\documentstyle[twoside,amssymb]{article}
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\markboth{\hfil Existence and multiplicity of solutions \hfil EJDE--1998/10}%
{EJDE--1998/10\hfil Klaus Pf\/l\"uger \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~10, pp. 1--13. \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} \\
Existence and multiplicity of solutions to a  $p$-Laplacian
equation with nonlinear boundary condition 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J65, 35J20.
\hfil\break\indent
{\em Key words and phrases:} p-Laplacian, nonlinear boundary condition, 
variational methods, \hfil\break\indent
unbounded domain, weighted function space.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted March 5, 1998. Published April 10, 1998.} }
\date{}
\author{Klaus Pf\/l\"uger} % Pflueger

\maketitle

\begin{abstract} 
We study the nonlinear elliptic boundary value problem 
\begin{eqnarray*} 
 & A u = f(x,u) \quad \mbox{in }\Omega\,,\\
 & Bu = g(x,u)\quad \mbox{on }\partial \Omega\,,
\end{eqnarray*}
where $A$ is an operator of $p-$Laplacian type, $\Omega$ is  
an unbounded domain in ${\Bbb R}^N$ with non-compact boundary, and 
$f$ and $g$ are subcritical nonlinearities. 
We show existence of a  nontrivial nonnegative weak solution
when both $f$ and $g$ are superlinear. Also we show
existence of at least two nonnegative solutions when one of the two 
functions $f$, $g$ is  sublinear and the other one superlinear. 
The proofs are based on  variational methods applied to weighted 
function spaces.
\end{abstract}

\newtheorem{theo}{Theorem}           
\newtheorem{lemma}{Lemma}   
\newcommand{\bewende}{\hfill\mbox{$\Box$}\protect}
\newcommand{\R}{{\Bbb R}} 
\newcommand{\vi}{\varphi}
\newcommand{\eps}{\varepsilon}
\newcommand{\n}{\mbox{\sf n}}
\newcommand{\dn}{\partial_{\n}}
\newcommand{\diver}{\mathop{\rm div}}
                                               
\section{Introduction}

The objective of this paper is to study the nonlinear elliptic boundary value 
problem
\begin{eqnarray}
&-\diver (a(x) |\nabla u|^{p-2} \nabla u) 
 =   f(x,u) \quad  \mbox{ in } \quad \Omega \subset \R^N, &   \label{p1}  \\
&\n \cdot  a(x) |\nabla u|^{p-2} \nabla u + b(x) |u|^{p-2}u 
 =  g(x,u) \quad  \mbox{ on } \quad \Gamma = \partial \Omega, &  \label{p2}
\end{eqnarray}
where $\Omega$ is an unbounded domain with noncompact, smooth boundary $\Gamma$
(for example a cylindrical domain), and $\n$ is the unit outward normal vector on 
$\Gamma$. We assume throughout that $1<p<N $, 
$0 <a_0 \leq a \in L^{\infty}(\Omega)$ and $b$ is a positive and continuous 
function defined on $\R^N$. 
The $p-$Laplace operator in (\ref{p1}) is a special case of the divergence--form operator 
$-\diver (a(x, \nabla u))$ which appears in many nonlinear diffusion problems, 
in particular in the mathematical modeling of non--Newtonian fluids.
For a  discussion of some physical background see \cite{3}. 
 The boundary condition (\ref{p2}) 
describes a flux through the boundary which depends in a nonlinear manner 
on the solution itself. For some physical motivation of such boundary conditions see 
for example \cite{6}. 

The energy functional corresponding to (\ref{p1}), (\ref{p2}) is defined as 
$$
J(u) = \frac{1}{p} \int_{\Omega} a(x) |\nabla u|^p \, dx + \frac{1}{p} \int_{\Gamma} 
b(x) |u|^p\, d\Gamma - \int_{\Omega} F(x,u)\, dx - \int_{\Gamma} G(x,u)\, d\Gamma \, ,
$$
where $F$ and $G$ denote the primitive functions of $f$ and $g$  
with respect to the second variable, i.\ e. 
$F(x,u) = \int_0^u f(x,s) \, ds, \; G(x,u) = \int_0^u g(x,s) \, ds $.
Then the weak solutions of (\ref{p1}), (\ref{p2}) are the critical points of 
$J$. We remark that, according to the regularity theorem 
of \cite{8a}, every weak solution of (\ref{p1}), (\ref{p2}) belongs to 
$C_{\rm loc}^{1,\beta}(\Omega)$. In addition, in \cite{4} regularity up to the 
boundary was proved, but only under rather restrictive conditions on $g$. 

In this paper we consider problem (\ref{p1}), ({\ref{p2}) under several conditions on 
$f$ and $g$. If both functions are subcritical and superlinear with respect to 
$u$, then we prove existence of a nontrivial nonnegative solution (Theorem 2). 
In the case, where $f$ is sublinear and $g$ superlinear, we show that there 
exist at least two nonnegative solutions, one with positive energy, the other one 
with negative energy (Theorem 3). The same result holds in the case where $f$ 
is superlinear and $g$ sublinear (Theorem 4). 

Such kind of problems with combined concave and convex nonlinearities were studied 
recently by several authors, with the right hand side of (\ref{p1}) 
of the form $f+g$ and the boundary condition is $u = 0 $ on $\Gamma$. For a 
bounded domain $\Omega$ and $p=2$ see \cite{1}, for $1<p<N$ see \cite{2} and \cite{2a} 
(which also includes the critical case). For the $p-$Laplacian in an exterior domain 
see \cite{10}. Our proofs are based on weighted-norm estimates in Sobolev spaces, which 
imply some compactness properties of the functional $J$. For some related results 
on the existence of nontrivial solutions to equation (\ref{p1}) in $\R^{N}$ see for 
example \cite{2b}, \cite{3a}, \cite{3b}, \cite{5}.
We remark that the results in this paper are new even in the 
semilinear elliptic case $p=2$. 

