_{|z |\to \infty} V(z) >\inf_{z \in \mathbb{R}^N} V(z)\equiv V_0> 0\,.$$ In the case $N=1$ and $p=3$, Floer and Weinstein \cite{FW}, still imposing a global condition on $V$, have shown that the solution concentrates around of the critical point of $V$, as $\epsilon \to 0$. This result was extended by Oh \cite{O1,O2} and by Wang \cite{W} for higher dimensions $N \geq 3$. In the case $N\geq 3$, Ambrosetti-Badiale and Cingolani \cite{ABC}, basead on the Lyapunov-Schmidt reduction, showed a similar result with the concentration involving a local maximun of $V$. Del Pino and Felmer \cite{DF} assume only that $V$ has a local minima in a bounded set $\Lambda \subset \mathbb{R}^N$ with $$\inf_{\bar{V}} V < \inf_{\partial \Lambda} V\,$$ and some additional hypotheses on $f$. They use local variational techniques without any global restriction involving the minimun of $V$ to concluded that the solutions of (\ref{Peps}) with $N\geq 3$ concentrate around local minima of $V$. Ren and Wei \cite{We} also studied the behavior of solutions to (\ref{Peps}) on $\mathbb{R}^2$ with $\epsilon =1$ and $f(u)=u^{\tau}$, as $\tau \to \infty$. For the critical case the first author and Souto \cite{AS} have considered (\ref{Peps}) with $N\geq 3$ and $V$ having same global property given in \cite{R1} but with $f(u):= \lambda u^q + u^{2^* -1}$ where $\lambda > 0$ and $1< q< 2^* -1$, and they proved that the solutions also concentrate in the global minima of $V$. Later, the first author together with do \'O and Souto \cite{AOS} using the same arguments explored in \cite{DF} showed that similar fenomena holds for local minima of $V$ when $f$ has the growth found in \cite{AS}. For the case involving critical growth in $N=2$, we cite the paper by do \'O and Souto \cite{OS} that worked with local minima of $V$ studying also the concentration of solutions. Imposing among others assumption on $f$ and $V$, for instance that $V$ is a nonconstant function having a finite limit at infinity, Cao \cite{C} proved some existence result for (\ref{Peps}). For the situation involving supercritical growth when $N\geq 3$, we cite the work of the first author \cite{A}, where he studied problem (\ref{Peps}) assuming that $f(u)= u^p (p > 1)$ without any hypothesis on $p$ besides supposing that $V$ is radial and satisfies the following condition: There exist positive constants $R_1 0$ in $\Lambda^{c}=B_{R_2}^{c} \cup B_{R_1}$. \end{description} In \cite{A}, the author does not study the concentration phenomena, there the result obtained involves only the existence of positive solutions to (\ref{Peps}) for $\epsilon$ sufficiently small. Here we shall study problem (\ref{Peps}) with $N\geq 2$ and show the existence of positive