\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 31, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/31\hfil Multiplicity of solutions]
{Multiplicity of solutions for a quasilinear problem
with supercritical growth}
\author[G. M. Figueiredo\hfil EJDE-2006/31\hfilneg]
{Giovany M. Figueiredo}

\address{Giovany M. Figueiredo \newline
Universidade Federal do Par\'{a}\\
CEP 66.075-110, Bel\'{e}m-Par\'{a}, Brazil}
\email{giovany@ufpa.br}

\date{}
\thanks{Submitted October 26, 2005. Published March 16, 2006.}
\subjclass[2000]{35A15, 35H30, 35B40}
\keywords{Variational method; penalization method; Moser iteration method}

\begin{abstract}
 The multiplicity and concentration of positive solutions are
 established for the equation
 $$
 -\epsilon^{p}\Delta_{p}u+V(z)|u|^{p-2}u=|u|^{q-2}u
 +\lambda|u|^{s-2}u \quad\text{in }\mathbb{R}^N,
 $$
 where $1<p<N$, $\epsilon >0$, $p<q<p^* \leq s$,
 $p^{*}=\frac{Np}{N-p}$, $\lambda\geq 0$ and $V$ is a positive
 continuous function.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

This article concerns the multiplicity and
concentration of positive solutions for the problem
\begin{equation} \label{Plamb}
 \begin{gathered}
  -\epsilon^{p}\Delta_{p}u + V(z)|u|^{p-2}u= |u|^{q-2}u +
              \lambda |u|^{s-2}u
               \quad\text{in }\mathbb{R}^N \\
 u \in W^{1,p}(\mathbb{R}^N) \quad\text{with } 1<p<N  \\
 u(z) > 0, \quad\text{for } z \in \mathbb{R}^N,
 \end{gathered}
\end{equation}
$\epsilon > 0$, $p < q < p^{*} \leq s$,
$p^{*}=\frac{Np}{N-p}$, $\lambda \geq 0$ and $\Delta_{p}u$ is
the p-Laplacian operator; that is,
$$
\Delta_{p}u=\sum_{i=1}^{N}{\frac{\partial}{\partial{x_{i}}}
\Big(|\nabla{u}|^{p-2}\frac{\partial{u}}{\partial{x_{i}}}\Big)}.
$$
We assume that $V$ is a continuous function satisfying
\begin{equation}
V(x)\geq V_{0}=\inf_{x \in \mathbb{R}^{N}} V(x)>
0\quad \mbox{for } x\in  \mathbb{R}^{N}; \label{V1}
\end{equation}
Also assume that there exists an open and bounded domain
$\Omega \subset \mathbb{R}^{N}$ such that
\begin{equation}
V_0 < \min_{\partial\Omega}V. \label{V2}
\end{equation}

In recent years,  much attention has been paid to the
existence and multiplicity of solutions for both subcritical and
critical cases and to the concentration behavior of solutions for
problem
\begin{equation}
-\epsilon^{2}\Delta u +V(z)u=f(u) \quad\text{in } \mathbb{R}^{N},
\label{P*}
\end{equation}
when $\epsilon$ is small. Interesting  results may be found,
for example, in
\cite{Alves8,Bartsch,Benci1,Chabr1,Cingolani,Gui,Rabinow}
and their references.

 Cingolani \& Lazzo \cite{Lazzo1}, using Lusternik-Schnirelman
category and involving the sets
\begin{gather*}
M=\{x\in \Omega : V(x)=V_{0}\}, \\
M_{\delta}=\{x\in \mathbb{R}^{N}:dist(x,M)\leq \delta \},
\quad \delta>0,
\end{gather*}
showed a result of multiplicity of positive solutions for
 \eqref{P*}, where $\Omega=\mathbb{R}^{N}$,
$f(u)=|u|^{q-2}u$ with $q \in (2, 2^{*})$,  and
\begin{equation}
V_{\infty}=\liminf_{|x|\to\infty}V(x)>V_{0}=\inf_{\mathbb{R}^{N}}V(x)>0.
\label{V}
\end{equation}
Recall that for a closed subset $Y$  of a topological space
$X$, the Lusternik-Schnirelman category, denoted by
$\mathop{\rm cat}_{X}Y$, is the least number of closed and
contractible sets in $X$ which cover $Y$.