This paper is organized as follows: In the next section we prove some preliminary 
results concerning equivalent norms and traces in weighted Sobolev spaces. Section 3 
is devoted to the superlinear case (Theorem 2), and Section 4 contains the results 
on the mixed  case (Theorems 3 and 4).

 

\section{ Preliminaries: Weighted Sobolev Spaces}
 
Let $C_{\delta}^{\infty}(\Omega)$ be the space of 
$C_0^{\infty}(\R^N)$-functions  restricted on $\Omega$. 
We define the weighted Sobolev-space $E$ as the completion
of $C_{\delta}^{\infty}(\Omega)$ in the norm 
$$
\| u \|_E = \left( \int_{\Omega} | \nabla u (x) |^p + \frac{1}{(1+|x|)^p} 
| u(x)|^{p} \, dx  \right)^{1/p}  \, .
$$
First we prove the following weighted Hardy-type inequality.
\begin{lemma}   \label{t1}
Let  $1<p<N$. Then there exist positive constants $C_1$ and $C_2$,
such that for every $u \in E$
\begin{equation}   \label{eq1}
\int_{\Omega} \frac{1}{(1+|x|)^p} |u|^{p}\, dx \leq C_1 \int_{\Omega} |\nabla u |^p
\, dx + C_2 \int_{\Gamma} \frac{|\n \cdot x|}{(1+|x|)^p} |u |^p \, d\Gamma \, .
\end{equation}
\end{lemma}

\paragraph{Proof.}
Using the divergence theorem we obtain for $u \in C_{\delta}^{\infty}(\Omega)$
$$
\int_{\Omega} x \cdot \nabla \left( \frac{1}{(1+|x|)^p} |u|^{p} \right) dx = 
\int_{\Gamma} (\n \cdot x)  \frac{1}{(1+|x|)^p} |u|^{p}\, d\Gamma -
N \int_{\Omega} \frac{1}{(1+|x|)^p} |u|^{p}\, dx \, .
$$
This implies 
\begin{eqnarray*}
N \int_{\Omega} \frac{1}{(1+|x|)^p} |u|^{p}\, dx & \leq &
\int_{\Gamma} \frac{|\n \cdot x|}{(1+|x|)^p} |u|^p \, d\Gamma +
p \int_{\Omega} \frac{1}{(1+|x|)^p} |u|^{p}\, dx  \\
& & + p \int_{\Omega} \frac{1}{(1+|x|)^{p-1}} |u|^{p-1} |\nabla u |\, dx \, .
\end{eqnarray*}
Using H\"older's and Young's inequality, the last term can be estimated by 
\begin{eqnarray*}
\lefteqn{ p \, \left( \int_{\Omega}  \frac{1}{(1+|x|)^{p}} |u|^{p}\, dx \right)
^{(p-1)/p}  
\left( \int_{\Omega} |\nabla u |^{p}\, dx \right)^{1/p} }&& \hspace{6cm} \\
& \leq &\eps(p-1) \, \int_{\Omega}  \frac{1}{(1+|x|)^{p}} |u|^{p}\, dx +
\eps^{1-p} \, \int_{\Omega} |\nabla u |^{p}\, dx \, , 
\end{eqnarray*}
where $\eps > 0$ is an arbitrary real number. It follows that 
$$
(N - \eps(p-1) - p)\,  \int_{\Omega}  \frac{1}{(1+|x|)^{p}} |u|^{p}\, dx \leq 
\eps^{1-p} \, \int_{\Omega} |\nabla u |^{p}\, dx + \int_{\Gamma} 
\frac{|\n \cdot x|}{(1+|x|)^p} |u|^p \, d\Gamma \, ,
$$
and for $\eps$ small enough, the desired inequality follows by standard density arguments.
\bewende

Now denote by $L^r(\Omega; w_1)$ and $L^q(\Gamma; w_2)$ 
the weighted Lebesgue spaces with weight functions 
\begin{equation}
w_i(x) = (1+|x|)^{\alpha_i}\, , \quad i=1,2, \quad \alpha_i \in \R  \label{eq1a}
\end{equation}
and norm defined by 
$$
\| u \|_{r,w_1}^r  = \int_{\Omega} w_1 |u(x)|^{r} \, dx  \, , \qquad
\mbox{and} \qquad
\| u \|_{q,w_2}^q  = \int_{\Gamma} w_2 |u(x)|^{q} \, dx  \, .
$$
Then we have the following embedding and trace theorem.
\begin{theo} \label{t2}
If 
\begin{equation}
p \leq r \leq  \frac{pN}{N-p}  \quad \mbox{and} \quad       
-N < \alpha_1 \leq  r\, \frac{N-p}{p} - N   \, ,          \label{eq2} 
\end{equation}
then the embedding $E \hookrightarrow L^r(\Omega; w_1)$ is continuous.
If the upper bounds for $r$ in (\ref{eq2}) are strict, then the embedding is compact. 
If
\begin{equation}
p \leq q \leq  \frac{p(N-1)}{N-p}   \quad \mbox{and} \quad        
-N < \alpha_2 \leq  q\, \frac{N-p}{p} - N + 1  \, ,       \label{eq3}
\end{equation}
then the trace operator $E \to L^q(\Gamma; w_2)$ is continuous. 
If the upper bounds for $q$ in (\ref{eq3}) are strict, then the trace is compact. 
\end{theo} 
This theorem is a consequence of Theorem 2 and Corollary 6 of \cite{7}. 

