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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 95, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/95\hfil p-Laplacian problem with competing
nonlinearities]
{Existence results for a p-Laplacian problem with competing
nonlinearities and nonlinear boundary conditions}

\author[D. A. Kandilakis, M. Magiropoulos\hfil EJDE-2011/95\hfilneg]
{Dimitrios A. Kandilakis, Manolis Magiropoulos}  % in alphabetical order

\address{ Dimitrios A. Kandilakis \newline
Department of Sciences \\
Technical University of Crete \\
73100 Chania, Greece}
\email{dkan@science.tuc.gr}

\address{Manolis Magiropoulos \newline
Department of Sciences \\
Technological Educational Institute of Crete \\
71500 Heraklion, Greece}
\email{mageir@staff.teicrete.gr}

\thanks{Submitted July 14, 2010. Published July 28, 2011.}
\subjclass[2000]{35J60, 35J92, 35J25}
\keywords{Quasilinear elliptic problems;
 subcritical nonlinearities; \hfill\break\indent fibering method}

\begin{abstract}
 By using the fibering method we study the existence of non-negative
 solutions for a class of quasilinear elliptic problems in the
 presence of competing subcritical nonlinearities.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

 In this paper we study the problem
\begin{equation}
\begin{gathered}
\Delta_{p}u=a(x)|u|^{p-2}u-b(x)|u|^{q-2}u\quad \text{in }\Omega\\
|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}
=\lambda c(x)|u|^{p-2}u\quad\text{on }\partial\Omega,
\end{gathered}  \label{h}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$
with a sufficiently smooth boundary $\partial\Omega$, $\nu$ is the
outward unit normal vector on $\partial\Omega$, $1<q<p<N$,
$a(\cdot)$, $b(\cdot)\in L^{\infty}(\Omega)$ with $a(x)>\theta>0$,
$b(x)>0$ a.e., $c(x)\in$ $L^{\infty}(\partial\Omega)$, with
$c(x)>0$ a.e. As usual, $\Delta_{p}u=\operatorname{div}(|
\nabla u| ^{p-2}\nabla u)$ denotes the $p$-Laplacian operator.

 When $b\equiv0$, problem \eqref{h} appears naturally in the study of
the Sobolev trace inequality. Since the embedding
$W^{1,p}(\Omega)\subseteq L^{p}(\Omega)$ is compact there exists
a constant $\lambda_1$ such that
\[
\lambda_1^{1/p}\| u\| _{L^{p}(\partial\Omega)}
\leq\| u\| _{W^{1,p}(\Omega)}.
\]
The functions at which equality holds; that is,
\begin{equation}
\lambda_1:=\inf_{u\in W^{1,p}(\Omega)\backslash\{0\}}
\frac{\| u\| _{W^{1,p}(\Omega)}^{p}}{\| u\|
_{L^{p}(\partial\Omega)}^{p}}, \label{2}
\end{equation}
are called extremals and are the solutions to the problem
\begin{equation}
\begin{gathered}
\Delta_{p}u=a(x)|u|^{p-2}u\quad\text{in }\Omega\\
|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}
=\lambda_1c(x)|u|^{p-2}u\quad\text{on }\partial\Omega,
\end{gathered}  \label{1}
\end{equation}
For more details we refer the reader to \cite{Mart-Ros}.

Problems of the form
$\Delta_{p}u=\pm\lambda |u|^{p-2}u+f(x,u)$ with Dirichlet boundary
conditions has been extensively studied, see for example
\cite{Chip-chl,Garcia-Peral,Hu,Mihail,Ter}.
Recently, this problem with nonlinear boundary conditions has been
considered in \cite{pino-fl,Far-Ian,So-Wa-Zh}.

In this paper we employ Pohozaev's fibering method in order to show
that if $\lambda<\lambda_1$, then \eqref{h} admits a nonnegative
solution. In the case $\lambda=\lambda_1$, the fibering method
is no longer applicable, so we
introduce the term $\varepsilon d(\cdot)|u|^{s-2}u$ in the equation, where
$\varepsilon>0$ and $d(\cdot)\in L^{\infty}(\Omega),d(\cdot)>0$ a.e.,
and examine the behavior of the solutions $u_{\varepsilon}$ as
$\varepsilon\to0$. It turns out that
$\|u_{\varepsilon}\|_{W^{1,p}(\Omega)}\to+\infty$ and
the energy of the solutions diverges to $-\infty$.

