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\markboth{\hfil Multiple positive solutions \hfil EJDE--1997/13}%
{EJDE--1997/13\hfil C. O. Alves  \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 13, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Multiple positive solutions for equations involving
critical Sobolev exponent in ${\Bbb R}^N$
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J20, 35J25.
\hfil\break\indent
{\em Key words and phrases:} Mountain Pass Theorem, Ekeland Variational Principle.
\hfil\break\indent
\copyright 1997 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted April 22,1997. Published August 19, 1997.} }
\date{}
\author{C. O. Alves}
\maketitle

\begin{abstract} 
This article concerns with the problem 
$$
-\mbox{div}(|\nabla u|^{m-2}\nabla u) = 
\lambda h  u^q+u^{m^*-1},\quad\mbox{in}\quad {\Bbb R}^N\,. 
$$
Using the Ekeland Variational Principle and the
Mountain Pass Theorem, we show the existence of $\lambda ^*>0$ such that
there are at least two non-negative solutions for each $\lambda$ in 
$(0,\lambda ^*)$.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}

\section{Introduction}
In this work, we study the existence of solutions for the problem
$$
\left\{ \begin{array}{l}
-\Delta_mu=\lambda hu^q+u^{m^*-1},\;{\Bbb R}^N \\ 
u\geq 0,\;u\neq 0,\;u\in D^{1,m}({\Bbb R}^N)
\end{array}
\right. \leqno(P) 
$$
where $\Delta_mu=\mbox{div}\,(\left|\nabla u\right|^{m-2}\nabla u)$,
 $\lambda >0$, $N>m\geq 2$, $m^*=Nm/(N-m)$, $0<q<m-1$, $h$
is a nonnegative function with $L^\Theta ({\Bbb R}^N)$ with 
$\Theta =\frac{Nm}{Nm-(q+1)(N-m)}$, and 
$$
D^{1,m}({\Bbb R}^N)=\{u\in L^{m^*} ({\Bbb R}^N) \mid 
\frac{\partial u}{\partial x_i}\in L^m({\Bbb R}^N)\}$$ 
endowed with the norm $\| u\| =\left(\int\left|\nabla
u\right|^m\right)^{1/m}$.

The case $q = 0$, $m = 2$ was studied by Tarantello \cite{r:2}, and a more general case with $m\geq 2$  by Cao, LI \& Zhou \cite{r:1}. In these two references, \cite{r:1} and \cite{r:2}, it is proved that (P) has  multiple solutions.
In the case  $m=2$, $h\in L^p({\Bbb R}^N)$ with $p_1\leq p\leq p_2$ and $1<q<2^*-1$, Pan \cite{r:3} proved the existence of a positive solution for (P). In the more general case, $m\geq 2$, $h\in L^\Theta ({\Bbb R}^N)$,
Gon\c{c}alves \& Alves \cite{r:4} proved the existence of a positive solution for (P).  

By a solution to (P), we mean a function $u\in D^{1,m}({\Bbb R}^N)$,
$u\geq 0$ and $u\neq 0$ satisfying
\[
\int \left| \nabla u\right| ^{m-2}\nabla u\nabla \Phi =\lambda \int hu^q\Phi
+\int u^{m^*-1}\Phi ,\;\;\;\forall \Phi \in D^{1,m}({\Bbb R}^N). 
\]
Hereafter, $\int$, $D^{1,m}$, $L^p$ and $|.|_p$ stand for 
$\int_{{\Bbb R}^N}$, $D^{1,m}({\Bbb R}^N)$, $L^p({\Bbb R}^N)$ and $|.|_{L^p}$
respectively.

In the search of solutions we apply minimizing arguments to
the energy functional
\begin{equation}
I(u)=\frac 1m\int \left| \nabla u\right|^m-\frac \lambda {q+1}\int
h\left(u^+\right)^{q+1}-\frac 1{m^*}\int \left(u^+\right)^{m^*}
\label{eq:e1}
\end{equation}
associated to (P), where $u^+(x)=\max \{u(x),0\}$. 
Note that the condition $h\in L^\Theta$  
implies that $I\in C^1\left( D^{1,m},{\Bbb R}\right)$.

