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\markboth{\hfil Perturbed eigenvalue problems \hfil EJDE--1997/11}%
{EJDE--1997/11\hfil Jo\~{a}o Marcos B. do \'{O}\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 11, pp. 1--15. \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} \\
Solutions to perturbed eigenvalue problems of the $p$-Laplacian in
${\Bbb R}^N$
\thanks{ {\em 1991 Mathematics Subject Classification:}
35A15, 35J60.\hfil\break\indent
{\em Key words and phrases:} Elliptic Equations on unbounded Domains,
$p$-Laplacian, \hfil\break\indent
Mountain Pass Theorem, Palais-Smale Condition, First eigenvalue,
\hfil\break\indent
\copyright 1997 Southwest Texas State University and University of
North Texas.\hfil\break\indent
Work partially supported by CNPq/Brazil.\hfil\break\indent
Submitted January 24, 1997. Published July 15, 1997.} }
\date{}
\author{Jo\~{a}o Marcos B. do \'{O}}
\maketitle
\begin{abstract}
Using a variational approach, we investigate the existence of solutions
for non-autonomous perturbations of the
$p-$Laplacian eigenvalue problem
$$
-\Delta _pu=f(x,u)\quad \mbox{in}\quad {\Bbb R}^N\,.
$$
Under the assumptions that the primitive $F(x,u)$ of $f(x,u)$
interacts only with the first eigenvalue, we look for solutions in
the space $D^{1,p}({\Bbb R}^N)$.
Furthermore, we assume a condition that measures how different the
behavior of the function $F(x,u)$ is from that of the $p-$power of $u$.
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newcommand{\diver}{\mathop{\rm div}}
\newcommand{\supp}{\mathop{\rm supp}}
\section{Introduction}
In this paper we study the existence of solutions for
non-autonomous perturbations of the $p$-Laplacian eigenvalue problem
\begin{equation}
-\Delta _pu\equiv -\diver\left( |\nabla u|
^{p-2}\nabla u\right) =f(x,u)\quad\mbox{in}\quad {\Bbb R}^N\,, \label{1}
\end{equation}
where $u\in D^{1,p}({\Bbb R}^N)$.
We assume that the primitive $F(x,u)$ of
the nonlinearity $f(x,u)$ interacts only with the first eigenvalue of some
$p$-Laplacian eigenvalue problems with weights naturally associated with
$F(x,u)$. We also assume that $f:{\Bbb R}^N\times {\Bbb R}\rightarrow
{\Bbb R}$ is a continuous function satisfying the growth condition,
$$
|f\left( x,u\right)|\leq a(x)|u|^r+b(x)|u|^s,\quad
\forall(x,u)\in {\Bbb R}^N\times {\Bbb R} \leqno{(f)}
$$
where $a,b$ are continuous functions,
$a\in L^\infty ({\Bbb R}^N)\cap L^{r_0}({\Bbb R}^N)$ and
$b\in L^\infty ({\Bbb R}^N)\cap L^{s_0}({\Bbb R}^N)$ with
$0\leq r\leq p-1\leq s
0,\quad \forall (x,u)\in {\Bbb R}%
^N\times ({\Bbb R-\{}0\}),
\]
\item{$(F_2)$} For some $p0,
\]
for all $u\in D^{1,p}({\Bbb R}^N)-\{0\}$.
For more details about this eigenvalue problem, see e. g.
\cite{Allegretto-Huang}.
Now, we are ready to present the main results of this article.
\begin{theorem}
Assume that $(f)$, $(F_1)$, and $(F_2)$ are satisfied. Furthermore,
suppose
\begin{description}
\item{($F_3$)} There exist a function $\alpha \in L^\infty ({\Bbb R%
}^N)\cap L^{\frac Np}({\Bbb R}^N)$ and a positive Conant $\delta$,
such that
\[
F(x,u)\leq \frac 1p\alpha (x)|u|^p\quad \forall x\in {\Bbb R}^N,\
\forall |u|\leq \delta ,
\]
where either $\alpha \leq 0$ or $\lambda _1(\alpha )>1$,
\item{($F_4$)} There exist a function $\omega \in L^\infty ({\Bbb R}%
^N)\cap L^{\frac Np}({\Bbb R}^N)$ and a positive constant $R$,
such that
\[
F(x,u)\geq \frac 1p\omega (x)|u|^p\quad \forall x\in {\Bbb R}^N,\
\forall |u|\geq R,
\]
where $\omega>0$ on a subset of positive measure and
$\lambda_1(\omega )<1$.