 Alves \& Souto \cite{Alves10} showed an  existence and
concentration result for  \eqref{P*} with
$f(u)=u^{q-1}+u^{2^{*}-1}$ assuming  that condition \eqref{V}
holds.

 Alves \& Figueiredo \cite{AlvesGio1} (see also \cite{giovany})
proved a multiplicity result for
\begin{equation}
-\epsilon^{p}\Delta_{p}u+V(z)|u|^{p-2}u=f(u) \ \ in \ \
\mathbb{R}^N \label{P**}
\end{equation}
using again Lusternik-Schinirelman category and assuming that
condition \eqref{V} holds, $2\leq p<N$ and $f$ belongs to a large class
which includes the model $f(u)=|u|^{q-2}u$ with
$ q \in (p,p^{*})$. Moreover, the authors showed that each solution of
${(P_{**})}$ has a phenomenon of concentration near a point of minimum
of the potential $V$. The case with critical growth  was proved
in \cite{Gio2}.

 del Pino \& Felmer  \cite{Pino} proved that if the conditions
\eqref{V1} and \eqref{V2} hold, problem \eqref{P*} has a positive
solution for small $\epsilon$, which has a phenomenon of
concentration near of one minimum point of potential $V$.

 Alves \& Figueiredo \cite{AlvesGio2}, using the penalization method
and Lusternik-Schnirelman category theory, showed again a
multiplicity and concentration result for \eqref{P**}, using
now the conditions \eqref{V1} and \eqref{V2} with $1<p<N$.

In this work, motivated by \cite{AlvesGio2} and by some ideas
developed \cite{rabino}, \cite{moser} and \cite{Chabr}, we
prove the multiplicity and concentration of positive solutions to
\eqref{Plamb} using Lusternik-Schnirelman category. For
 $\lambda=0$ and $p=2$, we   have the result obtained
in \cite{Lazzo1}. Hence the results of this paper complete those
\cite{Lazzo1} in three senses:  because we deal with $1<p<N$
instead of $p=2$,  because we do  restrict the
behavior of $V$ at infinity,  and because we have
$f(u)=|u|^{q-2}u+\lambda|u|^{s-2}u$ with $s\geq p^{*}$. Moreover,
in the present paper, we continue the study of \cite{AlvesGio2}
and \cite{Gio2}, because we consider supercritical nonlinearities.
To our knowledge there is no results on existence of
solutions to problem $(P_{\lambda })$ via the  penalization method, and
multiplicity results with supercritical growth via the
Lusternik-Schnirelman category theory.

Our main result for problem \eqref{Plamb} is the following.

\begin{theorem}\label{thmD}
Suppose that the function $V$ satisfies \eqref{V1}-\eqref{V2}.
Then, for any $\delta>0$, there exists
$\overline{\epsilon}=\overline{\epsilon}(\delta)>0$ and
$\lambda_{0}>0$ such that \eqref{Plamb} has at least
$\mathop{\rm cat}_{M_{\delta}}M$ positive solutions for all
$\epsilon\in(0,\overline{\epsilon})$ and for all
$\lambda \in [0,\lambda_{0}]$. Moreover, if $u_{\epsilon}$ is a positive
solution of \eqref{Plamb} and $\eta_{\epsilon}\in \mathbb{R}^{N}$
a global maximum point of $u_{\epsilon}$, then
$$
\lim_{\epsilon\to 0}V(\eta_{\epsilon})=V_{0}.
$$
\end{theorem}

To solve problem \eqref{Plamb}, we first consider a
truncated problem which involves only a subcritical Sobolev
exponent. We show that any positive solution of truncated problem is
a positive solution of \eqref{Plamb}.