As a corollary of Lemma \ref{t1} and Theorem \ref{t2}  we obtain
\begin{lemma}   \label{t3}
Let $b$ satisfy $c/(1+|x|)^{p-1} \leq b(x) \leq C/(1+|x|)^{p-1}$ 
for some constants $0 < c \leq C$. Then 
$$
\| u \|_b^p = \int_{\Omega} a(x) |\nabla u |^p\,dx + \int_{\Gamma} b(x) |u|^p\,d\Gamma  
$$ 
defines an equivalent norm on $E$. 
\end{lemma}

\paragraph{Proof.} The inequality $\| u \|_E \leq C_1 \| u \|_b$ follows directly from 
Lemma \ref{t1}, while from Theorem \ref{t2} (setting $p=q$ and $\alpha_2 = -(p-1)$)
we obtain 
\begin{eqnarray*}
\| u \|_b^p 
& \leq &  \| a \|_{L^{\infty}} \int_{\Omega} | \nabla u |^p dx + C \int_{\Gamma} 
|u|^p (1+ |x|)^{-(p-1)} d\Gamma    \\
& \leq & \| a \|_{L^{\infty}} \int_{\Omega} | \nabla u |^p dx + C_2 \| u \|_E^p ,
\end{eqnarray*}
which shows the desired equivalence.
\bewende

\paragraph{ Remark.}
In special geometries the lower bound for $b$ required in Lemma \ref{t3} can be 
improved. In view of Lemma \ref{t1} it is sufficient to assume 
$ b(x) \geq |\n \cdot x| / (1+|x|)^{p} $, where 
$\n \cdot x = |\n | | x| \cos \gamma$ and $\gamma$ is the angle between 
$x$ and $\n$. For a cylindrical domain $\Omega = B \times \R$, where 
$B \subset \R^{N-1}$ is bounded, we obtain $| \cos \gamma | \leq C_B / |x| $, 
with a constant $C_B$ depending only on the diameter of $B$. This shows that 
in cylindrical domains, Lemma \ref{t3} holds under the weaker assumption
$$
\frac{c}{(1+|x|)^{p}} \leq b(x) \leq \frac{C}{(1+|x|)^{p-1}} \, .
$$  
We shall assume throughout the paper that $b$ satisfies the assumption of 
Lemma~\ref{t3} so that we can use $\| \cdot \|_b$ as an equivalent norm in $E$.

\section{The superlinear case} 

We make the following assumptions
\begin{description}  
\item{A1 } $ f$ and $g$ are Carath\'{e}odory functions on 
$\Omega \times \R$ and  $\Gamma \times \R $, respectively, 
$ f(\cdot ,0) = g(\cdot ,0) = 0 \, $ and  
\begin{eqnarray*}
|f(x,s)| \leq  f_{0}(x) + f_{1} (x)|s|^{r-1} & , & \;   p\leq r<pN/(N-p) ,  \\
|g(x,s)| \leq  g_{0}(x) + g_{1} (x)|s|^{q-1} & , & \;   p\leq q<p(N-1)/(N-p), 
\end{eqnarray*}
where $f_i$, $g_i$ are nonnegative, measurable functions which satisfy the 
following hypotheses: There exist $\alpha_1, \, \alpha_2$, 
$-N < \alpha_1 < r \frac{N-p}{p} - N $, 
$-N < \alpha_2 < q \frac{N-p}{p} - N + 1 $, such that, with 
$ w_i$ defined as in (\ref{eq1a}),  we have  
\begin{eqnarray*}
0 \leq f_i(x) \leq C_{f} w_1 \; \mbox{ a.\ e. }  ,&& 
f_0 \in L^{r/(r-1)}(\Omega; w_1^{1/(1-r)}) ,   \\
0 \leq g_i(x) \leq C_{g} w_2 \; \mbox{ a.\ e. }  ,&& 
g_0 \in L^{q/(q-1)}(\Gamma; w_2^{1/(1-q)}) \, .
\end{eqnarray*}
\item{A2 }
$ \lim_{s \to 0} f(x,s)/|s|^{p-1} = \lim_{s \to 0} g(x,s)/|s|^{p-1} = 0$ 
uniformly in $x$.
\item{A3 }  
There exists $ \mu > p$ such that 
$ \mu F(x,s) \leq  f(x,s) s $,  $\mu G(x,s) \leq g(x,s) s$ 
for a.\ e. $ x \in \Omega $, resp. $x \in \Gamma$ and every $s \in \R$.
\item{A4 } One of the following conditions holds: 
\begin{description} 
\item{ a) }  There is a nonempty open set $O \subset \Omega$ with $F(x,s) > 0$ 
for $(x,s) \in O \times (0,\infty) $  
\item{ b) } There is a nonempty open set $U \subset \Gamma$ with $G(x,s) > 0$ 
for $(x,s) \in U \times (0,\infty) $ and $G$ satisfies 
$\bar{\mu} G(x,s) \leq g(x,s) s$ with some  $\bar{\mu} >r$. 
\item{ c) }  $G(x,s) > 0$ for $(x,s) \in U \times (0,\infty) $ and 
and there exist an open, nonempty subset $V \subset \Omega $, 
$\overline{V} \cap U \neq \emptyset$ and a constant $C_F$, such that  
$F(x,u) \geq  -C_F $ on $ V \times (0,\infty)$.
\end{description}
\end{description}
We denote by $N_f$, $N_F$, $N_{g}$, $N_{G}$ the corresponding 
Nemytskii operators. Under the assumptions above we have the following result.  
\begin{lemma}  \label{t5}
The operators  
\begin{eqnarray*} 
N_f : L^{r}(\Omega; w_1) \to  L^{r/(r-1)}(\Omega; w_1^{1/(1-r)}) \, , && 
N_F : L^{r}(\Omega; w_1) \to  L^{1}(\Omega)  \, , 
\\
N_{g} : L^{q}(\Gamma; w_2) \to  L^{q/(q-1)}(\Gamma; w_2^{1/(1-q)}) \, , &&
N_{G} : L^{q}(\Gamma; w_2) \to  L^{1}(\Gamma) 
\end{eqnarray*}
are bounded and continuous.   
\end{lemma}