To show the existence of at least two critical points of the energy
functional, we shall use the Ekeland Variational Principle \cite{r:5}, and the
Mountain Pass Theorem of Ambrosetti \& Rabinowitz \cite{r:15} without 
the Palais-Smale condition.
Using the Ekeland Variational Principle, we obtain a
solution $u_1$  with $I(u_1)<0$, and by the Mountain Pass Theorem we
prove the existence of a second solution $u_2$ with $I(u_2)>0$. 
Techniques for finding the solutions $u_1$ and $u_2$ are borrowed from Cao, Li
\& Zhou \cite{r:1}. Then we combine these techniques with arguments developed 
by Chabrowski \cite{r:8}, Noussair, Swanson \& Jianfu \cite{r:9}, 
Jianfu \& Xiping \cite{r:10}, Azorero \& Alonzo \cite{r:11}, 
Gon\c {c}alves \& Alves \cite{r:4} and Alves, Gon\c {c}alves \& 
Miyagaki \cite{r:12} to obtain the following result 


\begin{theorem} There exists a constant $\lambda ^*>0$, such that
(P) has at least two solutions,  $u_1$ and $u_2$,
satisfying
\[
I(u_1)<0<I(u_2)\quad \forall \lambda \in (0,\lambda ^*)\,. 
\]
\end{theorem}


\section{Preliminary Results}
In this section we establish some results needed
for the proof of Theorem 1.

\paragraph{Definition.} A sequence $\{u_n\}\subset D^{1,m}$ is called a $(PS)_c$
sequence, if 
$I(u_n)\rightarrow c$ and $I'(u_n)\rightarrow 0$. 


\begin{lemma} If $\{u_n\}$ is a $(PS)_c$ sequence,  then $\{u_n\}$ is bounded, and $\{u_n^+\}$ is
a $(PS)_c$ sequence.
\end{lemma}

\paragraph{Proof.}
Using the hypothesis that $\{u_n\}$ is a $(PS)_c$ sequence, there exist $n_o$
and $M>0$ such that
\begin{equation}
I(u_n)-\frac 1{m^*}I'(u_n)u_n\leq M+\| u_n\|
\quad \forall n\geq n_o\,.  \label{eq:e2}
\end{equation}
Now, using (1) and the H\"{o}lder's inequality, we have
\begin{equation}
I(u_n)-\frac 1{m^*}I'(u_n)u_n\geq \frac 1N\left\| u_n\right\|
^m+c_1\left\| u_n\right\| ^{q+1}  \label{eq:e3}
\end{equation}
where $c_1$ is a constant depending of $N, m,q,\| h\| _\Theta $
and $\Theta$.
It follows from (2) and (3) that $\{u_n\}$ is bounded.
Now, we shall show that $\{u_n^+\}$ is a also $(PS)_c$ sequence.
Since $\{u_n\}$ is bounded, the
sequence $u_n^{-}=u_n-u_n^+$ is also bounded. Then 
\[
I'(u_n)u_n^{-}\rightarrow 0 
\]
and we conclude that
\begin{equation}
\left\| u_n^{-}\right\| \rightarrow 0.  \label{eq:e4}
\end{equation}
From (4) we achieve that
\begin{equation}
\left\| u_n\right\| =\left\| u_n^+\right\| +o_n(1).  \label{eq:e5}
\end{equation}
Therefore, by (4) and (5)
\[
I(u_n)=I(u_n^+)+o_n(1) 
\]
 and
\[
I'(u_n)=I'(u_n^+)+o_n(1), 
\]
which consequently implies that $\{u_n^+\}$ is a $(PS)_c$ sequence.
\hfil $\Box $

From Lemma 1, it follows that  any $(PS)_c$ sequence can be considered as a sequence of nonnegative functions.


\begin{lemma} If $\{u_n\}$ is a $(PS)_c$ sequence with 
$u_n\rightharpoonup u$ in $D^{1,m}$,  then $I'(u)=0$, and there exists a constant $M>0$ depending of $N, m, q, | h| _\Theta$ and $\Theta$, 
such that 
\[
I(u)\geq -M\lambda ^\Theta 
\]
\end{lemma}

