% fprad.tex - version of March 16, 2008 \documentclass[english,10pt]{article} \usepackage[latin1]{inputenc} \usepackage{amsmath,amsthm,amssymb} \usepackage{graphicx} \usepackage{color} %\usepackage[notref,notcite]{showkeys} \newcommand{\cb}{\color{blue}} \newcommand{\be}{\begin{eqnarray}} \newcommand{\ee}{\end{eqnarray}} \renewcommand{\arraystretch}{1.7} \textwidth159mm \textheight22cm \hoffset-20mm \voffset-14mm \font\sans=cmss10 \def\proof{{\it Proof.}\ } \def\endproof{\nolinebreak\hfill $\square$ \par\vskip3mm} \def\pt#1{{\it Proof of Theorem \ref{#1}.}} \def\eq#1{(\ref{#1})} \def\th#1{Theorem \ref{#1}} \def\neweq#1{\begin{equation}\label{#1}} \def\endeq{\end{equation}} \def\weak{\rightharpoonup} \def\ep{\varepsilon} \def\la{\lambda} \def\half{{1\over2}} \def\vp{\varphi_1} \def\RR{{\mathbb R} } \def\NN{{\bf N} } \def\supp{{\rm supp}} \def\di{\displaystyle} \def\ri{\rightarrow} \def\intom{\int_\Omega} \def\ii{\^\i } \newtheorem{theorem}{Theorem}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{rem}{Remark}[section] \title{\sc Existence and non--existence results\\ for quasilinear elliptic exterior problems\\ with nonlinear boundary conditions \thanks{{\bf 2000 Mathematics Subject Classification}: 35J60; 58E05.\newline {\bf Key words}: quasilinear elliptic equation, existence and non--existence results, exterior domain, nonlinear boundary condition.} } \author{\sc Roberta Filippucci, Patrizia Pucci and Vicen\c{t}iu R\u{a}dulescu} \date{} \begin{document} \maketitle \begin{abstract} Existence and non--existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non--linearities. \end{abstract} \section{Introduction and the main results} Let $\Omega$ be a smooth exterior domain in $\RR^N$, that is, $\Omega$ is the complement of a bounded domain with $C^{1,\delta}$ boundary ($0<\delta <1$). Assume that $p$ is a real number satisfying $1
0$ and some
$\alpha =\alpha(R)\in (0,1)$. Problems of this type are motivated
by mathematical physics (see, e.g., Reed and Simon \cite{reed} and
Strauss \cite{str}), where certain stationary waves in nonlinear
Klein--Gordon or Schr\"odinger equations can be reduced to this
form.
Actually, a weak solution of \eqref{yu1} satisfies for all $\varphi\in E$ the identity
\be\label{sol}
\int_\Omega(a(x)|Du|^{p-2}Du\boldsymbol{\cdot}D\varphi+b(x)|u|^{p-2}u\varphi)dx=
\int_\Omega g(x)|u|^{r-2}u\varphi dx,\ee
where $E$ is the completion of $C^\infty_0(\Omega)$ under the underlying norm
$$\|u\|_{a,b}=\left(\int_\Omega[a(x)|Du|^p+b(x)|u|^p]dx\right)^{1/p}.$$
By Lemma~2 of \cite{yu} every weak solution $u$ of \eqref{yu1} is
in $L^q(\Omega)$ for every $q\in[p^*,\infty)$ and approaches 0 as $|x|\to\infty$.
Of course $E\sim H^{1,p}_0(\Omega)$ whenever $0 p$, see, for instance, Section~4.8 of Pucci and Serrin \cite{ps2}
and the comments thereby. Property {\it(d)} follows using similar
arguments as in the proof of Lemma~2 of \cite{yu}, which is based on
Theorem~1 of Serrin \cite{s}.\qed
\section{Proof of Theorem~\ref{t2}}
Taking $\varphi=u$ in \eqref{sol1}, we see that any weak solution $u$ of \eqref{1}
satisfies the equality \eqref{2}, and the conclusion $(i)$ of Theorem~\ref{t2}
follows at once.