\paragraph{Proof.} We only prove the statements for $N_g$ and $N_G$, since the 
arguments for  $N_f$ and $N_F$ are similar. Let $q^{\prime} = q/(q-1)$ 
and $u \in  L^{q}(\Gamma; w_2)$. Then, by Assumption A1,  
\begin{eqnarray*} 
\int_{\Gamma} | N_g (u) |^{q^{\prime}} w_2^{1/(1-q)} d\Gamma 
& \leq & 
2^{q^{\prime}-1} \left( \int_{\Gamma} g_0^{q^{\prime}} w_2^{1/(1-q)} d\Gamma +  
\int_{\Gamma} g_1^{q^{\prime}} |u|^q  w_2^{1/(1-q)} d\Gamma  \right)  \\
& \leq &
2^{q^{\prime}-1} \left( C + C_g \, \int_{\Gamma}  |u|^q  w_2  d\Gamma \right)  ,
\end{eqnarray*}
which shows that $N_g$ is bounded. In a similar way we obtain 
\begin{eqnarray*} 
\int_{\Gamma} | N_G (u) | d\Gamma 
& \leq & 
\int_{\Gamma} g_0  | u | d\Gamma + \int_{\Gamma} g_1 |u|^q d\Gamma \\
& \leq & 
\left( \int_{\Gamma} g_0^{q^{\prime}} w_2^{1/(1-q)} d\Gamma 
\right)^{\frac{1}{q^{\prime}}} 
\left( \int_{\Gamma} | u |^q w_2 \, d\Gamma \right)^{\frac{1}{q}} + 
C_g \, \int_{\Gamma} | u |^q w_2 \, d\Gamma
\end{eqnarray*}
and again we claim that $N_G$ is bounded. The continuity of these operators now 
follows from the usual properties of Nemytskii operators (cf. \cite{9}). 
\bewende

\begin{lemma}  \label{t6}
Under Assumptions A1--A4, $J$ is Fr\'{e}chet--differentiable 
on $E$ and satisfies the Palais--Smale condition. 
\end{lemma}

\paragraph{Proof.} We use the notation 
$I(u) = \frac{1}{p} \| u \|_b^p $, $K_F (u) = \int_{\Omega} F(x,u)\,dx $, 
$K_G (u) = \int_{\Gamma} G(x,u)\,d\Gamma $.  
Then the directional derivative of $J$ in direction $h \in E$ is 
$$
\langle J^{\prime} u, h \rangle = \langle I^{\prime} u, h \rangle - 
\langle K_F^{\prime} u, h \rangle - \langle K_G^{\prime} u, h \rangle \, ,
$$
where 
\begin{eqnarray*}
&\langle I^{\prime}(u), h \rangle
 =  \int_{\Omega} a(x) |\nabla u|^{p-2} \nabla u \nabla h  \, dx + 
\int_{\Gamma} b(x) |u|^{p-2} u h \, d\Gamma \, , & \\
&\langle K_F^{\prime} (u), h \rangle
= \int_{\Omega} f(x,u) h \, dx \, , \quad 
\langle K_G^{\prime} (u), h \rangle \; = \;  \int_{\Gamma} g(x,u) h \, d\Gamma 
\, . &
\end{eqnarray*}
Clearly, $I^{\prime} : E \to E^{\prime}$ is continuous. The operator 
$K_G^{\prime}$ is a composition of  operators 
$$
K_G^{\prime} : E \to L^q(\Gamma; w_2) \stackrel{N_g}{\longrightarrow}  
L^{q/(q-1)}(\Gamma; w_2^{1/(1-q)}) \stackrel{\ell}{\longrightarrow} E^{\prime} ,
$$
where $\langle \ell (v), h \rangle = \int_{\Gamma} v h \, d\Gamma $. 
Since 
$$
\int_{\Gamma} |v h| \, d\Gamma \leq \left( \int_{\Gamma} |v|^{q^{\prime}} 
w_2^{1/(1-q)} d\Gamma \right)^{1/q^{\prime}} \left( \int_{\Gamma} 
|h|^q w_2  \, d\Gamma \right)^{1/q} , 
$$ 
$\ell$ is continuous by Theorem \ref{t2}. As a composition of continuous operators, 
$K_G^{\prime}$ is continuous, too. Moreover, by our assumptions on $w_2$ 
(see A1), the trace operator $E \to L^q(\Gamma; w_2) \, $ is compact and 
therefore, $K_G^{\prime}$ is also compact. In a  similar way we obtain that 
$ K_F^{\prime}$ is compact and the Fr\'{e}chet-differentiability of $J$ follows.