solutions imposing assumptions on the function $f$. We will explore the geometric conditions V1 and V2 in order to conclude that growth of $f$ can be made in some sense free". We will show that in dimension $N\geq 3$, if such conditions on $V$ hold the function $f$ can have an exponential growth. The main fact is that the geometry of $V$ implies that we do not need any additional restrictions from above on growth of $f$. Similarly, for $N=2$ the function $f$ can have the behavior like $\exp( \beta u^s)$ with $\beta > 0$ and $s \geq 2$, which is known in the literature as supercritical growth in $\mathbb{R}^2$. Thus, the growth above implies that (\ref{Peps}) can not be solved directly by applying the usual variational methods, because in this case the energy functional related to problem (\ref{Peps}) is not well defined on the suitable Sobolev spaces $H^1(\mathbb{R}^N)$ or $H^1_{{\rm rad}}(\mathbb{R}^N)$. To show the main result, we use similar arguments to those used in \cite{DF} and \cite{A}. The strategy consists of exploring the special deformation on the nonlinearity $f$ and some properties on the radial functions. Before to write our main result, we fix the hypotheses on $f$. In our work we assume that the function $f$ is continuous and verifies the following conditions \begin{description} \item{\bf F3:} $\displaystyle \frac{f(t)}{t}$ is non-decreasing with respect to $t$, for $t>0$ \item{\bf F4:} $\displaystyle \lim_{t \to 0} \frac{f(t)}{t} =0$. \end{description} \begin{thm} \label{thm1} Assume Conditions F1-F4, V1, V2. Then, there exists $\epsilon_o > 0$ such that for all $\epsilon \in (0,\epsilon_o)$, problem (\ref{Peps}) has a classical solution $u_{\epsilon} \in H^1(\mathbb{R}^N)$ with $$u_{\epsilon}(z) \to 0, \quad \mbox{as } |z|\to \infty\,.$$ \end{thm} \paragraph{Remark:} Theorem \ref{thm1} improves and complements the results showed in \cite{A} and \cite{C} respectively, because in our work we study the behavior on other nonlinearities and our approach treats at same time the cases $N\geq 3$ and $N=2$. \smallskip Hereafter, $\int_{U} f$ represents $\int_{U} f(z)dz$ and $$H^1_{{\rm rad}} = H_{{\rm rad}}^1(\mathbb{R}^N)=\{u \in H^1(\mathbb{R}^N): \mbox{ u is radially symmetric} \}\,.$$ \section{Preliminaries} In this section, we prove some auxiliary results for the proof of Theorem $\ref{thm1}$. Since we are concerned with positive solutions, we can assume in the sequel that $f(t) =0$ for $t \leq 0$. \begin{lemma} \label{lemma1} Let $g:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ be a continuous and radially symmetric function, that is, $g(z,u)=g(|z|,u)$, for all $z \in \mathbb{R}^N$ and $\in \mathbb{R}$. Given positive constants $a$ and $b$, let $$A=\{ z \in \mathbb{R}^N: a < |z |< b\} \quad\mbox{and}\quad G(z,t):=\int_{0}^{t}g(z,s)\,ds\,.