\section{Main results}

Our reference space is $W^{1,p}(\Omega)$ equipped with the norm
$\|u\| ^{p}=\int_{\Omega}[  |\nabla u|^{p}+a(x)|u|^{p}]  dx$,
which is equivalent to its usual one. In what follows, $\sigma(\cdot)$
is the surface measure on the boundary of $\Omega$.

The energy functional associated with \eqref{h} is
\begin{equation}
\Phi_{\lambda}(u)
:=\frac{1}{p}\int_{\Omega}[  |\nabla u|^{p}
dx+a(x)|u|^{p}]  dx-\frac{1}{q}\int_{\Omega}b(x)|u|^qdx
  -\frac{\lambda}{p}\int_{\partial\Omega}c(x)|u|^{p}d\sigma(x).
\label{S}
\end{equation}
Following \cite{Mart-Ros}, let $\lambda_1\in\mathbb{R}$
be the first positive eigenvalue of \eqref{1}, given by \eqref{2}.

\begin{theorem} \label{thm1}
Suppose that $1<q<p<N$ and $\lambda<\lambda_1$.
Then \eqref{h} admits a nonnegative solution.
\end{theorem}

\begin{proof}
We  employ the fibering method introduced in \cite{Poh}, see also
\cite{Dra-Poh} and \cite{Kand-Lyb}, in order to prove the existence of a
negative energy solution of \eqref{h}. Writing $u=rv$, $r>0$ and
$v\in W^{1,p}(\Omega)$, we have
\begin{equation} \label{F(r(v)v)}
\begin{split}
\Phi_{\lambda}(rv)
&  =\frac{r^{p}}{p}\int_{\Omega}|\nabla v|^{p}
dx+\frac{r^{p}}{p}\int_{\Omega}a(x)|v|^{p}dx-\frac{r^q}{q}\int_{\Omega
}b(x)|v|^qdx \\
&\quad  -\frac{\lambda r^{p}}{p}\int_{\partial\Omega}c(x)|v|^{p}d\sigma
(x).
\end{split}
\end{equation}
For $u\neq0$ to be a critical point, it should hold $\frac{\partial
\Phi_{\lambda}(rv)}{\partial r}=0$, from which we obtain
\begin{equation}
\begin{split}
& r^{p-q}\int_{\Omega}|\nabla v|^{p}dx+r^{p-q}\int_{\Omega}a(x)|v|^{p}
dx
 -\lambda r^{p-q}\int_{\partial\Omega}c(x)|v|^{p}d\sigma(x)\\
&=\int_{\Omega }b(x)|v|^qdx,
\end{split} \label{3}
\end{equation}
ensuring the existence of a unique $r=r(v)>0$ satisfying \eqref{3}.
By the implicit function theorem \cite[Thm. 4.B]{Zeid}, the function
$v\to r(v)$ is continuously differentiable for $v\neq 0$. Notice that
\begin{equation}
r(kv)kv=r(v)v\quad \text{for }k>0.\label{l}
\end{equation}
In view of \eqref{F(r(v)v)} and \eqref{3},
\[
\Phi_{\lambda}(r(v)v)=\big(  \frac{1}{p}-\frac{1}{q}\big)
r(v)^q \int_{\Omega}b(x)|v|^qdx<0.
\]
Consider the functional
\[
H(u):=\int_{\Omega}|\nabla u|^{p}dx+\int_{\Omega}a(x)|u|^{p}dx-\lambda
\int_{\partial\Omega}c(x)|u|^{p}d\sigma(x).
\]
By the way we chose $\lambda$, for $u\in W^{1,p}(\Omega)$,
$H(u)\geq0$ (equality holds exactly when $u=0$).
Define $V=\{  v\in W^{1,p} (\Omega):H(v)=1\}$.
Evidently, $(  H'(v),v)  \neq0$
for $v\in V$. In view of \cite[Lemma 3.4]{Dra-Poh},
 any conditional critical
point of $\widehat{\Phi}_{\lambda}(v)=\Phi_{\lambda}(r(v)v)$
subject to $H(v)=1$, provides a critical point $r(v)v$ of
$\Phi_{\lambda}$. Notice that $V$ is bounded. To see this,
let $\varepsilon>0$ be such that $\lambda +\varepsilon<\lambda_1$.
Then, for $v\in V$, by the definition of $\lambda_1$,
\[
\lambda+\varepsilon<\frac{{\int_{\Omega}}|\nabla v|^{p}+
{\int_{\Omega}}
a|v|^{p}}{{\int_{\partial\Omega}}
c(x)|v|^{p}d\sigma(x)},
\]
which implies that
\[
1={\int_{\Omega}}|\nabla v|^{p}+
{\int_{\Omega}} a|v|^{p}-\lambda
{\int_{\partial\Omega}}
c(x)|v|^{p}d\sigma(x)>\varepsilon
{\int_{\partial\Omega}}
c(x)|v|^{p}d\sigma(x).
\]
Thus ${\int_{\partial\Omega}}c(x)|v|^{p}d\sigma(x)$, $v\in V$,
is bounded. Consequently, $V$ is a bounded
set. Because of the embedding
$W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$,
\eqref{3} guarantees that $r(V)$ is bounded. Consequently,
$I=\{
\Phi_{\lambda,\mu}(r(v)v):v\in V\}  $ is a bounded interval in
$\mathbb{R}$ with endpoints $a$ and $b$, $a<b\leq0$.
We are now going to show that $a\in
I$. To this end, let $\{  v_{n}\}  _{n\in\mathbf{N}}$ be a sequence
in $V$, with $\Phi_{\lambda}(r(v_{n})v_{n})\to a$. Without loss of
generality, we may assume that $v_{n}\to v$ weakly in $W^{1,p}
(\Omega)$.
 We may also assume that $r(v_{n})\to r\in\mathbb{R}$.
Thus $r(v_{n})v_{n}\to rv$ weakly in $W^{1,p}(\Omega)$. Since
$\Phi_{\lambda}(\cdot)$ is weakly lower semicontinuous,
\begin{equation}
\Phi_{\lambda}(rv)\leq\liminf_{n\to\infty}
\Phi_{\lambda}(r(v_{n})v_{n})=a,\label{g}
\end{equation}
ensuring that $rv\neq0$. Because of the compactness of the Sobolev
and trace embeddings, $r(v_{n})v_{n}\to rv$ strongly in $L^q(\Omega)$,
$L^{p}(\partial\Omega)$, respectively. Taking into account the lower
semicontinuity of the norm in \eqref{3}, we have
\begin{equation}
r^{p-q}H(v)\leq\int_{\Omega}b(x)|v|^qdx.\label{5}
\end{equation}
Combining \eqref{3} and \eqref{5}, we get $r\leq r(v)$.
Our purpose is to prove equality. Let us assume the contrary;
that is $r<r(v)$. We define
$F(y)=\Phi_{\lambda}(yv)$, $y\geq0$. For $y\in[ r$, $r(v)]$, we have
\begin{equation}
F'(y)=y^{q-1}\Big(  y^{p-q}H(v)-\int_{\Omega}b(x)|v|^qdx\Big)
,\label{6}
\end{equation}
which is negative everywhere, but at $y=$\ $r(v)$.
Thus $F(y)$ decreases strictly in the considered interval, giving
\begin{equation}
\Phi_{\lambda}(r(v)v)<\Phi_{\lambda}(rv)\leq a,\label{7}
\end{equation}
because of \eqref{g}. Notice that for suitable $k\geq1$, $kv\in V$.
Then, combining \eqref{l} and \eqref{7}, we obtain
\[
\Phi_{\lambda}(r(kv)kv)=\Phi_{\lambda}(r(v)v)<\Phi_{\lambda}(rv)
\leq a,
\]
which is a contradiction. So, $r=r(v)$, and necessarily
$\Phi_{\lambda}(r(kv)kv)=a$. This means that $kv$ is a conditional
critical point of $\widehat{\Phi}_{\lambda}(\cdot)$ subject
to $H(v)=1$, and, consequently,
$r(kv)kv=r(v)v$ is a critical point of
$\Phi_{\lambda}(\cdot)$. Since for a
minimizer $w$, $|w|$ is also a minimizer, we may assume
$v\geq0$, and $r(v)v$ is a nontrivial nonnegative solution of \eqref{h}.
\end{proof}