\end{description}
Then, problem (\ref{1})\thinspace has a nontrivial solution, provided
that $0p$.
\end{description}
Indeed, by $(AR)$, $f(x,u)u-pF(x,u)\geq (\theta -p)F(x,u)$ and, moreover, $F(x,u)\geq \min
\{F(x,1),F(x,-1)\}|u|^\theta >0,\quad \forall x\in {\Bbb R}^N$ and $%
|u|\geq 1$. Thus, $(F_1)$ holds if we assume also that there exist $%
\mu \leq \theta $ and a measurable function $a\in L^\infty ({\Bbb R}^N)$
such that $F(x,u)\geq a(x)|u|^\mu >0$, for all $x\in {\Bbb R}^N$ and
$|u|\leq 1$.
On the other hand, Example~1 shows a function which satisfies
condition \thinspace $(F_1)$, but does not satisfy $(AR)$.
Requirements similar to $(F_1)-(F_2)$ were considered by
Costa and Miyagaki in \cite{Costa-Miyagaki} (see also \cite
{Costa-Magalhaes1, Costa-Magalhaes2}) where $a$ is constant.
They obtained a nontrivial solution for autonomous
perturbations of the $p$-Laplacian on a unbounded cylindrical domain,
$\Omega =\Omega _0\times {\Bbb R}^{N-K}$, with $\Omega _0$ bounded domain
of ${\Bbb R}^K$. Their solution, in the space $W_0^{1,p}(\Omega )$, is
obtained using the mountain pass argument combined with the
concentration-compactness principle of Lions.
In order to get the geometry of the mountain-pass, the number
\[
\lambda _1=\inf \{\int_\Omega |\nabla u|^pdx/\int_\Omega |
u|^pdx:u\in W_0^{1,p}(\Omega )-\{0\}\},
\]
has been explored, and it is known (\cite{Burton,Esteban}) to be equal to
the first eigenvalue of $-\Delta _p$ acting on $W_0^{1,p}(\Omega _0)$.
In fact, the following condition for crossing the first
eigenvalue has been used,
$$\limsup\nolimits_{u\rightarrow 0}pF(u)/|u|
^p\leq \xi <\lambda _1<\eta \leq \liminf\nolimits_{|u|
\rightarrow \infty }pF(u)/|u|^p\,.$$
For the non-autonomous perturbations of the $p$-Laplacian studied here,
the crossing of the first eigenvalue is expressed by requirements such as $(F_3)$
and $(F_4)$. These conditions give
the geometric shape required by the Mountain Pass Theorem. Condition
$(F_3)$ together with the growth condition $(f)$ lead to the fact that the
origin is a local minimum of the associated functional, while assumption $%
(F_4)$ implies that the functional is not bounded below.
The interaction of the potential $F(x,u)$ with the
first eigenvalue in elliptic eigenvalue problems with weights
in bounded domains have been considered by de Figueiredo and
Massab\'{o} in \cite{Figueiredo-Massabo}.
\paragraph{Remark 2.}
The same procedures as in Theorem 1, along with obvious
modifications, show the existence of a nontrivial solution to the
non-autonomous perturbation of the $p$-Laplacian studied by Yu in
\cite{Yu}:
\[
l(u)\equiv -\diver(\left( a(x)|\nabla u|^{p-2}\nabla u+b(x)|u|
^{p-2}u\right) =f(x,u)\mbox{ in }\Omega \mbox{ and } u|
_{\partial \Omega }=0,
\]
where $\Omega $ is a $C^{1,\delta }$ $(0<\delta <1)$ exterior domain in $%
{\Bbb R}^N$, $00$, we choose $r_n$ sufficiently large such that
\begin{equation}
\max \{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\|
b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\}<\epsilon /2C\,. \label{D2}
\end{equation}
On the other hand, since the restriction operator $u\longmapsto u|
_{B(0,r_n)}$ is continuous from $D^{1,p}({\Bbb R}^N)$ into $D^{1,p}(B(0,r_n)$
and $K_n$ is weakly continuous, up to a subsequence, we have
\begin{equation}
|K_n(u_k)-K_n(u)|<\epsilon /2\,. \label{D3}
\end{equation}
Combining (\ref{D2}) and (\ref{D3}) we conclude that $K$ is weakly
continuous.