\paragraph{Proof.}
If $\{u_n\}$ is a $(PS)_c$ sequence with $u_n\rightharpoonup u$,
using arguments similar to those found in \cite{r:4}, \cite{r:10} and \cite{r:9}, we can obtain a
subsequence, still denoted by $u_n$, satisfying
\begin{eqnarray}
u_n(x)&\rightarrow& u(x) \quad\mbox{a.e. in}\quad {\Bbb R}^N  \label{eq:e6}\\
\nabla u_n(x)&\rightarrow& \nabla u(x)\quad\mbox{a.e. in}\quad {\Bbb R}^N \label{eq:e7} \\
u(x)&\geq& 0\quad\mbox{a.e. in}\quad {\Bbb R}^N.  \label{eq:e8}
\end{eqnarray}
From (6), (7) and using the hypothesis that $\{u_n\}$ is bounded in $D^{1,m}$,
 we get
\begin{equation}
I'(u)=0\,,  \label{eq:e9}
\end{equation}
which in implies $I'(u)u=0$, and 
\[
\left\| u\right\| ^m=\lambda \int hu^{q+1}+\int u^{m^*}\,. 
\]
Consequently
\[
I(u)=\lambda \left( \frac 1m-\frac 1{q+1}\right) \int hu^{q+1}+\frac 1N\int
u^{m^*}.
\]
Using H\"{o}lder and Young Inequalities, we obtain
\[
I(u)\geq -\frac 1N\left| u\right| _{m^*}^{m^*}-M\lambda ^\Theta +\frac
1N\left| u\right| _{m^*}^{m^*}=-M\lambda ^\Theta 
\]
where $M=M(N,m,q,\Theta ,\| h\| _\Theta )$. \hfil$\Box$ 

For the remaining of this article, we will denote by $S$ the best
Sobolev constant for the imbedding $D^{1,m}\hookrightarrow L^{m^*}$.

\begin{lemma}
 Let $\{u_n\}\subset D^{1,m}$ be a $(PS)_c$ sequence with 
\[
c<\frac 1NS^{N/m}-M\lambda ^\Theta \,,
\]
where $M>0$ is the constant given in Lemma 2. Then, there exists a subsequence
$\{u_{n_j}\}$ that converges strongly in $D^{1,m}$.
\end{lemma}

\paragraph{Proof}
By Lemmas 1 and 2, there is a subsequence, still denoted by 
$\{u_n\}$ and a function $u\in D^{1,m}$ such that $u_n\rightharpoonup u$. 
Let $w_n=u_n-u$. Then by a lemma in Brezis \& Lieb \cite{r:18}, we have
\begin{eqnarray}
\left\| w_n\right\| ^m&=&\left\| u_n\right\| ^m-\left\| u\right\| ^m+o_n(1)
\label{eq:e10}\\
\| w_n\| _{m^*}^{m^*}&=&\left| u_n\right| _{m^*}^{m^*}-\left|
u\right| _{m^*}^{m^*}+o_n(1)\,.  \label{eq:e11}
\end{eqnarray}
Using the Lebesgue theorem (see Kavian \cite{r:19}), it follows  that
\begin{equation}
\int hu_n^{q+1}\longrightarrow \int hu^{q+1}.  \label{eq:e12}
\end{equation}
From (10), (11) and (12), we obtain
\begin{equation}
\left\| w_n\right\| ^m=\left| w_n\right| _{m^*}^{m^*}+o_n(1)
\label{eq:e13}
\end{equation}
and
\begin{equation}
\frac 1m\left\| w_n\right\| ^m-\frac 1{m^*}\left| w_n\right|
_{m^*}^{m^*}=c-I(u)+o_n(1).  \label{eq:e14}
\end{equation}
Using the hypothesis that $\{w_n\}$ is bounded in $D^{1,m}$, there exists 
$l\geq 0$ such that
\begin{equation}
\left\| w_n\right\| ^m\rightarrow l\geq 0.  \label{eq:e15}
\end{equation}
From (13) and (15), we have
\begin{equation}
\left| w_n\right| _{m^*}^{m^*}\rightarrow l,  \label{eq:e16}
\end{equation}
and using the best Sobolev constant $S$ and recalling that
\begin{equation}
\| w_n\| ^m\geq S\left( \int \left| w_n\right| ^{m^*}\right)
^{m/m^*}\,,  \label{eq:e17}
\end{equation}
we deduce from (15), (16) and (17) that
\begin{equation}
l\geq Sl^{ m/m^*}.  \label{eq:e18}
\end{equation}
Now, we claim that $l=0$. Indeed, if $l>0,$ from (18)
\begin{equation}
l\geq S^{N/m}\,. \label{eq:e19}
\end{equation}
By (14), (15) and (16), we have
\begin{equation}
\frac 1Nl=c-I(u).  \label{eq:e20}
\end{equation}
From (19), (20) and Lemma 2 we get
\[
c\geq \frac 1NS^{N/m}-M\lambda ^\Theta \,, 
\]
but this result contradicts the hypothesis. Thus, $l=0$ and we conclude that
\[
u_n\rightarrow u\quad\mbox{in}\quad D^{1,m}\,.
\]