\smallskip
We next show that the $C^1$ energy functional $J_\lambda:X\to\RR$ satisfies the
assumptions of the Mountain Pass theorem of Ambrosetti and Rabinowitz
\cite{ar}. Fix $w\in X\setminus\{0\}$. Since $p0$ such that
\neweq{5}
C\|v\|_{L^r(\Omega;g)}^p\leq\|v\|_{a,b}^p\end{equation}
for any $v\in X$. Thus, by \eq{2} and
\eq{5}, we have
$C\|u\|_{L^r(\Omega;g)}^p\leq
\lambda \|u\|_{L^r(\Omega;g)}^r.$
Since $p
0$$ at point $t_0=\left({\alpha
r}/{\beta q}\right)^{1/(q-r)}>0,$ we immediately have $\alpha
t^r-\beta t^q\le C(q,r)\alpha^{q/(q-r)}\beta^{r/(r-q)}$, where
$C(q,r)=({q-r})\left(r^r/q^q\right)^{1/(q-r)}.$ Returning to
\eqref{6} and using the above inequality, with
$t=\|u\|_{L^q(\Omega)}$,
$\alpha=\lambda\|g\|_{L^{q/(q-r)}(\Omega)}/r$ and $\beta=1/2q$, we
deduce that
$$J_\lambda(u)\ge\frac1p\|u\|_{a,b}^p+\frac1{2q}\|u\|_{L^q(\Omega)}^q-
C(\lambda,q,r,g),$$
where $C(\lambda,q,r,g)=2^{r/(q-r)}(q-r)\left(\lambda\|g\|_{L^{q/(q-r)}(\Omega)}\right)^{q/(q-r)}/qr$.
This implies the claim.
\smallskip
Let $n\mapsto u_n$ be a minimizing sequence of $J_\lambda$ in $X$, which is bounded in $X$
by Step 2. Without
loss of generality, we may assume that $(u_n)_n$ is non--negative,
converges weakly to some $u$ in $X$ and converges also pointwise.
\smallskip
\noindent{\it Step 3. The non--negative weak limit $u\in X$ is a weak solution of
\eqref{1}.} To prove this, we shall show that
$$J_\lambda(u)\le\liminf_{n\to\infty}J_\lambda(u_n).$$
By the weak lower semicontinuity of the norm $\|\cdot\|$ we have
$$\frac1p\|u\|_{a,b}^p+\frac1q\|u\|_{L^q(\Omega)}^q\le\liminf_{n\to\infty}
\left(\frac1p\|u_n\|_{a,b}^p+\frac1q\|u_n\|_{L^q(\Omega)}^q\right).$$
Next, the boundedness of $(u_n)_n$ in $X$ implies with the same
argument that
$$\|u\|_{L^r(\Omega;g)}=\lim_{n\to\infty}\|u_n\|_{L^r(\Omega;g)}$$
by \eqref{cc}. Hence $u$ is a global minimizer of $J_\lambda$ in $X$.
\smallskip
\noindent{\it Step 4. The weak limit $u$ is a non--negative weak solution of
\eqref{1} if $\lambda>0$ is sufficiently large.} Clearly $J_\lambda(0)=0$. Thus, by Step 3
it is enough to show that there exists $\Lambda>0$ such that
$$\inf_{u\in X} J_\lambda(u)<0\quad\mbox{for all }\lambda>\Lambda.$$
Consider the constrained minimization problem \be\label{8}
\Lambda:=\displaystyle\inf\left\{\frac 1p\|w\|_{a,b}^p+\frac
1q\|w\|_{L^q(\Omega)}^q\,:\, w\in X\mbox{ and
}\|w\|_{L^r(\Omega;g)}^r=r\right\}.\ee Let $n\mapsto v_n\in X$ be
a minimizing sequence of \eqref{8}, which is clearly
bounded in $X$, so that we can assume, without loss of generality,
that it converges weakly to some $v\in X$, with
$\|v\|_{L^r(\Omega;g)}^r=r$ and
$$\di\Lambda=\frac1p\,\|v\|_{a,b}^p+\frac1q\,\|v\|_{L^q(\Omega)}^q$$
by the weak lower semicontinuity of $\|\cdot\|$.