To prove that $I$ $\in C^1\left( D^{1,p}({\Bbb R}^N)\right) $, we must show
that $K\in C^1\left( D^{1,p}({\Bbb R}^N)\right) $ and that
\[
K' (u)v=\int_{{\Bbb R}^N}f(u,x)vdx,
\]
since the first term in $I$ is $C^1$ and its Fr\'{e}chet
derivative is the first term in (\ref{D1}). For fixed $u\in D^{1,p}({\Bbb R%
}^N)$ and given $\epsilon >0$, we must show that there exists $\delta
=\delta (u,\epsilon )$ such that
\[
|\int_{{\Bbb R}^N}F\left( x,u+v\right) dx-\int_{{\Bbb R}%
^N}F(x,u)dx-\int_{{\Bbb R}^N}f(u,x)vdx|\leq \epsilon \|
v\| _{D^{1,p}},
\]
for all $v\in D^{1,p}({\Bbb R}^N)$ with $\| v\| _{D^{1,p}}\leq
\delta $. Notice that if $\| v\| _{D^{1,p}}\leq 1$,
\begin{eqnarray*}
\lefteqn{|\int_{{\Bbb R}^N}F\left( x,u+v\right)\,dx-\int_{{\Bbb R}^N}
F(x,u)dx-\int_{{\Bbb R}^N}f(u,x)v\,dx|} && \\
&\leq & |K_n(u+v)-K_n(v)-K_n' (u)v|\\
&&+ C \| v\| _{D^{1,p}}\{\| a\| _{L^{r_0}({\Bbb R}
^N-B(0,r_n))}(\| u_k\| _{D^{1,p}}^r+\| u\|
_{D^{1,p}}^r)\\
&&+ \| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}(\|
u_k\| _{D^{1,p}}^s+\| u\| _{D^{1,p}}^s)\}\,.
\end{eqnarray*}
Since $a\in L^{r_0}({\Bbb R}^N)$ and $b\in L^{s_0}({\Bbb R}^N)$, as before,
we can choose $r_n$ sufficiently large such that $\max \{\|
a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\| b\| _{L^{s_0}(%
{\Bbb R}^N-B(0,r_n))}\}<\epsilon /2C$.
Using that $K_n\in C^1\left(D^{1,p}\left( B(0,r_n)\right) \right) $,
there exists $\delta =\delta
(u,\epsilon )$ such that, for all $v\in D^{1,p}({\Bbb R}^N)$ with $\|
v\| _{D^{1,p}}\leq \delta $,
\[
|K_n(u+v)-K_n(v)-K_n' (u)v|\leq \epsilon \|
v\| _{D^{1,p}},
\]
and the proof that $I$ is Fr\'{e}chet differentiable is complete.
To prove that $K' $ is continuous we use the estimate
\begin{eqnarray*}
|K' \left( u_k\right) -K' \left( u\right) |
&\leq& |K_n' \left( u_k\right) -K_n' \left( u\right) |\\
&&+ C\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}(\|
u_k\| _{D^{1,p}}^r+\| u\| _{D^{1,p}}^r)\\
&&+ C\| b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}(\|
u_k\| _{D^{1,p}}^s+\| u\| _{D^{1,p}}^s)
\end{eqnarray*}
together with the facts that $K_n' $ is continuous, and that
$\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))}$ and $\| b\| _{L^{s_0}(%
{\Bbb R}^N-B(0,r_n))}$ converge to zero as $n\rightarrow \infty $.
Finally, to prove compactness, we use the diagonal method. Let $%
u_k\rightharpoonup u$ weakly in $D^{1,p}({\Bbb R}^N)$ as $k\rightarrow
\infty $. Since $a\in L^{r_0}({\Bbb R}^N)$ and $b\in L^{s_0}({\Bbb R}^N)$ we
can choose an increasing and unbounded sequence of positive
real numbers $(r_n)$ such that
\[
\max \{\| a\| _{L^{r_0}({\Bbb R}^N-B(0,r_n))},\|
b\| _{L^{s_0}({\Bbb R}^N-B(0,r_n))}\}\leq 1/2n.
\]
On the other hand, for each natural number $n$, we have a subsequence $%
(u_{kn})$, of $(u_k)$, such that
\[
\| K_n' (u_{kn})-K_n' (u)\| \leq 1/2n,
\]
since $K_n'$ is compact. Thus, combining these two estimates,
\[
\| K' (u_{kn})-K' (u)\| \leq C/n.