Now let  $u_k \in E$ be a Palais--Smale sequence, i.\ e. $| J(u_k) | \leq C$ 
for all $k$ and $J^{\prime}(u_k) \to 0 $ as $k \to \infty$. For $k$ large 
enough we have $| \langle J^{\prime}(u_k), u_k \rangle | \leq \| u_k \|_b$ 
and by Assumption A3
\begin{eqnarray*}
C + \| u_k \|_b 
& \geq & 
J(u_k) - \frac{1}{\mu} \langle J^{\prime}(u_k), u_k \rangle \\
& \geq & 
\left( \frac{1}{p} - \frac{1}{\mu} \right) \, \| u \|_b^p \, .
\end{eqnarray*}
This shows that $u_k$ is bounded in $E$. To show that $u_k$ contains a 
Cauchy sequence we use the following inequalities for $\xi, \zeta \in \R^N$ 
(see \cite{3}, Lemma 4.10):
\begin{eqnarray}
&| \xi - \zeta |^p 
 \leq  C ( |\xi |^{p-2} \xi - |\zeta |^{p-2}\zeta)(\xi - \zeta) \, , 
\quad \mbox{for }  p \geq 2, &\label{eq21}   \\
&| \xi - \zeta |^2 
 \leq  C ( |\xi |^{p-2} \xi - |\zeta |^{p-2}\zeta)(\xi - \zeta)
( |\xi | + |\zeta | )^{2-p} \, , \quad \mbox{for }  1< p < 2\,.& \label{eq22} 
\end{eqnarray}
Then we obtain in the case $p \geq 2$:
\begin{eqnarray*}
\| u_n - u_k \|^p_b 
&  =   &  
\int_{\Omega} a(x) |\nabla u_n - \nabla u_k |^{p}  dx + 
\int_{\Gamma} b(x) |u_n - u_p |^{p}  d\Gamma        \\
& \leq & 
C \Big( \langle I^{\prime}(u_n), u_n - u_k \rangle - 
\langle I^{\prime}(u_k), u_n - u_k \rangle  \Big)  \\
&  =   &  C \Big( \langle J^{\prime}(u_n), u_n - u_k \rangle - 
\langle J^{\prime}(u_k), u_n - u_k \rangle  + 
\langle K_F^{\prime}(u_n) \\
&&+ K_G^{\prime}(u_n), u_n - u_k \rangle    
- \langle K_F^{\prime}(u_n) + K_G^{\prime}(u_k), 
u_n - u_k \rangle \Big) \\
& \leq &  
C \Big( \| J^{\prime}(u_n) \|_{E^{\prime}} + \| J^{\prime}(u_k) \|_{E^{\prime}} +
\| K_F^{\prime}(u_n) - K_F^{\prime}(u_k) \|_{E^{\prime}}     \\
&   &
+ \| K_G^{\prime}(u_n) - K_G^{\prime}(u_k) \|_{E^{\prime}}
\Big) \| u_n - u_k \|_b \,.
\end{eqnarray*}
Since $ J^{\prime}(u_k) \to 0$ and $ K_F^{\prime},  K_G^{\prime}$ are 
compact, there exists a subsequence of $u_k$ which converges in $E$. 

If $1 < p < 2$, then we use (\ref{eq22}) and H\"older's inequality to obtain the
estimate 
$$
\| u_n - u_k \|^2_b  \leq  C \Big| \langle I^{\prime}(u_n), u_n - u_k \rangle - 
\langle I^{\prime}(u_k), u_n - u_k \rangle  \Big|
\Big( \| u_n \|_b^{2-p} + \| u_k \|_b^{2-p} \Big)   .
$$
Since $\| u_n \|_b$ is bounded,  the same arguments as above lead to a 
convergent subsequence.
\bewende

\begin{theo}  \label{t7}
There exists a nontrivial nonnegative solution of (\ref{p1}), (\ref{p2}) in $E$.
\end{theo}

\paragraph{Proof.} We shall use the Mountain--Pass lemma \cite{8} to obtain a solution. 
First we observe that, from Assumption A1 and A2, for every $\eps > 0$
there is a $C_{\eps}$ such that  
$|  F(x, u) | \leq \eps f_0(x)| u|^p + C_{\eps}f_{1}(x)|u|^r $,
and  $| G(x,u) | \leq \eps g_0(x) |u|^p + C_{\eps} g_{1}(x)|u|^q $.
Consequently
\begin{eqnarray*}
J(u) 
& \geq &
\frac{1}{p} \| u \|^p_b  - \int_{\Omega} \left( \eps f_0 (x) |u|^p + 
C_{\eps}f_{1}(x) |u|^r \right) dx \\
&&- \int_{\Gamma} \left( \eps g_0 (x) 
|u|^p + C_{\eps} g_{1}(x) |u|^q \right) d\Gamma  \\
& \geq &
\| u \|_b^p  - \eps C_1  \| u \|_b^p - C_{\eps} C_2  (\| u \|_b^r  + \| u \|_b^q )
\end{eqnarray*}
and for $ \eps$ and $\| u \|_b = \rho$ sufficiently small, 
the right hand side is strictly greater than $0$. It remains to show 
that there exists $u_0 \in E, \; \| u_0 \|_b > \rho $ such that 
$J(u_0) \leq 0$. 

In the case A4\ a), we choose a nontrivial nonnegative function $\vi \in C_0^{\infty}(O)$. 
From A3 we see that $F(x,s) \geq C_1 s^{\mu} - C_2$ on $O \times (0,\infty)$. 
Then, for $t \geq 0$,  
$$
J(t \vi) \leq  \frac{1}{p} \, t^p \| \vi \|^p_b  - C_1 t^{\mu} \int_{O} 
\vi^{\mu} dx + C_2 | O | \, .
$$
Since $\mu > p$, the right hand side tends to $-\infty$ as $t \to \infty$
and for sufficiently large $t_0$, $u_0 = t_0 \vi$ has the desired properties.