In attempting to obtain the existence of a solution to problem
\eqref{h} for $\lambda=$ $\lambda_1$, following a similar procedure,
we encounter an unsurpassable difficulty, due to the fact that \eqref{3} does no longer
guarantee the existence of a suitable $r(v)$. In order to study this
situation, we add an additional term in \eqref{h}, with the problem taking the
following form
\begin{equation}
\begin{gathered}
\Delta_{p}u=a(x)|u|^{p-2}u-b(x)|u|^{q-2}u
+\varepsilon d(x)|u|^{s-2}u\quad\text{in }\Omega, \\
|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}
=\lambda_1c(x)|u|^{p-2}u\quad\text{on }\partial\Omega,
\end{gathered}\label{8}
\end{equation}
where, $\varepsilon>0$, $q<s<p^{\ast}$, and
$d(\cdot)\in L^{\infty}(\Omega)$ with
$d(\cdot)>0$ a.e. in $\Omega$.
The energy functional is
\begin{equation}
F_{\lambda_1,\varepsilon}(u):=\Phi_{\lambda_1}(u)+\frac{\varepsilon}
{s}D(u), \label{9}
\end{equation}
where
\[
D(u):=\ \int_{\Omega}d(x)|u|^{s}dx.
\]


\begin{theorem} \label{thm2}
Suppose that $1<q<s<p^{\ast}$, $\varepsilon>0$ and
$d(\cdot)\in L^{\infty}
(\Omega)$ with $d(\cdot)>0$ a.e. in $\Omega.$Then problem \eqref{8} admits a
nonnegative solution $u_{\varepsilon}$ for every $\varepsilon>0$. Furthermore,
$F_{\lambda_1,\varepsilon}(u_{\varepsilon})\to-\infty$ and
$\|u_{\varepsilon}\|\to+\infty\ $as $\varepsilon\to0$.
\end{theorem}