\]
Therefore, the diagonal subsequence $(u_{kk})$ leads to
$K' (u_{kk})\rightarrow K' (u)$. $\hfill\Box$
In the next lemmas we prove that the functional $I$ satisfies a compactness
condition of the Palais-Smale type which was introduced by Cerami in \cite
{Cerami}.
\paragraph{Definition}
Let $J:E\rightarrow {\Bbb R}$ be a $C^1$ functional. We say that
$J$ satisfies condition $(C)$ if every sequence $(u_n)$ in $E$ such that
\begin{equation}
\begin{array}{lll}
(i) & & J(u_n)\rightarrow c \\
(ii) & & (1+\| u_n\| _{D^{1,p}})\|J'(u_n)\|_{(D^{1,p})^{*}}\rightarrow 0,
\end{array}
\label{C1}
\end{equation}
possesses a convergent subsequence.
\medskip
Using this compactness condition, Bartolo, Benci and Fortunato in \cite
{Bartolo-Benci-Fortunato} obtained rather general minimax results.
\begin{lemma}[Compactness Condition I]
\label{Compactness1} Assume that $(f)$, $(F_1),\ (F_2)$, and $(F_3)$
are satisfied with $01$, the sequence $(u_n)$
is bounded, since $\mu >\frac Np(q-p)$ is equivalent to $tq0$. Now, integrating the above inequality
over an interval $[u,U]\subset (0,+\infty )$, we obtain
\[
\frac{F(x,U)}{U^p}-\frac{F(x,u)}{u^p}\geq \frac{a(x)}{\mu -p}\left( \frac
1{U^{p-\mu }}-\frac 1{u^{p-\mu }}\right) .
\]
From the hypothesis $\mu
0$ and $e\in E,$
with $\| e\| >\rho $, one has $\sigma \leq \inf_{\|
u\| =\rho }I(u)$ and $I(e)<0$. Then $I$ has a critical value $c\geq
\sigma $, characterized by
\[
c=\inf_{\gamma \in \Gamma }\max_{0\leq \tau \leq 1}I(\gamma (\tau )),
\]
where $\Gamma =\{\gamma \in C([0,1],E):\gamma (0)=0,\ \gamma (1)=e\}.$
\end{theorem}
\paragraph{Remark 4.}
It is not difficult to see that the same proof of the standard Mountain-Pass
Theorem (cf. \cite{Rabinowitz}) applies to the present
context; since the deformation theorem, Theorem 1.3 in \cite
{Bartolo-Benci-Fortunato}, is obtained with condition $(C)$ in a Banach
space framework. It is worth to observe also that this version of the
Mountain-Pass Theorem can be obtained as a consequence of Lemma 5 in \cite
{JMarcos}.
The proof of Theorem 1 follows from Lemma \ref{Compactness1}, where
the compactness condition is proved, and the next lemma, where the
geometric conditions are checked.
\begin{lemma}[Mountain-Pass Geometry]
\label{Geometry}Suppose $(f)$, $(F_3)$, and $(F_4)$ hold. Then there exist
positive constants $\sigma $ and $\rho $ such that $I(u)\geq \sigma $ if $%
\| u\| _{D^{1,p}}=\rho $. Moreover, there exists $\varphi \in
D^{1,p}({\Bbb R}^N)$ such that $I(t\varphi )\rightarrow -\infty $ as $%
t\rightarrow \infty .$
\end{lemma}
\paragraph{Proof.} Using the growth condition $(f)$ and $(F_3)$,
there exists a positive constant $C_\delta $ such that
\[
F(x,u)\leq \frac{\alpha (x)|u|^p}p+C_\delta |u|^{p^{*}}.
\]
Hence
\begin{eqnarray*}
I(u) &\geq &\frac 1p\int_{{\Bbb R}^N}|\nabla u|^pdx-\frac 1p\int_{%
{\Bbb R}^N}\alpha (x)|u|^pdx-C_\delta \int_{{\Bbb R}^N}|u|
^{p^{*}}dx \\
&\geq &\frac 1p(1-\lambda _1^{-1}(\alpha ))\| u\| _{D^{1,p}}^p-%
\widehat{C}_\delta \| u\| _{D^{1,p}}^{p^{*}},
\end{eqnarray*}
for some positive constant $\widehat{C}_\delta $, where in the last
inequality we have used the variational characterization of the first
eigenvalue $\lambda _1(\alpha )$ and the Sobolev inequality. Since $\lambda
_1(\alpha )>1$, we can fix positive constants $\sigma $ and $\rho $ such
that $I(u)\geq \sigma $ if $\| u\| _{D^{1,p}}=\rho .$
Let us prove the second assertion. Consider $\varepsilon >0$ such that $%
\lambda _1(\omega )+\varepsilon <1$, and choose $\varphi \in C_0^\infty (%
{\Bbb R}^N)-\{0\}$ satisfying
\[
\int_{{\Bbb R}^N}|\nabla \varphi |^pdx\leq (\lambda _1(\omega
)+\varepsilon )\int_{{\Bbb R}^N}\omega |\varphi |^pdx.