In the case A4\ b), we choose a nonnegative $\vi \in C_{\delta}^{\infty}(\Omega)$ such that 
$\mbox{supp} \vi \cap \Gamma \subset U$  is not empty. Again from 
$G(x,s) \geq C_3 s^{\bar{\mu}} - C_4$ on $U \times (0,\infty)$ and 
Assumption A1 we claim 
$$
J(t \vi) \leq  \frac{1}{p} \, t^p \| \vi \|^p_b  + C_5  \int_{\Omega} t \vi + 
t^r \vi^r dx - C_3 t^{\bar{\mu}} \int_{U} \vi^{\bar{\mu}} d\Gamma  + C_4 | U | \, .
$$
Since $\bar{\mu} > r \geq p$, we obtain $J(t \vi) \to -\infty$ as $t \to \infty $.

In the case A4\ c), we take $\vi \in C_{\delta}^{\infty}(\Omega)$ with 
$\mbox{supp} \vi \cap \overline{\Omega} \subset \overline{V}$ 
and $\mbox{supp} \vi \cap U \neq \emptyset $. Then 
$$
J(t \vi) \leq  \frac{1}{p} \, t^p \| \vi \|^p_b  + C_F | V | - C_3 t^{\mu} 
\int_{U} \vi^{\mu} d\Gamma  + C_4 | U | 
$$
and again we claim  $J(t \vi) \to -\infty$ as $t \to \infty $.

Since $J$ satisfies the Palais--Smale condition and $J(0) = 0$, the 
Mountain--Pass Lemma shows that there is a nontrivial critical point 
of $J$ in $E$ with critical value
$$
c = \inf_{\gamma \in P} \max_{t\in [0,1]} J(\gamma(t)) > 0 \, ,
$$
where $P = \{ \gamma \in C([0,1], E) \mid \gamma(0) = 0, \, \gamma(1) = u_0 \}$.

To obtain a nonnegative solution by this procedure, we introduce the truncated
functions  $\bar{f}$ and $\bar{g}$ such that $\bar{f}(x,s) = \bar{g}(x,s) = 0 $ 
for all $s\leq 0$. Then the arguments above 
remain true and we obtain a critical point $u$ of the truncated functional 
$\bar{J}$, i.\ e. $\langle \bar{J}^{\prime}(u), h \rangle = 0$ for 
all $h \in E$. In particular, setting $u_{-}(x) = \max \{ -u(x), 0 \}$ 
and $h = u_{-}$, we claim that $u \geq 0$. Since any nonnegative solution of 
the truncated problem is also a solution of the original equation, we have found 
a nonnegative solution of (\ref{p1}), (\ref{p2}). 
\bewende

\section{Combined Sub- and Superlinear Nonlinearities}

In this part we introduce an additional parameter into equation 
(\ref{p1}), i.\ e.\ we study 
$$
-\diver (a(x) |\nabla u|^{p-2} \nabla u)  =  \lambda f(x,u) \quad  \mbox{ in } 
\quad \Omega  \eqno (1)_{\lambda}
$$
with the same boundary condition (\ref{p2}) as before. 
Here, we assume  the following 
\begin{description}  
\item{B1 } 
Let $g$ satisfy  Assumptions A1--A3 with $g_0 \equiv 0$ and 
$|f(x,s)| \leq  f_{1} (x)|s|^{r-1}, \quad  1 \leq r < p $, 
where $f_1$ is nonnegative, measurable and there exists $\alpha_1$, 
$-N < \alpha_1 < r \frac{N-p}{p} - N $, such that for 
$w_1 (x) = (1 + |x|)^{\alpha_1} $, we have 
$f_1  \in L^{p/(p-r)}(\Omega; w_1^{r/(r-p)}) $.
\item{B2 } 
$ |f(x,s)| \geq f_{2} (x)|s|^{\bar{r}-1}, \quad 1 \leq \bar{r} \leq r $,
with $f_2 > 0$ in some nonempty open set $O \subset \Omega$.
\item{B3 }
There is a nonempty open set $U \subset \Gamma$ with $G(x,s) > 0$ 
for $(x,s) \in U \times (0,\infty) $.
\end{description}
The Nemytskii operators $N_g$ and $N_G$ have the same properties as in Lemma \ref{t5}, 
while for $N_f$ and $N_F$ we obtain

\begin{lemma}  \label{t8}
The operators  $N_f : L^{p}(\Omega; w_1) \to  L^{p/(p-1)}(\Omega; w_1^{1/(1-p)})$,
and $N_F : L^{p}(\Omega; w_1) \to  L^{1}(\Omega)$
are bounded and continuous.   
\end{lemma}

\paragraph{Proof.} Since the first statement is trivial if $r=1$, we may assume that 
$r>1$. From B1 we obtain with H\"older's inequality (setting $p^{\prime} = p/(p-1)$)
\begin{eqnarray*}
\int_{\Omega} | f(x,u) |^{p^{\prime}} w_1^{1/(1-p)} dx  
& \leq &     \int_{\Omega} |f_1|^{p^{\prime}} w_1^{r/(1-p)} 
|u |^{p^{\prime}(r-1)} w_1^{(r-1)/(p-1)} dx \\
& \leq & \left( \int_{\Omega} |f_1|^{p/(p-r)} w_1^{r/(r-p)} \right)^{\frac{p-r}{p-1}}
\left( \int_{\Omega} |u|^{p} w_1  \right)^{\frac{r-1}{p-1}}  \\
& \leq & C \, \| u \|_{p,w_1}^{p(r-1)/(p-1)} \, .
\end{eqnarray*}
For $N_F$ we obtain
\begin{eqnarray*}
\int_{\Omega} | F(x,u) | dx 
& \leq &  
\int_{\Omega} | f_1 | w_1^{-r/p} |u |^r w_1^{r/p}  dx  \\
& \leq & 
\left( \int_{\Omega} | f_1 |^{p/(p-r)}  w_1^{r/(r-p)} dx \right)^{(p-r)/p}
\left( \int_{\Omega} | u |^{p}  w_1 dx \right)^{r/p}    \\
& \leq & C \, \| u \|_{p,w_1}^{r} \, .
\end{eqnarray*}

\vspace*{-6mm}  \bewende

\vspace*{3mm}
\noindent The differentiability  for $J$ now follows as above. 

