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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 37, 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-2006/37\hfil
$p$-Laplacian with nonlinear boundary conditions]
{Multiple solutions for the $p$-Laplace equation with nonlinear
boundary conditions}
\author[J. Fern\'andez B.\hfil EJDE-2006/37\hfilneg]
{Juli\'an Fern\'andez Bonder}

\address{Juli\'an Fern\'andez Bonder \newline
Departamento  de Matem\'atica, FCEyN \\
UBA (1428) Buenos Aires, Argentina}
 \email{jfbonder@dm.uba.ar}
\urladdr{http://mate.dm.uba.ar/$\sim$jfbonder}

\date{}
\thanks{Submitted January 10, 2006. Published March 21, 2006.}
\thanks{Supported by grant TX066 from Universidad de Buenos Aires,
grants 03-05009 and \hfill\break\indent 03-10608 from  ANPCyT PICT,
project 13900-5 from Fundacion Antorchas and \hfill\break\indent
CONICET (Argentina)}

\subjclass[2000]{35J65, 35J20}
\keywords{$p$-Laplace equations; nonlinear boundary conditions;
\hfill\break\indent variational methods}

\begin{abstract}
 In this note, we show the existence of at least three nontrivial
 solutions to the  quasilinear elliptic equation
 $$
 -\Delta_p u + |u|^{p-2}u = f(x,u)
 $$
 in a smooth bounded domain $\Omega$ of $\mathbb{R}^N$
 with nonlinear boundary conditions
 $|\nabla u|^{p-2}\frac{\partial u}{\partial\nu} = g(x,u)$
 on $\partial\Omega$. The proof is based on variational arguments.
\end{abstract}

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

\section{Introduction}

Let us consider the  nonlinear elliptic problem
\begin{equation}\label{e1.1}
\begin{gathered}
-\Delta_p u + |u|^{p-2}u = f(x,u) \quad \mbox{in } \Omega\\
|\nabla u|^{p-2}\frac{\partial u}{\partial\nu} = g(x,u) \quad
\mbox{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$,
$\Delta_p u = \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$
is the $p$-laplacian and $\partial/\partial\nu$ is the outer
unit normal derivative.

Problem \eqref{e1.1} appears naturally in several branches of pure and applied
mathematics, such as the study of optimal constants for the Sobolev trace
embedding (see \cite{FdP, FBR, FBMR, FBLDR}); the theory of quasiregular and
quasiconformal mappings in Riemannian manifolds with boundary (see \cite{E,
To}); non-Newtonian fluids, reaction diffusion problems, flow through porus
media, nonlinear elasticity, glaciology, etc. (see \cite{AD,AEK, ACh, D}).


The purpose of this note, is to prove the existence of at least
three nontrivial solutions for \eqref{e1.1} under adequate
assumptions on the sources terms $f$ and $g$. This result extends
previous work by the author \cite{FB, FBR1}.

Here, no oddness condition is imposed in $f$ or $g$ and a
positive, a negative and a sign-changing solution are found. The
proof relies on the Lusternik--Schnirelman method for non-compact
manifolds (see \cite{S}).

For a related result with Dirichlet boundary conditions, see
\cite{St} and more recently \cite{BL, ZCL}. The approach in this
note follows the one in \cite{St}.

Throughout this work, by (weak) solutions of \eqref{e1.1} we
understand critical points of the associated energy functional
acting on the Sobolev space $W^{1,p}(\Omega)$:
\begin{equation}\label{Phi}
\Phi(v)=\frac1p\int_{\Omega} |\nabla v|^p + |v|^p\, dx - \int_{\Omega} F(x,v)\,
dx - \int_{\partial\Omega} G(x,v)\, dS,
\end{equation}
where $F(x,u) = \int_0^u f(x,z)\, dz$, $G(x,u) = \int_0^u g(x,z)\,
dz$ and $dS$ is the surface measure.

We will denote
\begin{equation}\label{FyG}
\mathcal{F}(v)=\int_{\Omega} F(x,v)\, dx\quad\mbox{and}\quad
\mathcal{G}(v)=\int_{\partial\Omega} G(x,v)\, dS,
\end{equation}
so the functional $\Phi$ can be rewritten as
$$
\Phi(v) = \frac1p \|v\|_{W^{1,p}(\Omega)}^p - \mathcal{F}(v) - \mathcal{G}(v).
$$

\section{Assumptions and statement of the results}

The precise assumptions on the source terms $f$ and $g$ are as
follows:

\begin{enumerate}
\item[(F1)] $f:\Omega\times\mathbb{R}\to\mathbb{R}$, is a measurable function with
respect to the first argument and continuously differentiable with
respect to the second argument for almost every $x\in\Omega$.
Moreover, $f(x,0)=0$ for every $x\in\Omega$.