$$ If $u_{n} \rightharpoonup u$ weakly in $H_{{\rm rad}}^1$, then $$\int_{A} g(z,u_{n})u_{n} \to \int_{A}g(z,u)u \ \ \mbox{and} \ \ \int_{A} G(z,u_{n}) \to \int_{A} G(z,u),\ \mbox{as} \to \infty\,.$$ \end{lemma} \paragraph{Proof.} Since $u_{n} \rightharpoonup u$ weakly in $H_{{\rm rad}}^1$, there exists a positive constant $C$, such that $\|u_{n} \|\leq C$. Using Straus's inequality (see \cite{K} or \cite{St}), $$|u_{n}(z)|\leq \frac{2\pi\|u_{n}\|}{|z \mid^{1/2}},\ \forall z \in \mathbb{R}^N\setminus \{0\} \label{eq:eq1}$$ we obtain $$|u(z) |\leq \frac{2\pi C}{a^{1/2}} \equiv \bar{a} \in L^1(A),\ \forall z \in \mathbb{R}^N\setminus \{0\}.$$ From this, we have $$|g(z,u_{n}) u_{n} |\leq \max_{(z,t) \in A \times [-\bar{a}, \bar{a}]}g(z,t) \bar{a} \ \equiv \ \bar{c} \in L^1(A),\ \forall z \in \mathbb{R}^N\setminus \{0\}.$$ Similarly, $$|G(z,u_{n})| \leq \hat{c} \in L^1(A),\ \forall z \in \mathbb{R}^N\setminus \{0\}.$$ Then from the Lebesgue dominated convergence theorem, we conclude the present proof. \hfill$\diamondsuit$\medskip Let $$g(z,t) = \chi_{\Lambda}(z) f(t) + (1 - \chi_{\Lambda})(z) \bar{f}(t),$$ where $\chi_{\Lambda}$ denotes the characteristic function on $\Lambda$, $$\bar{f}(t) = \left\{ \begin{array}{rr} f(t) & t \leq a\,, \\[3pt] \frac{V_0t}{k} & t > a, \end{array} \right.$$ and $a$ is a positive constant so that $\frac{f(a)}{a} =\frac{V_0}{k}$ with $k > \max\{\frac{\theta}{\theta - 2}, 2\}$. It is easy to see that $g$ satisfies not only the condition F2, with $f$ replaced by $g$, but also the following conditions \begin{description} \item{\bf G2:} $0 \leq \theta G(z,t) \leq g(z,t) t$ for all $z \in \Lambda$, $t\in \mathbb{R}$. \item{\bf G3:} $0 \leq 2 G(z,t) \leq g(z,t) t \leq \frac{V(z) t^2}{k}$ for $z \in \Lambda^{c}$, $t \in \mathbb{R}$. \end{description} In the sequel, we denote by G1, the condition F2 with $f$ replaced by $g$. Now we shall state the crucial auxiliary result. \begin{thm} \label{thm2} Assume Conditions V1, V2, and G1--G3. Then the problem $$\label{Plambda} - \Delta u + V(z) u =g(z,u), \quad\mbox{in }\mathbb{R}^N$$ admits a positive solution. \end{thm} To prove this theorem, we first fix notation and prove some technical results. We work in the Hilbert space $$E =\{ u \in H_{{\rm rad}}^1(\mathbb{R}^N): \int_{\mathbb{R}^N} V u^2 < \ \infty \}$$ endowed by the norm $$\|u \| = \left( \int_{\mathbb{R}^N}( |\nabla u \mid^2 +Vu^2) \right)^{1/2}\,.$$ We shall find critical points on $E$ of the $C^1$ functional $$I(u) =\int_{\mathbb{R}^N} \frac{1}{2}(\mid \nabla u \mid^2 +V u^2) - \int_{\mathbb{R}^N} G(z,u)$$ whose Fr\'echet derivative is $$\langle I'(u),v \rangle = \int_{\mathbb{R}^N}( \nabla u \cdot \nabla v +V u v - g(z,u) v), \quad u, v \in E\,.