Hereafter, we will work with the following problem equivalent to
\eqref{Plamb}, which is obtained under change of variable
$z=\epsilon x$
\begin{equation} \label{Plambst}
\begin{gathered}
-\Delta_{p}u + V(\epsilon x)|u|^{p-2}u=|u|^{q-2}u +
  \lambda|u|^{s-2}u \quad\text{in } \mathbb{R}^N        \\
        u \in W^{1,p}(\mathbb{R}^N) \quad\text{with } 1<p<N \\
        u(x) > 0,\quad  \forall   x \in \mathbb{R}^{N} .
\end{gathered}
\end{equation}

\section{Truncated Problem}

First of all, we have to note that because $f$ has supercritical
growth we cannot use directly variational techniques because  of the lack of compactness of the Sobolev immersions.

So we construct a suitable truncation of $f$ in order to use
variational methods or more precisely, the Mountain Pass Theorem.
This truncation was used in  \cite{rabino} (see also
\cite{Chabr} and \cite{giovany}).

Let $K>0$, be a constant to be determined later,  and
$\widehat{f}_{K}:\mathbb{R} \to \mathbb{R}$ given by
 \[
  \widehat{f}_{K}(t) =
\begin{cases}
              0 &\text{if } t < 0 \\
 t^{q-1}+\lambda t^{s-1} &\text{if } 0 \leq t < K \\
 (1+\lambda K^{s - q})t^{q-1} &\text{if } t\geq K.
\end{cases}
\]
Consider $\alpha,\gamma\in\mathbb{R}$ such that $\alpha<1<\gamma$
and $\eta \in C^{1}([\alpha K ,\gamma K])$ with $\alpha$ and
$\gamma$ independent of $K$ and $\eta$ satisfying
\begin{gather*}
 \eta(t)\leq\widehat{f}_{K}(t)\quad \mbox{for all } t\in
[\alpha K ,\gamma K], \\
\eta(\alpha K)=\widehat{f}_{K}(\alpha K),\quad
\eta(\gamma K)=\widehat{f}_{K}(\gamma K), \\
\eta'(\alpha K)=\widehat{f}'_{K}(\alpha K), \quad
\eta'(\gamma K)=\widehat{f}'_{K}(\gamma K), \\
t\mapsto\frac{\eta(t)}{t^{p-1}}
\mbox{ is increasing for all }t\in [\alpha K,\gamma K].
\end{gather*}
Now using the functions $\eta$ and $\widehat{f}_{K}$, we define
\[
  f_{K}(t) =\begin{cases}
  \eta(t) &\text{if } t \in [\alpha K ,\gamma K], \\
  \widehat{f}_{K}(t) &\text{if } t \not\in [\alpha K ,\gamma K]
\end{cases}
\]
and the truncated problem
\begin{equation} \label{Tlamb}
\begin{gathered}
-\Delta_{p}u + V(\epsilon x)|u|^{p-2}u = f_{K}(u) \\
u \in W^{1,p}(\mathbb{R}),\quad u > 0 \quad\text{in } \mathbb{R}^{N}.
\end{gathered}
\end{equation}
It is easy to check that $f_{K} \in C^{1}(\mathbb{R})$, and that
\begin{gather*}
f_{K}(t)=0,\quad \mbox{for all } t < 0, \label{fK1} \\
f_{K}(t)\leq(1+\lambda K^{s-q})t^{q-1}\quad \mbox{for all }
 t\geq 0,\label{fK2} \\
F_{K}(t)\leq\frac{1}{q}(1+\lambda K^{s-q})t^{q}\quad
\mbox{for all } t\geq 0,
\quad F_{K}(t)=\int^{t}_{0}f_{K}(\xi)d\xi, \label{fK3}
\end{gather*}
there exists $\theta \in \mathbb{R}$ such that $p<\theta$ and
\begin{equation}
0 < \theta F_{K}(t) \leq f_{K}(t)t \quad \mbox{for all } t>0,
\label{fK4}
\end{equation}
the function
\begin{gather}
t\mapsto \frac{f_{K}(t)}{t^{p-1}}
 \mbox{ is increasing for all } t>0, \label{fK5} \\
f'_{K}(t)t^{2}-(p-1)f_{K}(t)t\geq (q -p )t^{q}. \label{fK6}
\end{gather}


\begin{remark}\label{final} \rm
Note that if $u_{\epsilon,\lambda}$ is a positive solution of
\eqref{Tlamb} such that there exists $K_{0} > 0$, where for each
$K \geq K_{0}$, there exists $\lambda_{0}(K)>0$ such that
$|u_{\epsilon,\lambda}|_{L^{\infty}(\mathbb{R}^{N})}\leq \alpha K$
for all $\epsilon \in (0, \bar{\epsilon})$ and for all $\lambda
\in [0, \lambda_{0}]$, then $u_{\epsilon,\lambda}$ is a positive
solution of \eqref{Plambst}.
\end{remark}