\item[(F2)] There exist constants $p<q<p^*=Np/(N-p)$,
$s>p^*/(p^*-q)$, $t=sq/(2+(q-2)s)>p^*/(p^*-2)$ and functions $a\in
L^s(\Omega)$,  $b\in L^t(\Omega)$, such that for $x\in\Omega$,
$u,v\in \mathbb{R}$,
\begin{gather*}
 |f_u(x,u)|\le a(x)|u|^{q-2} + b(x),\\
 |(f_u(x,u)-f_u(x,v))u|\le (a(x)(|u|^{q-2}+|v|^{q-2})+b(x))|u-v|.
\end{gather*}

\item[(F3)] There exist constants $c_1\in (0,1/(p-1))$, $c_2>p$, $0<c_3<c_4$,
such that for any $u\in L^q(\Omega)$
\begin{align*}
 c_3 \|u\|_{L^q(\Omega)}^q
&\le c_2 \int_{\Omega} F(x,u)\, dx\le
\int_{\Omega} f(x,u) u\, dx\\
&\le  c_1\int_{\Omega} f_u(x,u) u^2\, dx\le c_4 \|u\|_{L^q(\Omega)}^q.
\end{align*}

\item[(G1)] $g:\partial\Omega\times\mathbb{R}\to\mathbb{R}$ is a measurable
function with respect to the first argument and continuously
differentiable with respect to the second argument for almost
every $y\in\partial\Omega$. Moreover, $g(y,0)=0$ for every
$y\in\partial\Omega$.

\item[(G2)] There exist constants $p<r<p_*=(N-1)p/(N-p)$,
$\sigma>p_*/(p_*-r)$, $\tau=\sigma r/(2+(r-2)\sigma)>p_*/(p_*-2)$
and functions $\alpha\in L^{\sigma}(\partial\Omega)$, $\beta\in
L^{\tau}(\partial\Omega)$, such that for $y\in\partial\Omega$,
$u,v\in \mathbb{R}$,
\begin{gather*}
 |g_u(y,u)|\le \alpha(y)|u|^{r-2} + \beta(y),\\
 |(g_u(y,u)-g_u(y,v))u|\le
(\alpha(y)(|u|^{r-2}+|v|^{r-2})+\beta(y))|u-v|.
\end{gather*}

\item[(G3)] There exist constants $k_1\in (0,1/(p-1))$, $k_2>p$, $0<k_3<k_4$,
such that for any $u\in L^r(\partial\Omega)$
\begin{align*}
 k_3 \|u\|_{L^r(\partial\Omega)}^r
&\le k_2 \int_{\partial\Omega} G(x,u)\, dS\le
\int_{\partial\Omega} g(x,u) u\, dS\\
&\le  k_1\int_{\partial\Omega} g_u(x,u) u^2\, dx\le k_4
\|u\|_{L^r(\partial\Omega)}^r.
\end{align*}
\end{enumerate}



\begin{remark} \rm
Assumptions (F1)--(F3) imply, since the immersion
$W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ with $1<q<p^*$ is
compact, that $\mathcal{F}$ is $C^1$ with compact derivative. Analogously,
(G1)--(G3) implies the same facts for $\mathcal{G}$ by the compactness of
the immersion $W^{1,p}(\Omega)\hookrightarrow L^r(\partial\Omega)$
for $1<r<p_*$.
\end{remark}

The main result of the paper reads as follows.

\begin{theorem}
Under assumptions {\em (F1)--(F3), (G1)--(G3)}, there
exist three different, nontrivial, (weak) solutions of \eqref{e1.1}.
Moreover these solutions are, one positive, one negative and the
other one has non-constant sign.
\end{theorem}


\section{Proof of the Theorem}

The proof uses the same approach as in \cite{St}. That is, we will
construct three disjoint sets $K_i\ne\emptyset$ not containing $0$
such that $\Phi$ has a critical point in $K_i$. These sets will be
subsets of smooth manifolds $M_i\subset W^{1,p}(\Omega)$ that will
be constructed by imposing a sign restriction and a normalizing
condition.