$$ Next, we shall prove some lemmas related to this functional. \begin{lemma} \label{lemma2} $I$ satisfies the following conditions \begin{description} \item [{\bf (i)}] There exist $\rho,\beta > 0$ such that $I(u) \geq \beta$ for $\|u \|= \rho$ \item [{\bf (ii)}] There exists $e \in E$ with $\|e \|>\rho$ such that $I(e) < 0$. \end{description} \end{lemma} \paragraph{Proof.} Part (i): From F4, given $\epsilon > 0$, there exists $\delta > 0$ such that $$F(t) \leq \frac{\epsilon t^2}{2},\quad |t|\leq \delta.$$ Thus $$\label{ro} \int_{\Lambda} F(u) \leq \frac{\epsilon}{2}\int_{\Lambda}u^2, \ \mbox{as}\ \ ||u|| \leq \rho, \ \rho \ \mbox{small enough}$$ Now, using condition G3 and $(\ref{ro})$, we have \begin{eqnarray} I(u) & = & (\int_{\Lambda} + \int_{\Lambda^{c}}) (\frac{1}{2} ( |\nabla u \mid^2 + V(z)u^2) - G(z,u))dz \nonumber \\ & \geq & \frac{1}{2}\int_{\mathbb{R}^N}( |\nabla u \mid^2 + V u^2) - \int_{\Lambda} F(u) - \frac{1}{2k}\int_{\Lambda^{c}} V u^2 \nonumber \\ & \geq & \frac{1}{2} \int_{\mathbb{R}^N} |\nabla u \mid^2 + \frac{1}{2}(1 - \frac{1}{k}) \int_{\mathbb{R}^N} V u^2 - \int_{\Lambda} F(u) \nonumber \\ & \geq & \frac{1}{2} \int_{\mathbb{R}^N} ( |\nabla u \mid^2 + (1 - \frac{1}{k}) V u^2) - \frac{\epsilon}{2} \int_{\Lambda} u^2 \nonumber \\ & \geq & C_1\|u \|^2 - \frac{\epsilon}{2} \int_{\Lambda} u^2\,. %\nonumber \label{eq:eq2} \end{eqnarray} Recalling that $$\int_{\Lambda} u^2 \leq C \int_{\mathbb{R}^N}(|\nabla u|^2 + V u^2) ,$$ from (\ref{eq:eq2}) we have $$I(u) \geq C_2 \|u \|^2,\quad\mbox{for } ||u||=\rho.$$ The proof of part $(i)$ is complete. \smallskip \noindent Verification of part (ii): Choose $\psi \in C_0^{\infty}(\Lambda)$, so that $\psi > \psi_0 > 0$ for all $x \in \mbox{K} \subset \mathop{\rm supp}\psi$. Then, by condition F2 there exists a positive constant $C_1$, such that $$F(t\psi) \geq C (t\psi)^{\theta}, \ \ t \geq t_0, \ \forall z \in K,\ \ t_0 > 0.$$ Using this inequality, we get \begin{eqnarray} I(t\psi) & = & \frac{t^2}{2} \|\psi \|^2 - \int_{\Lambda} G(z,t\psi) \nonumber \\ & \leq & \frac{t^2}{2} \|\psi \|^2 - \int_{K} F(t \psi)\nonumber \\ & \leq & \frac{t^2}{2} \|\psi \|^2 - C_1 t^{\theta} ,\ \mbox{for} \ t \geq t_0. \label{eq:100} \end{eqnarray} This proves $(ii)$ and it completes the proof of Lemma~$\ref{lemma2}$. \hfill$\diamondsuit$ \smallskip Now, by using Ambrosetti and Rabinowitz Mountain Pass Theorem \cite{AR}, there exists a $(PS)_{c}$ sequence $\{u_{n} \}$; that is, $$I(u_{n}) \to c \quad \mbox{and} \quad I'(u_{n}) \to 0,$$ where $c=\inf_{h\in \Gamma} \max_{t\in [0,1]} I(h(t))$ and $$\Gamma =\{ h \in C([0,1], E): h(0)= 0, h(1) = e \}.