\section{Multiplicity and Concentration of positive solutions
for Truncated Problem}

The result below is related to the multiplicity and concentration
of solutions for  \eqref{Tlamb} and its proof can be
found in \cite[Theorem 1.1]{AlvesGio2} or \cite{giovany}.

\begin{theorem}\label{super11}
Suppose that $V$ verify \eqref{V1}\eqref{V2}. Then,
for any $\delta>0$, there exists
$\overline{\epsilon}=\overline{\epsilon}(\delta,\lambda,K)>0$ such
that $(T_\lambda)$ has at least $\mathop{\rm cat}_{M_{\delta}}M$ positive
solutions for all $\epsilon\in(0,\overline{\epsilon})$ and for
each $\lambda>0$. Moreover, if $u_{\epsilon,\lambda}$ is a
positive solution of \eqref{Tlamb} and $\eta_{\epsilon}\in
\mathbb{R}^{N}$ a global maximum point of $u_{\epsilon,\lambda}$,
then
$$
\lim_{\epsilon\to 0}V(\eta_{\epsilon})=V_{0}.
$$
\end{theorem}

\section{Multiplicity of positive solutions for \eqref{Plambst}}

We recall that the weak solutions of \eqref{Tlamb} are the
critical points of the functional
$$
I_{\epsilon,\lambda}(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}|\nabla
u|^{p}+ \frac{1}{p}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u|^{p}-\int_{\mathbb{R}^{N}}F_{K}(u),
$$
which is well defined for $u \in W_{\epsilon}$, where
$$
W_{\epsilon}=\{u \in
W^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(\epsilon
x)|u|^{p}<\infty\}
$$
endowed with the norm
$$
\|u\|^{p}_{\epsilon}= \int_{\mathbb{R}^{N}}|\nabla
u|^{p}+ \int_{\mathbb{R}^{N}}V(\epsilon x)|u|^{p}.
$$
Let us also denote by $E_{V_{0},\lambda}$ the energy
functional associated to the problem
\begin{equation} \label{PV0lamb}
\begin{gathered}
 -\Delta_{p}u + V_{0}|u|^{p-2}u = f_{K}(u)         \\
 u \in W^{1,p}(\mathbb{R}),\quad u > 0 \ in \ \mathbb{R}^{N},
\end{gathered}
\end{equation}
that is,
$$
E_{V_{0},\lambda}(u)
=\frac{1}{p}\int_{\mathbb{R}^{N}}|\nabla
u|^p + \frac{1}{p}\int_{\mathbb{R}^{N}}V_{0}|u|^p -
\int_{\mathbb{R}^N}F_{K}(u),
$$
Here we will establish a preliminary estimative for
$\|u_{\epsilon,\lambda}\|_{\epsilon}$.

\begin{lemma}\label{super2}
For any solution $u_{\epsilon,\lambda}$ of \eqref{Tlamb}, there
exists $\bar{C}>0$, such that
\begin{eqnarray*}
\|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C},
\end{eqnarray*}
for $\epsilon > 0$ sufficiently  small and uniformly in $\lambda$.
\end{lemma}