In fact, let
\begin{gather*}
M_1 = \{ u\in W^{1,p}(\Omega):  \int_{\partial\Omega} u_+\,
dS>0 ,\;
 \|u_+\|_{W^{1,p}(\Omega)}^p = \langle \mathcal{F}'(u), u_+\rangle +
\langle \mathcal{G}'(u), u_+\rangle \},
\\
M_2 = \{ u\in W^{1,p}(\Omega): \int_{\partial\Omega} u_-\,
dS>0 ,\;
 \|u_-\|_{W^{1,p}(\Omega)}^p = \langle \mathcal{F}'(u), u_-\rangle +
\langle \mathcal{G}'(u), u_-\rangle \},\\
M_3 = M_1\cap M_2,
\end{gather*}
where $u_+ = \max\{u,0\}$, $u_-=\max\{-u,0\}$ are the positive and
negative parts of $u$, and $\langle\cdot,\cdot\rangle$ is the
duality pairing of $W^{1,p}(\Omega)$.

Finally we define
\[
 K_1 = \{ u\in M_1\ |\ u\ge 0\},\quad
 K_2 = \{ u\in M_2\ |\ u\le 0\},\quad
 K_3 = M_3.
\]

For the proof of the main theorem, we need the following Lemmas.

\begin{lemma}\label{lema1}
There exist $c_j>0$ such that, for every $u\in K_i$, $i=1,2,3$,
$$
\|u\|_{W^{1,p}(\Omega)}^p\le c_1\Big(\int_{\Omega} f(x,u)u\, dx +
\int_{\partial\Omega} g(x,u)u\, dS\Big) \le c_2 \Phi(u)\le
c_3\|u\|_{W^{1,p}(\Omega)}^p.
$$
\end{lemma}

\begin{proof}
Since $u\in K_i$, we have
$$
\|u\|_{W^{1,p}(\Omega)}^p = \int_{\Omega} f(x,u)u\, dx
 + \int_{\partial\Omega} g(x,u)u\, dS.
$$
This proves the first inequality.
Now, by (F3) and (G3)
\begin{gather*}
\int_\Omega F(x,u)\, dx \le \frac{1}{k_2} \int_\Omega f(x,u)u\, dx,\\
\int_{\partial\Omega} G(x,u)\, dS \le \frac{1}{c_2} \int_{\partial\Omega}
g(x,u)u\, dS.
\end{gather*}
So, for $C=\max\{\frac{1}{k_2};\frac{1}{c_2}\}<\frac{1}{p}$, we have
$$
\Phi(u) \le \big(\frac{1}{p}-C\big) \|u\|_{W^{1,p}(\Omega)}^p.
$$
This proves the third inequality.

To prove the middle inequality we proceed as follows:
\begin{align*}
\Phi(u) &= \frac{1}{p}\|u\|_{W^{1,p}(\Omega)}^p - \int_{\Omega} F(x,u)\, dx -
\int_{\partial\Omega} G(x,u)\, dS\\
& = \frac{1}{p}\Big(\int_\Omega f(x,u)u\, dx + \int_{\partial\Omega}
g(x,u)u\,dS\Big) - \Big(\int_{\Omega} F(x,u)\, dx +
\int_{\partial\Omega} G(x,u)\, dS\Big)\\
&\ge (\frac{1}{p}-C) \Big(\int_\Omega f(x,u)u\, dx + \int_{\partial\Omega}
g(x,u)u\, dS\Big).
\end{align*}
This completes the proof.
\end{proof}

\begin{lemma}\label{lema2}
There exists $c>0$ such that
\begin{gather*}
\|u_+\|_{W^{1,p}(\Omega)}\ge c\quad \mbox{for } u\in K_1,\\
\|u_-\|_{W^{1,p}(\Omega)}\ge c\quad \mbox{for } u\in K_2, \\
\|u_+\|_{W^{1,p}(\Omega)},\, \|u_-\|_{W^{1,p}(\Omega)}\ge c\quad
\mbox{for } u\in K_3.
\end{gather*}
\end{lemma}

\begin{proof}
By the definition of $K_i$, by (F3) and (G3), we have
\begin{align*}
\|u_\pm\|_{W^{1,p}(\Omega)}^p
&= \int_{\Omega} f(x,u)u_\pm\, dx +
\int_{\partial\Omega} g(x,u)u_\pm\, dS \\
&\le c (\|u_\pm\|_{L^q(\Omega)}^q + \|u_\pm\|_{L^r(\partial\Omega)}^r).
\end{align*}
Now the proof follows by the Sobolev immersion Theorem and by the
Sobolev trace Theorem, as $p<q,r$.
\end{proof}