\begin{proof}
Following a similar reasoning, we obtain the counterpart of \eqref{3},
\begin{equation}
\begin{split}
&r^{p-q}\big[  \int_{\Omega}|\nabla v|^{p}dx+\int_{\Omega}a(x)|v|^{p}
dx-\lambda_1\int_{\partial\Omega}c(x)|v|^{p}d\sigma(x)\big]\\
&+\varepsilon r^{s-q}\int_{\Omega}d(x)|v|^{s}dx\\
&=\int_{\Omega}b(x)|v|^q dx.
\end{split}\label{10}
\end{equation}
The function $R(y)=Hy^{p-q}+\varepsilon Dy^{s-q}-B$, with
$H\geq0$, $D$,
$B>0$, has a unique root in $(0,+\infty)$, since it is
strictly increasing, $R(0)=-B$ and $R(y)\to+\infty$, for
$y\to+\infty$. Thus, for $v\in W^{1,p}(\Omega)$ there exists a unique
positive $r_{\varepsilon}(v)$ satisfying \eqref{10}.
The so defined function
$v\to r_{\varepsilon}(v)$ is once more continuously differentiable for
$v\neq0$, by another application of the implicit function theorem. In
addition, it is easily checked that \eqref{l} remains true.
 We notice also that, due to \eqref{10}, if $v\neq0$,
\begin{equation}
F_{\lambda_1,\varepsilon}(r_{\varepsilon}(v)v)
=\big(  \frac{1}{p}-\frac{1}{q}\big)
r_{\varepsilon}(v)^{p}H(v)+\varepsilon\big(  \frac{1}{s}
-\frac{1}{q}\big)  r_{\varepsilon}(v)^{s}D(v)<0. \label{Fneg}
\end{equation}
 We define next the positive functional (except at $u=0$),
\begin{equation}
L(u):=H(u)+D(u). \label{L}
\end{equation}
Consider the set
\[
W=\{  v\in W^{1,p}(\Omega):L(v)=1\}  .
\]
Because of our hypothesis on $d(\cdot)$, $( L'(v),v)  >D(v)>0$
for $v\in W$. As usual, the conditional critical points of
$\widehat {F}_{\lambda_1,\varepsilon}(v)
=F_{\lambda_1,\varepsilon}(r_{\varepsilon }(v)v)$ subject to
$L(v)=1$ provide critical points $r_{\varepsilon}(v)v$ of
$F_{\lambda_1}$. We claim that $W$ is bounded. Indeed, if not,
there would exist $v_{n}\in W,n\in \mathbb{N}$, such that
$\|v_{n}\|\to+\infty$. Let $v_{n}:=t_{n}u_{n}$ with
$t_{n}>0$ and $\|u_{n}\|=1$. Since $u_{n}$, $n\in \mathbb{N}$,
is bounded, by passing to a subsequence if necessary, we may
assume that $u_{n}\to u_0$ weakly in $W^{1,p}(\Omega)$ and
strongly in $L^{p}(c,\partial\Omega)$ and $L^{s}(\Omega)$.
By \eqref{L},
\begin{align*}
&t_{n}^{p}\big[  \int_{\Omega}|\nabla u_{n}|^{p}dx+\int_{\Omega}
a(x)|u_{n}|^{p}dx\\
&-\lambda_1\int_{\partial\Omega}c(x)|u_{n}|^{p} d\sigma(x)\big]
+t_{n}^{s}\int_{\Omega}d(x)|u_{n}|^{s}dx=1,
\end{align*}
and so
\begin{equation}
0\leq\int_{\Omega}|\nabla u_{n}|^{p}dx+\int_{\Omega}a(x)|u_{n}|^{p}
dx-\lambda_1\int_{\partial\Omega}c(x)|u_{n}|^{p}d\sigma(x)\leq\frac{1}
{t_{n}^{p}}\to0 \label{lim1}
\end{equation}
and
\begin{equation}
0<\int_{\Omega}d(x)|u_{n}|^{s}dx\leq\frac{1}{t_{n}^{s}}\to0.
\label{lim2}
\end{equation}
By \eqref{lim2}, $u_0=0$. On the other hand, since $\|u_{n}\|=1$,
\eqref{lim1} yields
\[
\lambda_1\int_{\partial\Omega}c(x)|u_0|^{p}d\sigma(x)=1
\]
and so $u_0\neq0$, a contradiction, thereby proving the claim.
We can now continue as in the previous case. Namely, we notice
that by the way it was defined, $r_{\varepsilon}(v)$ is bounded on