\begin{proof}  By  \cite[Theorem 1.1]{AlvesGio2}
(see \cite{giovany} too), we have that all solutions
$u_{\epsilon,\lambda}$ from \eqref{Tlamb} verify the inequality
\[
I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},\lambda}+
h_{\lambda}(\epsilon),
\]
where $c_{V_{0},\lambda}$ is the level Mountain Pass related of
functional $E_{V_{0},\lambda}$ and
$h_{\lambda}(\epsilon)\to 0$ as $\epsilon\to 0$
for each $\lambda \geq 0$. In this case, we may suppose that
\begin{eqnarray*}
I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},\lambda}+
1,
\end{eqnarray*}
for all $\epsilon\in (0,\bar{\epsilon}(K,\lambda))$. Since
$c_{V_{0},\lambda}\leq c_{V_{0},0}$, we have
\begin{equation} \label{G}
I_{\epsilon,\lambda}(u_{\epsilon,\lambda})\leq c_{V_{0},0}+ 1,
\end{equation}
for all $\epsilon\in (0,\bar{\epsilon}(K,\lambda))$ and for all
$\lambda \geq 0$.
Moreover,
\begin{align*}
I_{\epsilon,\lambda}(u_{\epsilon,\lambda})
&=I_{\epsilon,\lambda}(u_{\epsilon,\lambda}) -
\frac{1}{\theta}I'_{\epsilon,\lambda}(u_{\epsilon,\lambda})
u_{\epsilon,\lambda}\\
&=\big(\frac{1}{p}-\frac{1}{\theta}\big)
\|u_{\epsilon,\lambda}\|^{p}_{\epsilon}+
\int_{\mathbb{R}^{N}}\big[\frac{1}{\theta}f_{K}(u_{\epsilon,\lambda})u_{\epsilon,\lambda}
-F_{K}(u_{\epsilon,\lambda})\big].
\end{align*}
By \eqref{fK4},
\[
I_{\epsilon,\lambda}(u_\epsilon,\lambda)\geq
\big(\frac{1}{p}-\frac{1}{\theta}\big)\|u_{\epsilon,\lambda}\|^{p}_{\epsilon}
\]

\noindent Therefore, by \eqref{G},
$\|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C}$,
for $\epsilon\in (0,\bar{\epsilon}(K,\lambda))$ and for all
$\lambda \geq 0$, where
$$
\bar{C}=\Big[(c_{V_{0},0}+1)\big(\frac{\theta p}{\theta -
p}\big)\Big]^{1/p}.
$$
\end{proof}

Now, we use the Moser iteration technique \cite{moser}
(see also \cite{Chabr}) to prove that each solution found of
\eqref{Tlamb} is a solution of \eqref{Plambst}