To obtain the Palais--Smale condition for $J$, let  $u_k \in E$ be a  sequence
such that $ | J(u_k) | \leq C$  and $J^{\prime}(u_k) \to 0 $ as $k \to \infty$.
With Assumptions A3, B1 and H\"older's inequality we get 
\begin{eqnarray*}
\lefteqn{ J(u_k) - \frac{1}{\mu} \langle J^{\prime}(u_k), u_k \rangle } &&\\
& = & 
\left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p + 
\int_{\Omega} \frac{1}{\mu} f(x,u)u  - F(x,u) \, dx 
+ \int_{\Gamma} \frac{1}{\mu} g(x,u)u  - G(x,u) \, d\Gamma  \\
& \geq & 
\left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - 
\left( 1 + \frac{1}{\mu} \right) \int_{\Omega} f_1(x) |u_k|^r dx  \\
& \geq & 
\left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - 
\left( \int_{\Omega} f_1^{p/(p-r)} w_1^{r/(r-p)} dx \right)^{(p-r)/p} 
\left( \int_{\Omega} |u_k|^{p}  dx \right)^{r/p}   \\
& \geq & 
\left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - 
C_1 \| f_1 \|_{*} \| u_k \|_b^r \, ,
\end{eqnarray*}
where $\| f_1 \|_{*}$ is the weighted norm of $f_1$ in 
$L^{p/(p-r)}(\Omega; w_1^{r/(r-p)}) $. Since $r<p$ and 
$C + \| u_k \|_b  \geq  J(u_k) - \frac{1}{\mu} \langle J^{\prime}(u_k), u_k \rangle $, 
we claim that $u_k$ is bounded in $E$. The convergence of a subsequence of $u_k$ 
then follows as above from the compactness properties of $K_F^{\prime}$ and 
$K_G^{\prime}$. 

\begin{theo}  \label{t9}   
Under Assumptions B1--B3 there exists $\lambda^{*} > 0$, such that for every 
$0 < \lambda < \lambda^{*}$, there are at least two  nontrivial nonnegative 
solutions of (\ref{p1})$_{\lambda}$, (\ref{p2}).
\end{theo}

\paragraph{Proof.} First we show that for $\lambda \in (0, \lambda^{*})$, we can find 
$\rho > 0$ such that $J(u) \geq c > 0 \, $ if $\| u \|_b = \rho $. We denote by 
$C_{\Omega}, C_{\Gamma}$ the embedding and trace constants for the operators 
$E \hookrightarrow  L^p(\Omega; w_1)$ and  $E \to L^q(\Gamma; w_2) $, respectively.
We obtain 
\begin{eqnarray*}
J_{\lambda}(u) 
& \geq & 
\frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} \int_{\Omega} f_1(x) |u|^r  \, dx 
- \frac{1}{q} \int_{\Gamma} g_1(x) |u|^q  \, d\Gamma   \\
& \geq & 
\frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} \left( \int_{\Omega} 
f_1(x)^{p/(p-r)} w_1(x)^{r/(r-p)} dx \right)^{(p-r)/p}
\left( \int_{\Omega} |u|^p w_1 dx \right)^{r/p} \\
&&- \frac{1}{q} \int_{\Gamma} g_1(x) |u|^q  \, d\Gamma   \\
& \geq &
\frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} C_{\Omega} \| f_1 \|_{*} 
\| u \|_b^r - \frac{1}{q} C_{\Gamma} C_{g} \| u \|_b^{q} \, .
\end{eqnarray*}
If $\| u \|_b = \rho$, we obtain 
\begin{equation}  \label{eq32}
J_{\lambda}(u)  \geq \frac{1}{p}\, \rho^p  \left( 1  - \frac{p \lambda}{r} C_{\Omega}
\| f_1 \|_{*} \rho^{r - p}  - \frac{p}{q}\, C_{\Gamma} C_{g} \rho^{q-p} \right)
\end{equation}
Elementary calculations show that the right hand side is maximal for 
$$
\rho_m = \left( \frac{q(p-r) \lambda \, C_{\Omega} \| f_1 \|_{*} 
}{r(q-p) \, C_g C_{\Gamma}} \right)^{1/(q-r)} \, . 
$$
Inserting this into equation (\ref{eq32}), we find that the right hand side is 
zero for 
$$
\lambda = \lambda^{*} := \left[ \frac{p}{r}\, \| f_1 \|_{*} C_{\Omega}
C_0^{\frac{r-p}{q-r}} + \frac{p}{q}\, C_g C_{\Gamma}
C_0^{\frac{q-p}{q-r}} \right]^{\frac{r-q}{q-p}} \, ,
$$
where 
$$
C_0 = \left( \frac{\| f_1 \|_{*} C_{\Omega} (p-r)q}{C_g 
C_{\Gamma}(q-p)r} \right) \, ,
$$
and strictly greater than 0 for $\lambda < \lambda^{*}$. This shows that 
for every $\lambda < \lambda^{*}$, we find $\rho_{\lambda} > 0$ such that 
$J_{\lambda} \geq c_{\lambda} > 0$ for $\| u \|_b = \rho_{\lambda}$. 
The existence of a function $u_0 \in E, \, \| u_0 \|_b > \rho_{\lambda}$ and 
$J_{\lambda}(u_0) \leq 0$ now follows as in the proof of Theorem 2 (case A4\ b).
Then the Mountain-Pass Lemma again implies the existence of a nontrivial 
solution $u_1$ with $J_{\lambda}(u_1) \geq c_{\lambda}$. 