$W$ (we use now the embedding
$W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$).
Thus $I'=\{F_{\lambda_1,\varepsilon}(r_{\varepsilon}(v)v):v\in W\}$
is a bounded interval with endpoints $a'$ and $b'$,
$a'<b'\leq0$. Let $\{  v_{n}\}_{n\in\mathbb{N}}$ be a sequence of
$W$ with
$F_{\lambda_1,\varepsilon}(r_{\varepsilon }(v_{n})v_{n})\to a'$.
We may assume that $v_{n}\to v_{\varepsilon}$ weakly in
$W^{1,p}(\Omega)$, and $r_{\varepsilon} (v_{n})\to r_{\varepsilon}\in
\mathbb{R}$. Thus
$r_{\varepsilon}(v_{n})v_{n}\to r_{\varepsilon}v_{\varepsilon}$
weakly in $W^{1,p}(\Omega)$ , and consequently, at least for a
subsequence,
strongly in $L^{s}(\Omega)$.
Since $\Phi_{\lambda_1}(\cdot)$ is weakly lower semicontinuous,
so is $F_{\lambda_1,\varepsilon}(\cdot)$, and the obvious
counterpart of \eqref{g} ensures that
$r_{\varepsilon}v_{\varepsilon}\neq 0$.
Combining \eqref{10} with the lower semicontinuity of the involved
norms, the compactness of the Sobolev and trace embeddings
$W^{1,p}(\Omega )\hookrightarrow L^q(\Omega)$,
$W^{1,p}(\Omega)\hookrightarrow L^{s} (\Omega)$, and
$W^{1,p}(\Omega)\hookrightarrow$ $L^{p}(\partial\Omega)$,
respectively, we obtain
\begin{equation}
r_{\varepsilon}^{p-q}H(v_{\varepsilon})
+r_{\varepsilon}^{s-q}\varepsilon
D(v_{\varepsilon})\leq\int_{\Omega}b(x)|v_{\varepsilon}|^q
dx=B(v_{\varepsilon}). \label{11}
\end{equation}
Evidently, \eqref{10} and \eqref{11} ensure that
$r_{\varepsilon}\leq r_{\varepsilon}(v_{\varepsilon})$.
We are going to prove equality. Assuming
the contrary, the function
$G(y)=F_{\lambda_1,\varepsilon} (yv_{\varepsilon})$, $y>0$,
has its derivative
\[
G'(y)=y^{q-1}\big(  y^{p-q}H(v_{\varepsilon})+y^{s-q}\varepsilon
D(v_{\varepsilon})-B(v_{\varepsilon})\big)
\]
which is negative in $[r_{\varepsilon}$,
$r_{\varepsilon}(v_{\varepsilon})]$
except at $y=r_{\varepsilon}(v_{\varepsilon})$, where it is zero.
Thus $G(y)$ decreases strictly in the above interval, meaning
\begin{equation}
F_{\lambda_1,\varepsilon}(r_{\varepsilon}(v_{\varepsilon})
v_{\varepsilon})
<F_{\lambda_1,\varepsilon}(r_{\varepsilon}v_{\varepsilon})
\leq a', \label{14}
\end{equation}
since $F_{\lambda_1,\varepsilon}(r_{\varepsilon}v_{\varepsilon}
)\leq\liminf_{n\to+\infty} F_{\lambda_1,\varepsilon
}(r_{\varepsilon}(v_{n})v_{n})=a'$.
Next we choose a positive $k$,
such that $kv_{\varepsilon}\in W$. Since \eqref{l} holds,
we arrive at an obvious contradiction. Thus
 $r_{\varepsilon}=r_{\varepsilon}(v_{\varepsilon})$, and
$F_{\lambda_1,\varepsilon}(r_{\varepsilon}(kv_{\varepsilon})
kv_{\varepsilon})=a'$, thus obtaining a conditional critical point
of $\widehat{F}_{\lambda_1,\varepsilon}(\cdot)$ subject to
$L(v)=1$, and, consequently,
$u_{\varepsilon}:=r_{\varepsilon}(v_{\varepsilon})v_{\varepsilon}$
is a critical point\ of $F_{\lambda_1,\varepsilon}(\cdot)$.
Once more, we may assume $v_{\varepsilon}\geq0$, and so
$u_{\varepsilon}$ is a nontrivial nonnegative solution of \eqref{8}.