Thus, $J_\lambda(v)=\Lambda-\lambda<0$ for any $\lambda>\Lambda$.
\smallskip
Now put
$$\begin{aligned}\lambda^{*}:&=\sup\{\lambda>0\,:\,\mbox{problem \eqref{1} does not admit any weak
solution}\},\\
\lambda^{**}:&=\inf\{\lambda>0\,:\,\mbox{problem \eqref{1} admits a weak
solution}\}. \end{aligned}$$
Of course $\Lambda\ge\lambda^{**}\ge\lambda^{*}>0$. To complete the proof of Theorem~\ref{t1}
it is enough to argue the following essential facts: (a) problem \eqref{1} has
a weak solution for any $\lambda>\lambda^{**}$; (b) $\lambda^{**}=\lambda^{*}$
and problem \eqref{1} admits a weak
solution when $\lambda=\lambda^{*}$.
\smallskip
\noindent{\it Step 5. Problem \eqref{1} has a weak solution for
any $\lambda>\lambda^{**}$ and $\lambda^{**}=\lambda^{*}$.} Fix
$\lambda>\lambda^{**}$. By the definition of $\lambda^{**}$, there
exists $\mu\in(\lambda^{**},\lambda)$ such that that $J_\mu$ has a
non--trivial critical point $u_\mu\in X$. Of course, $u_\mu$ is a
sub--solution of \eqref{1}. In order to find a super--solution of \eqref{1}
which dominates $u_\mu$, we consider the constrained minimization
problem
$$\inf\left\{\frac1p\,\|w\|_{a,b}^p+\frac1q\,\|w\|_{L^q(\Omega)}^q-
\frac\lambda r\,\|w\|_{L^r(\Omega;g)}^r\,:\, w\in X\mbox{ and
}w\ge u_\mu\right\}.$$
Arguments similar to those used in Step 4
show that the above minimization problem has a solution
$u_\lambda\ge u_\mu$ which is also a weak solution of problem
\eqref{1}, provided $\lambda>\lambda^{**}$.
We already know
that $\lambda^{**}\ge\lambda^{*}$. But, by the definition of
$\lambda^{**}$ and the above remark, problem \eqref{1} has no
solutions for any $\lambda <\lambda^{**}$. Passing to the supremum, this forces
$\lambda^{**}=\lambda^{*}$ and completes the proof.
\smallskip
\noindent{\it Step 6. Problem \eqref{1} admits a non--negative weak
solution when $\lambda=\lambda^*$.} Let $n\mapsto\lambda_n$ be a
decreasing sequence converging to $\lambda^{*}$ and let $n\mapsto
u_n$ be a corresponding sequence of non--negative weak solutions of
\eqref{1}. As noted in Step 2, the sequence $(u_n)_n$ is bounded
in $X$, so that, without loss of generality, we may assume that it
converges weakly in $X$, strongly in $L^r(\Omega ;g)$, and
pointwise to some $u^*\in X$, with $u^*\geq 0$. By \eqref{sol1},
for all $\varphi\in X$,
$$\int_\Omega a(x)|Du_n|^{p-2}Du_n\boldsymbol\cdot D\varphi dx+
\int_\Gamma b(x)|u_n|^{p-2}u_n\varphi d\sigma
+\displaystyle{\int_\Omega}|u_n|^{q-2}u_n\varphi dx
=\lambda_n\int_\Omega g(x)|u_n|^{r-2}u_n\varphi dx,$$
and passing to the limit as $n\to\infty$ we deduce that
$u^*$ verifies \eqref{sol1} for $\lambda=\lambda^*$, as claimed.