On the other hand, for $\vi \in C_0^{\infty}(O)$ and $t > 0$ we obtain 
$$
J_{\lambda}(t \vi) \leq \frac{t^p}{p} \| \vi \|_b^p - \frac{t^{\bar{r}}}{\bar{r}}\, 
\int_{O} f_2(x) | \vi |^{\bar{r}} dx  \, . 
$$
This shows that $J_{\lambda}(t \vi) < 0$ for sufficiently small $t$ and 
consequently $J_{\lambda}$ attains its minimum in the ball 
$B_{\rho_{\lambda}} \subset E$. We claim that there is a second solution 
$u_2 \in B_{\rho_{\lambda}}$ with $J_{\lambda}(u_2) < 0$.

In addition, with the same truncation procedure  as in the proof of Theorem 2, 
we claim that there are two nonnegative solutions.
\bewende

Now we can prove the corresponding result for equation (\ref{p1}) with 
boundary condition
$$
\n \cdot  a(x) |\nabla u|^{p-2} \nabla u + b(x) |u|^{p-2}u 
 =  \lambda g(x,u) \quad  \mbox{ on } \quad \Gamma  
\eqno (2)_{\lambda}
$$
if we interchange  the roles of $g$ and 
$f$ in Assumptions B1--B3. That is, we assume now that $f$ satisfies Assumptions 
A1--A4\ a) (with $f_0 \equiv 0$) and $g$ satisfies 
\begin{description}  
\item{B4 } 
$| g(x,s)| \leq g_1(x) |s|^{q-1}$,  $1 \leq q < p$,
$g_1 \in L^{p/(p-q)}(\Gamma; w_2^{q/(q-p)})$,  
$| g(x,s)| \geq g_2(x) |s|^{\bar{q}-1}$,   $1 \leq \bar{q} \leq q$ 
and $g_2 > 0$ in some nonempty open set $ U \subset \Gamma $. 
\end{description} 
 
\begin{theo}  \label{t10}   
Let $f$ satisfy Assumptions A1--A4\ a) (with $f_0 \equiv 0$) and $g$ satisfy B4.
Then for every $0 < \lambda < \lambda^{*}$, there are at least two  nontrivial  
nonnegative solutions of (\ref{p1}), (\ref{p2})$_{\lambda}$.
\end{theo}

\paragraph{Proof.} 
First we claim as in Lemma \ref{t8} that
$$ 
N_g : L^{p}(\Gamma; w_2) \to  L^{p/(p-1)}(\Gamma; w_2^{1/(1-p)}) \; , \qquad
N_G : L^{p}(\Gamma; w_2) \to  L^{1}(\Gamma)
$$
are bounded and continuous. The estimate for $J_{\lambda}$ now reads
$$
J_{\lambda}(u) \geq  \frac{1}{p} \| u \|_b^p  - 
\frac{1}{r} C_{\Omega}  C_f \| u \|_b^{r} - 
\frac{\lambda}{q} C_{\Gamma} \| g_1 \|_{*} \| u \|_b^{q} ,   
$$
where $\| g_1 \|_{*}$ is the norm of $g_1$ in $L^{p/(p-q)}(\Gamma; w_2^{q/(q-p)}) $.
Now  $\lambda^{*}$ can be calculated as 
$$
\lambda^{*} := \left[ \frac{p}{q}\, \| g_1 \|_{*} C_{\Gamma}
\bar{C}_0^{\frac{q-p}{r-q}} + \frac{p}{r}\, C_f C_{\Omega}
\bar{C}_0^{\frac{r-p}{r-q}} \right]^{\frac{q-r}{r-p}} \, , \quad
\bar{C}_0 = \left( \frac{\| g_1 \|_{*} C_{\Gamma} (p-q)r}{C_f 
C_{\Omega}(r-p)q} \right) \, .
$$
The existence of $u_0$ with $\| u_0 \|_b > \rho_{\lambda}$ and $J(u_0) < 0 $ 
follows in the same way as in the proof of Theorem \ref{t7}, case A4\ a).
Finally, for a nonnegative $\vi \in C_{\delta}^{\infty}(\Omega)$ with 
$\mbox{supp}\,\vi \cap \Gamma \subset U$  not empty, we find 
$$
J_{\lambda}(t \vi) \leq \frac{t^p}{p} \| \vi \|_b^p + C \frac{t^r}{r} \| \vi \|_b^r  
- \frac{t^{\bar{q}}}{\bar{q}}  \int_{U} g_2(x) | \vi |^{\bar{q}} dx  \, . 
$$
Since $ \bar{q} < p \leq r $, $J_{\lambda}(t \vi) < 0$ for 
$t$ sufficiently small and we claim that $J_{\lambda}$ attains its minimum in 
$ B_{\rho_{\lambda}} \subset E$.
\bewende

We remark that, if $\Omega$ is of class $C^{1,\alpha}\, (\alpha \leq 1)$ and, 
in addition to B4, $g$ satisfies 
$$
| g(x,s) - g(y, t) | \leq C \Big( |x-y|^{\alpha} + |s-t|^{\alpha} \Big) , \qquad 
|g(x,s)| \leq C 
$$
for all $x,y \in \Gamma$, $s,t \in \R$, then the regularity result of \cite{4}, 
Thm.\ 2, shows that the solution $u$ belongs to $C^{1,\beta}(\overline{\Omega})$
for some $\beta > 0$.  

 
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\bigskip


{\sc Klaus Pfl\"uger }\\    
FB Mathematik, Freie Universit\"at Berlin\\
Arnimallee 3, 14195 Berlin, Germany \\
email: pflueger@math.fu-berlin.de


\end{document}