Next we study the behavior of the solutions
$u_{\varepsilon}=r_{\varepsilon }(v_{\varepsilon})v_{\varepsilon}$
as $\varepsilon\to0$. Let
$\varphi_1>0$ be the eigenfunction of \eqref{1} corresponding to
$\lambda_1$, with $L(\varphi_1)=1$. By \eqref{10},
\begin{equation}
r_{\varepsilon}(\varphi_1)^{s-q}=\frac{\int_{\Omega}b(x)|\varphi_1|^q
dx}{\varepsilon\int_{\Omega}d(x)|\varphi_1|^{s}dx}, \label{re}
\end{equation}
which implies that $r_{\varepsilon}(\varphi_1)\to+\infty$ as
$\varepsilon\to0$. In view of \eqref{Fneg} and \eqref{10}
\[
F_{\lambda_1,\varepsilon}(r_{\varepsilon}(\varphi_1)\varphi_1
)=\varepsilon\big(  \frac{1}{s}-\frac{1}{q}\big)  r_{\varepsilon}
(\varphi_1)^{s}D(\varphi_1)
=\big(  \frac{1}{s}-\frac{1}{q}\big)
r_{\varepsilon}(\varphi_1)^q\int_{\Omega}b(x)|\varphi_1|^qdx.
\]
Since $F_{\lambda_1,\varepsilon}(r_{\varepsilon}(v_{\varepsilon
})v_{\varepsilon})\leq F_{\lambda_1,\varepsilon}
(r_{\varepsilon}(\varphi _1)\varphi_1)$, we conclude that
$$
F_{\lambda_1,\varepsilon}(u_{\varepsilon})
=F_{\lambda_1,\varepsilon}(r_{\varepsilon}(v_{\varepsilon
})v_{\varepsilon})\to-\infty
$$
as $\varepsilon\to 0$.
 By \eqref{Fneg} we also get that $r_{\varepsilon}(v_{\varepsilon})\to
+\infty$ as $\varepsilon\to0$. Let $\widehat{v}$ be a weak
accumulation point of $v_{\varepsilon}$; that is,
 $\widehat{v}=w-\lim_{n\to+\infty} v_{\varepsilon_{n}}$ where
$\varepsilon_{n} \to 0$ as $n\to+\infty$.
Since $L(v_{\varepsilon_{n}})=1$,
necessarily
\[
0\leq\int_{\Omega}|\nabla v_{\varepsilon_{n}}|^{p}dx+\int_{\Omega
}a(x)|v_{\varepsilon_{n}}|^{p}dx-\lambda_1\int_{\partial\Omega
}c(x)|v_{\varepsilon_{n}}|^{p}d\sigma(x)\to0.
\]
Consequently, either $\widehat{v}=0$ or
$\widehat{v}=\gamma\varphi_1$ for
some $\gamma\neq0$. We cannot have $\widehat{v}=0$, because then, since
$v_{\varepsilon_{n}}\in W$, we would get that
$\int_{\Omega}d(x)|\widehat {v}|^{s}dx
=\lim\int_{\Omega}d(x)|v_{\varepsilon_{n}}|^{s}dx=1$. Therefore,
$\widehat{v}=\gamma\varphi_1$ and so
$\|u_{\varepsilon_{n}} \|
=r_{\varepsilon_{n}}(v_{\varepsilon_{n}})\|v_{\varepsilon_{n}}
\|\to+\infty$ as $n\to+\infty$.
\end{proof}