$$ \begin{lemma} \label{lemma3} The functional $I$ satisfies the $(PS)_{c}$ condition for all $c \in \mathbb{R}$. \end{lemma} \paragraph{Proof:} Firstly, from Conditions G2 and G3, we have \begin{eqnarray*} \lefteqn{\|u_{n} \| + M } \\ & \geq & I(u_{n}) - \frac{1}{\theta}I'(u_{n}) u_{n} \\ & = & (\frac{1}{2} - \frac{1}{\theta})\int_{\mathbb{R}^N}( |\nabla u_{n} \mid^2 + V u_{n}^2) + (\int_{\Lambda} + \int_{\Lambda^{c}})(\frac{g(z,u_{n})u_{n}}{\theta} - G(z,u_{n})) \\ & \geq & (\frac{1}{2} - \frac{1}{\theta})\int_{\mathbb{R}^N}( |\nabla u_{n} \mid^2 + V u_{n}^2) + \int_{\Lambda^{c}}(\frac{g(z,u_{n})u_{n}}{\theta} - G(z,u_{n})) \\ & \geq & (\frac{1}{2} - \frac{1}{\theta})(\int_{\mathbb{R}^N}( |\nabla u_{n} \mid^2 + V u_{n}^2) - \int_{\Lambda^{c}}g(z,u_{n})u_{n}) \\ & \geq & (\frac{1}{2} - \frac{1}{\theta})(\int_{\mathbb{R}^N} |\nabla u_{n} \mid^2 +(1 - \frac{1}{k})\int_{\mathbb{R}^N} V u_{n}^2)\,. \end{eqnarray*} By this inequality, there exists a constant $C>0$ such that $\|u_{n}\|+ M \geq C \|u_{n} \|^2$, which implies that $\{u_{n} \}$ is bounded in E. Therefore, up to subsequence, there exists $u \in E$ such that $$u_{n} \rightharpoonup u \mbox{ weakly in } E, \quad\mbox{and} \quad u_{n} \to u, \mbox{ a.e. in } \mathbb{R}^N.$$ % Now we state the following\\ \noindent{\bf Claim 1} Given $\epsilon > 0$, there exists a $R > 4R_2$ such that $$\limsup_{n \to \infty} \int_{|z |> R} (|\nabla u_{n} |^2 + V u_{n}^2) <\epsilon.$$ % Proof of claim 1: Arguing as in $\cite{A}$ and $\cite{DF}$, from Conditions G2 and G3, and taking a cut-off function $\eta_{R}\in C_0^{\infty}(\mathbb{R}^N)$ satisfying $$\eta_{R}=0 \ \mbox{in} \ B_{R/2},\quad \eta_{R} = 1, \ \mbox{in} \ B_{R}^{c}\quad \mbox{and} \quad |\nabla \eta_{R} | \leq \frac{C}{R},$$ we obtain \begin{eqnarray*} \lefteqn{I'(u_{n}) (u_{n}\eta_{R})} \\ & = & \int_{B_{R/2}^{c}}(|\nabla u_{n} \mid^2 + V u_{n}^2)\eta_{R} + \int_{B_{R} \setminus B_{R/2}} u_{n} |\nabla u_{n} |\nabla \eta_{R} - \int_{B_{R/2}^{c}} g(z,u_{n}) u_{n} \eta_{R} \\ & \geq & \int_{B_{R/2}^{c}}(|\nabla u_{n} \mid^2 + V u_{n}^2)\eta_{R} - |u_{n} |_2 |\nabla u_{n} |_2\frac{C}{R} - \frac{1}{k}\int_{B_{R/2}^{c}} V u_{n}^2 \eta_{R} + r(n)\,. \end{eqnarray*} where $r(n)$ is an $o(1)$-function as $n$ approaches $+\infty$. Since $I'(u_{n}) (u_{n}\eta_{R})=o(1)$, we have \begin{eqnarray*} (1 - \frac{1}{k})\int_{B_{R}^{c}}(|\nabla u_{n} \mid^2 + V u_{n}^2)\eta_{R} & \leq & (1 - \frac{1}{k})\int_{B_{R/2}^{c}}(|\nabla u_{n} \mid^2 + V u_{n}^2)\eta_{R} \\ & \leq & \frac{C}{R}( \mid u_{n}|_2 |\nabla u_{n} |_2) + o(1), \\ & \leq & \frac{C_1}{R} + o(1). \end{eqnarray*} So that the proof of Claim 1 follows by choosing $R > C_1/\epsilon$. \smallskip \noindent{\bf Claim 