\begin{lemma}\label{lema3}
There exists $c>0$ such that $\Phi(u)\ge c\|u\|_{W^{1,p}(\Omega)}^p$
for every $u\in W^{1,p}(\Omega)$ such that
$\|u\|_{W^{1,p}(\Omega)}\le c$.
\end{lemma}

\begin{proof}
By (F3), (G3) and the Sobolev immersions we have
\begin{align*}
\Phi(u) &= \frac{1}{p}\|u\|_{W^{1,p}(\Omega)}^p - \mathcal{F}(u) - \mathcal{G}(u)\\
& \ge \frac{1}{p}\|u\|_{W^{1,p}(\Omega)}^p - c(\|u\|_{L^q(\Omega)}^q +
\|u\|_{L^r(\partial\Omega)}^r)\\
&\ge \frac{1}{p}\|u\|_{W^{1,p}(\Omega)}^p -
c(\|u\|_{W^{1,p}(\Omega)}^q + \|u\|_{W^{1,p}(\Omega)}^r)\\
&\ge c\|u\|_{W^{1,p}(\Omega)}^p,
\end{align*}
if $\|u\|_{W^{1,p}(\Omega)}$ is small enough, as $p<q,r$.
\end{proof}

The following lemma describes the properties of the manifolds
$M_i$.

\begin{lemma}\label{lema4}
$M_i$ is a $C^{1,1}$ sub-manifold of $W^{1,p}(\Omega)$ of
co-dimension 1 $(i=1,2)$, 2 $(i=3)$ respectively. The sets $K_i$
are complete. Moreover, for every $u\in M_i$ we have the direct
decomposition
$$
T_u W^{1,p}(\Omega) = T_u M_i \oplus \mathop{\rm span}\{u_+, u_-\},
$$
where $T_u M$ is the tangent space at $u$ of the Banach manifold
$M$. Finally, the projection onto the first component in this
decomposition is uniformly continuous on bounded sets of $M_i$.
\end{lemma}


\begin{proof}
Let us denote
\begin{gather*}
 \bar M_1 = \Big\{u\in W^{1,p}(\Omega): \int_{\partial\Omega}
u_+\, dS>0\Big\},\\
 \bar M_2 = \Big\{u\in W^{1,p}(\Omega): \int_{\partial\Omega}
u_-\, dS>0\Big\},\\
 \bar M_3 = \bar M_1\cap \bar M_2.
\end{gather*}
Observe that $M_i\subset \bar M_i$.

By the Sobolev trace Theorem, the set $\bar M_i$ is open in
$W^{1,p}(\Omega)$, therefore it is enough to prove that $M_i$
is a smooth sub-manifold of $\bar M_i$. In order to do this,
we will construct a $C^{1,1}$ function
$\varphi_i:\bar M_i\to \mathbb{R}^d$ with $d=1\ (i=1,2)$, $d=2$
$(i=3)$ respectively and
$M_i$ will be the inverse image of a regular value of $\varphi_i$.

In fact, we define: For $u\in \bar M_1$,
$$
\varphi_1(u) = \|u_+\|_{W^{1,p}(\Omega)}^p
- \langle \mathcal{F}'(u), u_+\rangle - \langle \mathcal{G}'(u), u_+\rangle.
$$
For $u\in \bar M_2$,
$$
\varphi_2(u) = \|u_-\|_{W^{1,p}(\Omega)}^p - \langle \mathcal{F}'(u), u_-\rangle -
\langle \mathcal{G}'(u), u_-\rangle.
$$
For $u\in \bar M_3$,
$$
\varphi_3(u) = (k_1(u), k_2(u)).
$$
Obviously, we have $M_i = \varphi_i^{-1}(0)$. We need to show that $0$
is a regular value for $\varphi_i$. To this end we compute,
for $u\in M_1$,
\begin{align*}
\langle \nabla \varphi_1(u), u_+\rangle = & p\|u_+\|^p_{W^{1,p}(\Omega)}
- \int_{\Omega} f_u(x,u)u_+^2 + f(x,u)u_+\, dx\\
& - \int_{\partial\Omega} g_u(x,u)u_+^2 + g(x,u)u_+\, dS\\
 = & (p-1)\int_{\Omega}f(x,u)u_+\, dx - \int_{\Omega}
f_u(x,u)u_+^2\, dx\\
& + (p-1)\int_{\partial\Omega}g(x,u)u_+\, dS -
\int_{\partial\Omega} g_u(x,u)u_+^2\, dS.
\end{align*}
By (F3) and (G3) the last term is bounded by
$$
(p-1- c_1^{-1})\int_{\Omega}f(x,u)u_+\, dx +
(p-1- k_1^{-1})\int_{\partial\Omega}g(x,u)u_+\, dS.
$$
Recall that $c_1, k_1<1/(p-1)$. Now, by Lemma \ref{lema1},
this is bounded by
$$
-c\|u_+\|_{W^{1,p}(\Omega)}^p
$$
which is strictly negative by Lemma \ref{lema2}. Therefore, $M_1$
is a smooth sub-manifold of $W^{1,p}(\Omega)$. The exact same
argument applies to $M_2$.