\section{Existence of a first solution (Local Minimization)}
\begin{theorem}
There exists a constant $\lambda_1^*>0$ such that for 
$0<\lambda <\lambda _1^*$ Problem (P)
has a weak solution $u_1$ with $I(u_1)<0$. 
\end{theorem}

\paragraph{Proof.}
 Using arguments similar to those developed in \cite{r:1}, we have
\[
I(u)\geq \left( \frac 1m-\epsilon \right) \left\| u\right\| ^m+o\left(
\left\| u\right\| ^m\right) -C(\epsilon )\lambda ^{ m/(m-(q+1))}\,, 
\]
where $C(\epsilon )$ is a constant depending on $\epsilon >0$. The last
inequality implies that for small $\epsilon$, there exist constants $\gamma
,\rho $ and $\lambda _1^*>0$ such that 
\[
I(u)\geq \gamma >0\,,\quad \| u\| =\rho\,,
\quad\mbox{and}\quad 0<\lambda <\lambda _1^*\,. 
\]
Using the Ekeland Variational Principle, for the complete metric space
$\overline{B}_\rho (0)$ with  $d(u,v)=\left\| u-v\right\|$,
we can prove that there exists a $(PS)_{\gamma _o}$ sequence 
$\{u_n\}\subset\overline{B}_\rho (0)$ with
\[
\gamma _o=\inf \{I(u)\mid u\in \overline{B}_\rho (0)\}. 
\]
Choosing a nonnegative function $\Phi \in D^{1,m}\backslash \{0\}$, we have
that $I(t\Phi )<0$ for small $t>0$
and consequently $\gamma _o<0$.

Taking $\lambda _1^*>0,$ such that
\[
0<\frac 1NS^{ N/m}-M\lambda ^\Theta \quad\forall \lambda \in (0,\lambda_1^*) 
\]
from Lemma 3, we obtain a subsequence $\{u_{n_j}\}\subset \{u_n\}$ and 
$u_1\in D^{1,m}$, such that 
\[
u_{n_j}\rightarrow u\quad\mbox{in}\quad D^{1,m}\,. 
\]
Therefore,
\[
I'(u_1)=0\quad\mbox{and}\quad I(u_1)=\gamma _o<0\,, 
\]
which completes this proof.\hfil$\Box $

\section{Existence of a second solution (Mountain Pass)}
In this section, we shall use 
arguments similar to those explored by Cao, Li \& Zhou \cite{r:1}, Chabrowski \cite{r:8}, Noussair,
Swanson \& Jianfu \cite{r:9}, Jianfu \& Xiping \cite{r:10} and Gon\c calves \& Alves \cite{r:4} to obtain the following


\begin{theorem} There exists a constant $\lambda_2^*>0$
 such that for $0<\lambda <\lambda _2^*$ Problem (P) has a weak solution $u_2$ with $I(u_2)>0$.
\end{theorem}

\paragraph{Proof.}
 By arguments found in \cite{r:1} and \cite{r:4}, we can prove that
there exists $\delta _1>0$ such that for all $\lambda \in (0,\delta _1)$,
the functional $I$ has the Mountain Pass Geometry, that is:
\begin{description}
\item{(i)} There exist positive constants $r,\rho$ with 
 $I(u)\geq r>0$ for $\|u\|=\rho$ 
\item{(ii)} There exists $e\in D^{1,m}$ with
 $I(e)<0$ and $\| e\| >\rho$ \,.
\end{description}
Then by \cite{r:6}, there exists a $(PS)_{\gamma _1}$ sequence $\{v_n\}$
with
\[
\gamma_1 = \inf_{g\in \Gamma}\max_{t\in [0,1]}I(g(t)) 
\]
where
\[
\Gamma =\{g\in C([0,1],D^{1,m})\mid g(0)=0\quad\mbox{and}\quad g(1)=e\}\,.
\]
Using the next claim, which is a variant of a result found in \cite{r:1}, we can complete the proof of this theorem.