2:} \begin{description} \item[{\bf (i)}] $\int_{\mathbb{R}^N} g(z,u_{n}) u_{n} \to \int_{\mathbb{R}^N} g(z,u) u$, \item[{\bf (ii)}] $u$ is a critical point of $I$, that is, $I'(u) v = 0$ for all $v \in E$. \end{description} Assuming Claim 2, from $I'(u_{n}) u_{n} = o(1)$, it follows that \begin{eqnarray*} \|u_{n} \|^2 & = & \int_{\mathbb{R}^N} g(z,u_{n}) u_{n} + o(1) \\ & = & \int_{\mathbb{R}^N} g(z,u) u + o(1) \\ & = & \|u \|^2 + o(1)\,. \end{eqnarray*} Therefore, $u_{n} \to u$ strongly in $E$. \\[3pt] Proof of Claim 2 Part i): Note that \begin{eqnarray*} \int_{\mathbb{R}^2} (g(z,u_{n}) u_{n} - g(z,u) u) & = & (\int_{B_{R_1}} + \int_{B_{R} \setminus B_{R_1}} + \int_{B_{R}^{c}})(g(z,u_{n})u_{n} - g(z,u)u) \\ & = & I_1 + I_2 + I_{3}. \end{eqnarray*} We shall prove that each of these terms approaches zero as $n \to \infty$. From the boundedness of $B_{R_1} \subset \Lambda^{c}$, we have $u_{n} \to u,$ in $L^2(B_{R_1})$. By Condition G3 it follows that $I_1 \to 0$. From Lemma \ref{lemma1}, we conclude that $I_2 \to 0$. Finally, combining Claim 1 and condition G3, we get $I_{3} \to 0$. Then $(i)$ holds. \\[3pt] Proof of Claim 2 Part (ii): Since $I'(u_{n})v = o(1)$, it suffices to prove the following $$\int_{\mathbb{R}^N} g(z,u_{n})v \to \int_{\mathbb{R}^N} g(z,u) v,\ \mbox{as}\ \to \infty.$$ Arguing as before, splitting the integral in two,we obtain \begin{eqnarray*} \int_{\mathbb{R}^N} (g(z,u_{n}) - g(z,u))v & = & ( \int_{\Lambda} + \int_{\Lambda^{c}})(g(z,u_{n}) -g(z,u))v \\ & = & J_1 + J_2. \end{eqnarray*} From the behaviour of $u_{n}$, that is by ({\ref{eq:eq1}}), we have $$|u_{n}(x)|\leq \frac{C}{R_1^{1/2}} \equiv a \label{eq:eq4}$$ and since $g$ is a bounded function on $\Lambda$, applying Lebesgue's Dominated Convergence Theorem follows that $J_1 \to 0$, as $n \to \infty$. Now, from $(\ref{eq:eq4})$ and Conditions G3, we get $$\int_{\Lambda^{c}}(g(z,u_{n}) - g(z,u))^2 \leq \int_{\Lambda^{c}}(\frac{V_0(|u_{n}|+ |u \mid)}{k})^2 \leq \ \int_{\Lambda^{c}} C (|u_{n}\mid^2 + |u \mid^2) \leq C_1,$$ for some positive constant $C_1$. Now, using a Lemma from Brezis and Lieb \cite{K}, it follows that $J_2 \to 0$. This completes the proof of Lemma $\ref{lemma3}$. \hfill $\diamondsuit$ \paragraph{Proof of Theorem \ref{thm2}} From Lemmas $\ref{lemma2}$ and $\ref{lemma3}$, problem (\ref{Plambda}) has at least one positive weak solution $u \in E$. Similarly, for each $\epsilon > 0$, there exists $u_{\epsilon} \in E$ weak positive solution of (\ref{Plambda}), satisfying $$I'_{\epsilon}(u_{\epsilon}) v = 0, \quad \forall v \in E,$$ where $$I_{\epsilon}(u) = \int_{\mathbb{R}^N} \frac{1}{2}( \epsilon^2 |\nabla u \mid^2 + V u^2) - \int_{\mathbb{R}^N} G(z,u).