\begin{proof}[Proof of Theorem \ref{thmD}]
 We use the notation $u_{\epsilon,\lambda}:=u$.
For each $L>0$, we define
\begin{gather*}
u_{L} =\begin{cases}
              u &\text{if } u \leq L,        \\
        L &\text{if } u \geq L,
        \end{cases}\\
z_{L} = u_{L}^{p(\beta - 1)}u \quad \text{and} \quad
w_{L} = uu_{L}^{\beta-1}
\end{gather*}
with $\beta > 1$ to be determined later. Taking $z_{L}$ as a test
function, we obtain
\begin{align*}
\int_{\mathbb{R}^N}u_{L}^{p(\beta -1)}|\nabla u|^{p}
&= -p(\beta -1)\int_{\mathbb{R}^N}u_{L}^{p\beta -
p-1}u|\nabla
u|^{p-2}\nabla u \nabla u_{L}\\
&\quad + \int_{\mathbb{R}^N}f_{K}(u)uu_{L}^{p(\beta -1)}-
\int_{\mathbb{R}^N}V({\epsilon
x})|u|^{p}u_{L}^{p(\beta - 1)}.
\end{align*}
By \eqref{fK2},
\begin{equation} \label{w1}
\int_{\mathbb{R}^N}u_{L}^{p(\beta -1)}|\nabla u|^{p}
\leq C_{\lambda,K} \int_{\mathbb{R}^N}u^{q
}u_{L}^{p(\beta - 1)},
\end{equation}
where $C_{\lambda,K}= (1+\lambda K^{s-q})$.
 From Sobolev imbedding, H\"{o}lder inequalities and \eqref{w1},
\[ %\label{3eqb}
|w_{L}|^{p}_{p^{*}}\leq
C_{1}\beta^{p}C_{\lambda,K}\Big(\int_{\mathbb{R}^N}u^{p^{*}}
\Big)^{(q-p)/p^{*}}
\Big(\int_{\mathbb{R}^N}w_{L}^{pp^{*}/[p^{*}-(q-p)]}
\Big)^{[p^{*}-(q-p)]/p^{*}},
\]
where $p < \frac{pp^{*}}{p^{*}-(q-p)} < p^{*}$.
Recalling that $\|u_{\epsilon,\lambda}\|_{\epsilon}\leq \bar{C}$,
we have
\[
|w_{L}|^{p}_{p^{*}} \leq
C_{2}\beta^{p}C_{\lambda,K}\bar{C}^{(q-p)/p^*}
|w_{L}|^{p}_{\alpha^{*}}
\]
where $\alpha^{*}=\frac{pp^{*}}{p^{*}-(q-p)}$.
Note that if $u^{\beta}\in L^{\alpha^{*}}(\mathbb{R}^{N})$, using
the definition of $w_{L}$ and the fact that $u_{L}\leq u$, we
obtain
\[
 \Big(\int_{\mathbb{R}^{N}}\mid uu_{L}^{\beta -
1}\mid^{p^{*}}\Big)^{p/p^*} \leq
C_{3}\beta^{p}C_{\lambda,K}\Big(\int_{\mathbb{R}^{N}}u^{\beta
\alpha^{*}}\Big)^{p/\alpha^{*}}< + \infty.
\]
By Fatou's Lemma on the variable $L$, we get
\begin{equation}\label{w3}
|u|_{\beta p^{*}} \leq (C_{4}C_{\lambda,K})^{1/\beta
}\beta^{1/\beta}|u|_{\beta \alpha^{*}}.
\end{equation}
The assertion is obtained by iteration of estimative \eqref{w3}.
Namely, let $\chi=\frac{p^{*}}{\alpha^{*}}$; i.e.,
$p^{*}=\chi\alpha^{*}$. Then
\[
|u|_{\chi^{(m+1)}\alpha^{*}} \leq
C_{5}(C_{4}C_{\lambda,K})^{\sum_{i=1}^{m}\frac{\chi^{-i}}{p}}
\chi^{\sum_{i=1}^{m}i\chi^{-i}}\bar{C}.
\]
Passing to the limit as $m \to \infty$, we have
\[
|u|_{L^{\infty}(\mathbb{R}^{N})} \leq
C_{5}(C_{4}C_{\lambda,K})^{\sigma_{1}}\chi^{\sigma_{2}}\bar{C},
\]
where
$\sigma_{1}=\sum_{i=1}^{\infty}\frac{\chi^{-i}}{p}$ and
$\sigma_{2}=\sum_{i=1}^{\infty}i\chi^{-i}$.
To choose $\lambda_{0}$, we consider the
inequality
\[
\Big[C_{4}(1+\lambda
K^{s-q})\Big]^{\sigma_{1}}\chi^{\sigma_{2}}C_{5}\bar{C}\leq
\alpha K.
\]
We conclude that
\[
(1+\lambda K^{s-q})^{\sigma_{1}}\leq \frac{\alpha K
C_{6}}{C_{4}^{\sigma_{1}}\chi^{\sigma_{2}}\bar{C}}.
\]
We choose $\lambda_{0}$ verifying the inequality
\[
\lambda_{0}\leq \Big[\frac{(\alpha K
C_{6})^{\frac{1}{\sigma_{1}}}}{C_{4}\chi^\frac{\sigma_{2}}{\sigma_{1}}
\bar{C}^{1/\sigma_1}} -1\Big]
\frac{1}{K^{s-q}}
 \]
and fixing $K$ such that
\[
\Big[\frac{(\alpha K C_{6})^{1/\sigma_1}}{C_{4}
\chi^\frac{\sigma_{2}}{\sigma_{1}}\bar{C}^{1/\sigma_1}} -1\Big]
> 0,
\]
we have
$|u_{\lambda,\epsilon}|_{L^{\infty}(\mathbb{R}^{N})} \leq \alpha K$
for all $\epsilon \in (0,\bar{\epsilon}(K,\lambda))$ and
all  $\lambda \in [0,\lambda_{0}]$.
The result follows from Remark \ref{final}.
 \end{proof}

\subsection*{Acknowledgements}
We would like to thank to Professor C. O. Alves for his help
and encouragement, to the  anonymous referee for several
suggestions that improved this article.

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