\]
From $(F_4)$, we have
\begin{eqnarray*}
\int_{{\Bbb R}^N}F(x,t\varphi )\,dx
&=&\int_{\{x:|t\varphi (x)|\geq R\}\cap \supp\varphi }F(x,t\varphi)\,dx\\
&+&\int_{\{x:|t\varphi(x)|\leq R\}\cap \supp\varphi }F(x,t\varphi )\,dx\\
&\geq &\frac{|t|^p}p\int_{\{x:|t\varphi (x)|\geq R\}\cap
\supp\varphi }\omega (x)|\varphi |^p\,dx-C\,,
\end{eqnarray*}
for some positive constant $C$. $\,$Thus,
\begin{eqnarray*}
I(t\varphi ) &\leq &\frac{|t|^p}p\left[ \int_{{\Bbb R}^N}|\nabla
\varphi |^p\,dx-\int_{\{x:|t\varphi (x)|\geq R\}\cap
\supp\varphi }\omega |\varphi |^p\,dx\right] +C \\
&=&\frac{|t|^p}p\Big[ \int_{{\Bbb R}^N}|\nabla \varphi |
^pdx-\int_{{\Bbb R}^N}\omega |\varphi |^p\,dx\\
&+&\int_{\{x:|t\varphi
(x)|0$, such that
\[
\int_{\{x:|t\varphi (x)|0,\quad \forall x\in {\Bbb R}^N,\forall u\neq 0.
\]
It is not difficult to see that the remaining hypotheses of Theorem 1 are
satisfied. On the other hand, if $\mu >p,$%
\[
u\frac{\partial F}{\partial u}(x,u)-\mu F(x,u)=[(p-\mu )\ln |u|
+1]\frac 1p\omega (x)|u|^p<0
\]
for all $x$ $\in {\Bbb R}^N$ and $|u|$ sufficiently large.
Consequently, Ambrosetti-Rabinowitz condition $(AR)$ does not hold.
\paragraph{Example 2.}
Let $\psi :[1,+\infty )\rightarrow {\Bbb [}0,+\infty )$ be a continuous
nontrivial function satisfying $\int_1^{+\infty }\psi (t)dt<\infty $,
and let
$H(u)=d+\int_1^u[\psi (t)+t^{\mu -p-1}]\,dt$, $u\geq 1$, where $d$ is such
that $\lim_{u\rightarrow +\infty }H(u)=1$.
Let $F(x,u)=\omega (x)u^pH(u)/p$ for $x\in {\Bbb R}^N,\ u\geq 1$, where $\omega $ is a positive measurable
function in $L^\infty ({\Bbb R}^N)\cap L^{N/p}({\Bbb R}^N)$, such that $%
\lambda _1(\omega )\geq 1$. Notice that $\lim_{u\rightarrow +\infty
}pF(x,u)/u^p=\omega (x)$ and that
\[
u\frac{\partial F}{\partial u}(x,u)-pF(x,u)=\frac 1p\omega
(x)u^{p+1}H' (u)\geq \frac 1p\omega (x)u^\mu ,\quad x\in {\Bbb R}%
^N,\ u\geq 1.
\]
It is not difficult to see that $F$ can be extended as a function $%
F:{\Bbb R}^N\times {\Bbb R\rightarrow R}$ such that the conditions of
Theorem~2 are satisfied for all $u\in {\Bbb R}$ (cf. \cite{Costa-Magalhaes2}%
).
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\bigskip
{\sc Jo\~{a}o Marcos B. do \'{O}\newline
Departamento de Matem\'{a}tica, Universidade Federal da Para\'{\i}ba
58059.900 Jo\~{a}o Pessoa, Pb Brazil}\newline
E-mail address: jmbo@terra.npd.ufpb.br
\end{document}