$$ \section{Proof of Theorem \ref{thm1}} Let $\{ u_{\epsilon} \}$ be the sequence of positive weak solutions of (\ref{Plambda}) obtained in the previous section. The crucial result for this section is the following. \begin{lemma} \label{lemma5} $\|u_{\epsilon} \|_{H^1}\to 0$ as $\epsilon \to 0$. \end{lemma} \paragraph{Proof.} Note that $u_{\epsilon}$ satisfies $$I_{\epsilon}(u_{\epsilon}) = c_{\epsilon} \quad \mbox{and} \quad I'_{\epsilon}(u_{\epsilon}) v = 0, \ \forall v \in E_{\epsilon},$$ where $\displaystyle c_{\epsilon} = \inf_{\psi \in E_{\epsilon}} \max_{t \geq 0} I_{\epsilon}(t \psi)$ and $$E_{\epsilon}= \{ u \in H_{{\rm rad}}^1: \int_{\mathbb{R}^2} \frac{1}{2}( \epsilon^N |\nabla u \mid^2 + V u^2) < \infty \}.$$ Taking $\psi \in C_{o,{\rm rad}}^{\infty}(\Omega)$, a nonnegative function with $\mathop{\rm supp} \psi\subset \Omega$, there is an unique $t_{\epsilon} \in \mathbb{R}^{+}$ such that $$I_{\epsilon}(t_{\epsilon}\psi) = \max_{t\leq 0} I_{\epsilon}(t\psi),$$ so $$0 \leq c_{\epsilon} \leq I_{\epsilon}(t_{\epsilon}\psi) \leq \frac{t_{\epsilon}^2}{2}\int_{\Omega} \epsilon^2 |\nabla \psi|^2 - \int_{\Omega} F(t_{\epsilon} \psi).$$ On the other hand, we know that $$\epsilon^2\int_{\Omega}|\nabla \psi |^2= \int_{\Omega} \frac{f(t_{\epsilon}\psi)}{t_{\epsilon}} \psi, \label{eq:102}$$ choosing $\Omega_1 \subset \Omega$ such that $\psi(z) \geq \psi_0>0 \ \forall z \in \Omega_1$, it follows $$\epsilon^2\int_{\Omega}|\nabla \psi |^2 \geq \int_{\Omega_1} \frac{f(t_{\epsilon}\psi)}{t_{\epsilon}} \psi \geq \psi_0^2\int_{\Omega_1} \frac{f(t_{\epsilon}\psi)}{t_{\epsilon}\psi}, \label{eq:103}$$ thus from (\ref{eq:103}) and Conditions F1--F3 that $t_{\epsilon} \to 0$ as $\epsilon \to 0$. Now, remarking that $$\label{eq:104} c_{\epsilon} \leq I_{\epsilon}(t_{\epsilon}\psi)= (t^2_{\epsilon}/2)||\psi||^2 - \int_{\mathbb{R}^N}F(t_{\epsilon}\psi)\leq (t^2_{\epsilon}/2)||\psi||^2$$ and arguing as in the proof of Lemma $\ref{lemma3}$, we obtain \begin{eqnarray*} I_{\epsilon}(u_{\epsilon}) & = & I_{\epsilon}(u_{\epsilon}) - \frac{1}{\theta} I_{\epsilon}'(u_{\epsilon})u_{\epsilon} \\ & \geq & (\frac{1}{2} - \frac{1}{\theta}) ( \int_{\mathbb{R}^N} (\epsilon^2 |\nabla u_{\epsilon} \mid^2 + (1 - \frac{1}{k})V u_{\epsilon}^2) \\ & \geq & C \epsilon^2 \int_{\mathbb{R}^N}( |\nabla u_{\epsilon} \mid^2 + V u_{\epsilon}^2). \end{eqnarray*} Hence, combining this last inequality with (\ref{eq:104}), we have $$C \epsilon^2\int_{\mathbb{R}^N}|\nabla u_{\epsilon} \mid^2 + V u_{\epsilon}^2 \leq I_{\epsilon}(u_{\epsilon}) \leq \frac{t_{\epsilon}^2 \epsilon^2}{2}\int_{\Omega}|\nabla \psi \mid^2,$$ that is, $$\|u_{\epsilon} \|^2_{H^1} \leq C \|u_{\epsilon} \|^2 \leq \frac{t_{\epsilon}^2}{2}\int_{\Omega}|\nabla \psi \mid^2\,.