It remains to argue that $u^*\not=0$. A key ingredient in this
argument is the lower bound energy given in \eqref{esten}. Hence,
since $u_n$ is a non--trivial weak solution of problem \eq{1}
corresponding to $\lambda_n$, we have
$\|u_n\|_{a,b}^{p}\geq
\left({C^{r}}/{\lambda^p}\right)^{1/(r-p)}$ by
\eqref{esten}, where $C>0$ is the constant given in \eqref{5} and not depending
on $\lambda_n$. Next, since $\lambda_n\searrow\lambda^*$ as
$n\ri\infty$ and $\lambda^*>0$, it is enough to show that
\begin{equation}\label{ufinal}
\| u_n-u^*\|_{a,b}\ri 0\quad\mbox{as $n\ri\infty$}.\end{equation}
Since $u_n$ and $u^*$ are weak solutions
of \eqref{1} corresponding to $\lambda_n$ and $\lambda^*$, we have by \eqref{sol1},
with $\varphi=u_n-u^*$,
\begin{equation}\label{unustar}
\begin{aligned}
&\int_\Omega a(x)\left( |Du_n|^{p-2}Du_n
-|Du^*|^{p-2}Du^*\right)\boldsymbol\cdot D(u_n-u^*) dx\\
&\qquad+\int_\Gamma
b(x)\left(|u_n|^{p-2}u_n-|u^*|^{p-2}u^*\right)(u_n-u^*) d\sigma+
\int_\Omega (|u_n|^{q-2}u_n-|u^*|^{q-2}u^*)(u_n-u^*) dx\\
&=\int_\Omega
g(x)\left(\lambda_n\,|u_n|^{r-2}u_n-\lambda^*\,|u^*|^{r-2}u^*\right)(u_n-u^*)
dx.\end{aligned} \end{equation}
Elementary monotonicity properties imply that
$$\int_\Omega (|u_n|^{q-2}u_n-|u^*|^{q-2}u^*)(u_n-u^*) dx\ge 0\quad\mbox{and}\quad
\langle I'(u_n^*)-I'(u^*),u_n-u^*\rangle\ge 0,$$
where
$$I(u):=\|u\|_{a,b}^p/p.$$
Since $\lambda_n\searrow\lambda^*$ as $n\ri\infty$ and $X$
is compactly embedded in $L^r(\Omega;g)$, for all $p>1$ relation \eq{unustar}
implies
\neweq{weakco}
0\le \langle
I'(u_n^*)-I'(u^*),u_n-u^*\rangle\le
\int_\Omega
g(x)\left[\lambda_n\,u_n^{r-1}-\lambda^*\,(u^*)^{r-1}\right](u_n-u^*)
dx \ri 0\endeq
as $n\ri\infty$.
Now, we distinguish the cases $p\geq 2$ and $1
0,$$ whenever
$\|u\|=\varrho$ and $\varrho>0$ is sufficiently small. Set
$$\Gamma=\{\gamma\in C([0,1];X)\,:\,\gamma(0)=0,\,\,\gamma(1)\not=0\mbox{ and
} J_\lambda(\gamma(1))\le0\},$$ and put
$$c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J_\lambda(\gamma(t)).$$
Applying the Mountain Pass theorem without the Palais--Smale
condition we find a sequence $n\mapsto u_n\in X$ such that
\neweq{mplem1}J_\lambda(u_n)\to c\quad\mbox{and}\quad
J_\lambda'(u_n)\to0\end{equation} as $n\to\infty$. Moreover, since
$J_\lambda (|u|)\leq J_\lambda (u)$ for all $u\in X$, we can
assume that $u_n\geq 0$ for any $n\geq 1$. In what follows we
prove that $(u_n)_n$ is bounded in $X$. Indeed, since
$J_\lambda'(u_n)\to0$ in $X'$, then
$$\|u_n\|_{a,b}^p+\|u_n\|_{L^q(\Omega)}^q=\lambda\|u_n\|_{L^r(\Omega;g)}^r+o(1)\|u_n\|$$
as $n\to\infty$. Therefore, as $n\to\infty$
$$\begin{aligned}c+o(1)&=J_\lambda(u_n)=\left(\frac{1}p-\frac{1}r\right)\|u_n\|^p-\left(\frac{1}p-\frac{1}r\right)\|u_n\|_{L^q(\Omega)}^p+
\left(\frac{1}q-\frac{1}r\right)\|u_n\|_{L^q(\Omega)}^q
+o(1)\|u_n\|\\
&\ge
\left(\frac1p-\frac1r\right)\|u_n\|^p+o(1)\|u_n\|-K,\end{aligned}$$
where $K>0$ is an appropriate positive constant depending on $p$,
$q$ and $r$. Thus, since $q