Since trivially
$$
\langle \nabla \varphi_1(u), u_-\rangle = \langle \nabla \varphi_2(u),
u_+\rangle =0
$$
for $u\in M_3$, the same conclusion holds for $M_3$.

To see that $K_i$ is complete, let $u_k$ be a Cauchy sequence in
$K_i$, then $u_k\to u$ in $W^{1,p}(\Omega)$. Moreover,
$(u_k)_{\pm}\to u_{\pm}$ in $W^{1,p}(\Omega)$. Now it is easy to
see, by Lemma \ref{lema2} and by continuity that $u\in K_i$.

Finally, by the first part of the proof we have the decomposition
$$
T_u W^{1,p}(\Omega) = T_u M_i \oplus \mathop{\rm span}\{u_+,u_-\}.
$$
Now let $v\in T_u W^{1,p}(\Omega)$ be a unit tangential vector,
then $v = v_1 + v_2$ where $v_i$ are given by
\begin{gather*}
v_2 = (\nabla \varphi_i(u)|_{\mbox{\small span}\{u_+,u_-\}})^{-1}
\langle\nabla
\varphi_i(u), v\rangle \in \mathop{\rm span}\{u_+,u_-\},\\
v_1 = v-v_2\in T_u M_i.
\end{gather*}
 From these formulas and from the estimates given in the first
part of the proof, the uniform continuity follows.
\end{proof}

Now, we need to check the Palais-Smale condition for the
functional $\Phi$ restricted to the manifold $M_i$.

\begin{lemma}\label{lema5}
The functional $\Phi|_{K_i}$ satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof}
Let $\{u_k\}\subset K_i$ be a Palais-Smale sequence, that is
$\Phi(u_k)$ is uniformly bounded and $\nabla \Phi|_{K_i}(u_k)\to
0$ strongly. We need to show that there exists a subsequence
$u_{k_j}$ that converges strongly in $K_i$.

Let $v_j\in T_{u_j}W^{1,p}(\Omega)$ be a unit tangential vector
such that
$$
\langle \nabla \Phi(u_j), v_j\rangle = \|\nabla
\Phi(u_j)\|_{(W^{1,p}(\Omega))'}.
$$
Now, by Lemma \ref{lema4}, $v_j = w_j + z_j$ with $w_j\in
T_{u_j}M_i$ and $z_j\in \mathop{\rm span}\{(u_j)_+, (u_j)_-\}$.

Since $\Phi(u_j)$ is uniformly bounded, by Lemma \ref{lema1},
$u_j$ is uniformly bounded in $W^{1,p}(\Omega)$ and hence $w_j$ is
uniformly bounded in $W^{1,p}(\Omega)$. Therefore
$$
\|\Phi(u_j)\|_{(W^{1,p}(\Omega))'} = \langle \nabla \Phi(u_j),
v_j\rangle = \langle \nabla \Phi|_{K_i}(u_j), v_j\rangle\to 0.
$$

As $u_j$ is bounded in $W^{1,p}(\Omega)$, there exists $u\in
W^{1,p}(\Omega)$ such that $u_j \rightharpoonup u$, weakly in
$W^{1,p}(\Omega)$. As it is well known that the unrestricted
functional $\Phi$ satisfies the Palais-Smale condition (cf.
\cite{FBR1} and \cite{R}), the lemma follows.
See \cite{St} for the details.
\end{proof}


We obtain immediately the following result.

\begin{lemma}\label{lema6}
Let $u\in K_i$ be a critical point of the restricted functional
$\Phi|_{K_i}$. Then $u$ is also a critical point of the
unrestricted functional $\Phi$ and hence a weak solution to
\eqref{e1.1}.
\end{lemma}

With all this preparatives, the proof of the Theorem follows
easily.

\begin{proof}[Proof of the Theorem]
The proof now is a standard application of the
Lusternik--Schnirelman method for non-compact manifolds. See
\cite{S}.
\end{proof}


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\end{document}