$$ Therefore, the proof of Lemma \ref{lemma5} is complete. \hfill $\diamondsuit$ \smallskip Next, using an argument similar to those used in \cite{DF}, we will prove that $u_{\epsilon}$ is a solution of (\ref{Peps}). For each $\epsilon >0$, from $(\ref{eq:eq1})$ we have $$m_{\epsilon}^1 = \max_{\partial B_{R_1}} u_{\epsilon}(z) \to 0, \quad \mbox{as } \epsilon \to 0, \label{eq:3.1}$$ and $$m_{\epsilon}^2 = \max_{\partial B_{R_2}} u_{\epsilon}(z) \ \to \ 0, \quad \mbox{as } \epsilon \to 0. \label{eq:3.2}$$ Combining $(\ref{eq:3.1})$ and $(\ref{eq:3.2})$, there exists $\epsilon_o > 0$ such that $$m_{\epsilon}^{i} < a, \quad \forall \epsilon \in (0,\epsilon_o), \ i=1,2.$$ Now, since $(u_{\epsilon} - a)_{+} \in E_{\epsilon}$, we have $$\int_{\mathbb{R}^N \setminus \bar{\Lambda}} \epsilon^2 \mid \nabla (u_{\epsilon} - a)_{+}\mid^2 + V u_{\epsilon} (u_{\epsilon} - a)_{+} = \int_{\mathbb{R}^N \setminus \bar{\Lambda}} (g(z,u_{\epsilon})u_{\epsilon} (u_{\epsilon} - a)_{+}. \label{eq:3.3}$$ On the other hand, from $G3$, we obtain $$V u_{\epsilon} (u_{\epsilon} - a)_{+} - g(z,u_{\epsilon}) u_{\epsilon} (u_{\epsilon} - a)_{+} \geq 0, \quad \forall z \in \Lambda^{c},$$ which together with $(\ref{eq:3.3})$, we have $$\int_{\mathbb{R}^N \setminus \bar{\Lambda}} \epsilon^2 |\nabla (u_{\epsilon} - a)_{+}\mid^2 =0.$$ Therefore, $u_{\epsilon}(z) \leq a$ for all $z \in \mathbb{R}^N \setminus \bar{\Lambda}$. Using this, we conclude that $$g(z,u_{\epsilon}(z)) = f(u_{\epsilon}(z)), \quad \forall z \in \mathbb{R}^N \setminus \bar{\Lambda}\,.$$ So, for all $\epsilon \in (0,\epsilon_o)$, $u_{\epsilon}$ satisfies $$\int_{\mathbb{R}^N} (\epsilon^2 \nabla u_{\epsilon} \nabla \eta + V u_{\epsilon}\eta )= \int_{\mathbb{R}^N}f(u_{\epsilon}) \eta, \ \forall \eta \in E_{\epsilon}.$$ Thus, we infer that $f(u_{\epsilon}) \in L_{loc}^1(\mathbb{R}^N).$ On the other hand, using a result by Alves, de Moraes Filho and Souto (see \cite[Lemma1]{AMS}), we can conclude that $u_{\epsilon}$ satisfies (\ref{Peps}) in $D'(\mathbb{R}^N)$ and by the elliptic regularity (see e.g. \cite{AMS}), we have that $u_{\epsilon} \in C^2(\mathbb{R}^N )$. This completes the proof of Theorem \ref{thm1}. \paragraph{Acknowledgement} The first author would like to thank IMECC - UNICAMP, and in special to the Professor Djairo G. de Figueiredo for his help and encouragement. 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Alves } \\ Universidade Federal da Para\'{\i}ba \\ Departamento de Matem\'atica \\ 58109-970 - Campina Grande (PB), Brazil \\ e-mail: coalves@dme.ufpb.br \smallskip \noindent {\sc Olimpio H. Miyagaki} \\ Universidade Federal de Vi\c cosa \\ Departamento de Matem\'atica \\ 36571-000 Vi\c cosa-MG -Brazil \\ e-mail: olimpio@mail.ufv.br \end{document}