\paragraph{Claim.} There exists $\lambda _2^*>0$ such that for the  constant 
$M$ given by Lemma 2,
 \[
0<\gamma _1<\frac 1NS^{N/m}-M\lambda ^\Theta \quad \forall \lambda
\in (0,\lambda _2^*)\,. 
\]

Assuming this claim, by Lemma 3 there exists a subsequence 
$\{v_{n_j}\}\subset \{v_n\}$ and a function $u_2\in D^{1,m}$
such that 
$v_{n_j}\rightarrow u_2$.
Therefore,  
\[
I'(u_2)=0\quad\mbox{and}\quad I(u_2)=\gamma _1>0 \,.
\]
Which concludes the present proof. \hfil $\Box$

\paragraph{Verification of the above claim.} For $x\in {\Bbb R}^N$, let
\[
\Psi (x)=\frac{\left[ N\left( \frac{N-m}{m-1}\right) ^{m-1}\right] ^{%
(N-m)/m^2}}{\left[ 1+\left| x\right| ^{m/(m-1)}\right] \frac{N-m}m}\,.
\]
Then it is well known that (see \cite{r:16} or \cite{r:20}) 
{\large {\normalsize 
\begin{equation}
\left\| \Psi \right\| ^m=\left| \Psi \right| _{m^*}^{m^*}=S^{N/m}\,.
\label{eq:e21}
\end{equation}
Let $\delta _2>0$ such that
\[
\frac 1NS^{ N/m}-M\lambda ^\Theta >0\quad\forall \lambda \in
(0,\delta _2)\,. 
\]
Then from (1) and (21), we have
\[
I(t\Psi )\leq \frac{t^m}mS^{N/m}\,, 
\]
and there exists $t_o\in (0,1)$ with }}
\[
\sup_{0\leq t\leq t_o}I(t\Psi )<\frac 1NS^{N/m}-M\lambda ^\Theta
\quad\forall \lambda \in (0,\delta _2)\,.
\]
Moreover, from (1) and (21), we have
\[
I(t\Psi )=\left( \frac{t^m}m-\frac{t^{m^*}}{m^*}\right) S^{N/m}-%
\frac{\lambda t^{q+1}}{q+1}\int h\Psi ^{q+1}\,,
\]
and remarking that
\[
\left( \frac{t^m}m-\frac{t^{m^*}}{m^*}\right) \leq \frac 1N\quad\forall
t\geq 0,
\]
we obtain
\[
{\large {\normalsize I(t\Psi )\leq \frac 1NS^{N/m}-\frac{\lambda t^{q+1}%
}{q+1}\int h\Psi ^{q+1}\,;}}
\]
therefore,
\[
\sup_{t\geq t_o}I(t\Psi )\leq \frac 1NS^{N/m}-
\frac{\lambda t_0^{q+1}}{q+1}\int h\Psi ^{q+1}.
\]
Now, taking $\lambda >0$ such that 
\[
-\frac{\lambda t_0^{q+1}}{q+1}\int h\Psi ^{q+1}<-M\lambda ^\Theta 
\]
that is, 
\[
0<\lambda <\left( \frac{t_0^{q+1}\int h\Psi ^{q+1}}{M(q+1)}\right) ^{
1/(\Theta -1)}=\delta _3 
\]
we deduce that 
\[
\sup_{t\geq t_o}I(t\Psi )<\frac 1NS^{N/m}-M\lambda ^\Theta
\quad\forall \lambda \in (0,\delta _3)\,. 
\]
Choosing $\lambda _2^*=\min \{\delta _1,\delta _2,\delta _3\}$, we have 
\[
\sup_{t\geq 0}I(t\Psi )<\frac 1NS^{N/m}-M\lambda ^\Theta
\quad\forall \lambda \in (0,\lambda _2^*)\,. 
\]
and consequently 
\[
0<\gamma _1<\frac 1NS^{N/m}-M\lambda ^\Theta \quad\forall
\lambda \in (0,\lambda _2^*) 
\]
which proves the claim.

\paragraph{Proof of Theorem 1.} Theorem 1 is an immediate consequence of
Theorems~2 and 3.


\paragraph{Remark.} Using Lemma 3 and the same arguments explored by
Azorero \& Alonzo, in  the case $0<q<p$ \cite{r:11}, we can easily show that
for small $\lambda$ the following problem has infinitely many solutions
with negative energy levels.
$$ \begin{array}{rl}
-\Delta _mu&=\lambda h\left| u\right| ^{q-1}u+\left| u\right|
^{m^*-2}u,\quad\mbox{in}\quad {\Bbb R}^N \\ 
&u\in D^{1,m}
\end{array} \leqno(P)_*
$$
This result is obtained using the concept and properties of genus, and working
with a truncation of the energy functional associated with $(P)_*$.

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\end{thebibliography}
\bigskip
{\sc C. O. Alves\newline
Universidade Federal da Para\'{\i}ba - PB - Brazil}\newline
E-mail address: coalves@dme.ufpb.br


\end{document}