\begin{thebibliography}{00}


\bibitem{Chip-chl} M. Chipot, M. Chlebik, M. Fila, and I. Shafrir;
\emph{Existence of positive solutions of a semilinear elliptic
equation in $\mathbb{R}_{+}^{n}$ with a nonlinear boundary condition},
J. Math. Anal. Appl. 223, no. 2 (1998) 429--471.

\bibitem{Dra-Poh} P. Dr\'{a}bek and S. I. Pohozaev;
\emph{Positive solutions for the p-Laplacian: application of
the fibering method}, Proc. Roy. Soc. Edinburgh
Sect. A \textbf{127} (1997) 703-726.

\bibitem{pino-fl} M. del Pino and C. Flores;
\emph{Asymptotic behavior of best constants and extremals for
trace embeddings in expanding domains}, Commun. in
Partial Diff. Equations, 26(11\&12) (2001) 2189-2210.

\bibitem{Far-Ian} F. Faraci, A. Iannizzotto and C. Varga;
\emph{Infinitely many bounded solutions for the p-Laplacian
with nonlinear boundary conditions},
Monatsh. Math. (to appear).

\bibitem{FenBon} J. Fernandez Bonder;
\emph{Multiple solutions for the p-Laplace
equation with nonlinear boundary conditions},
Electron. J. Differential Equations 2006 no. 37 (2006) 1-7.

\bibitem{Garcia-Peral} J. P. Garcia Azorero and I. Peral Alonso;
\emph{Existence and nonuniqueness for the p-Laplacian:
Nonlinear eigenvalues}, Commun. in Partial
Diff. Equations, 12(12) (1987) 1389-1430.

\bibitem{Hu} B. Hu;
\emph{Nonexistence of a positive solution of the Laplace
equation with a nonlinear boundary condition},
 Differential Integral Equations 7, no. 2 (1994) 301--313.

\bibitem{Kand-Lyb} D. A. Kandilakis and A. N. Lyberopoulos;
\emph{Indefinite quasilinear elliptic problems with subcritical
and supercritical nonlinearities on unbounded domains},
J. Differential Equations 230 (2006) 337--361.

\bibitem{Mart-Ros} S. Martinez and J. D. Rossi;
\emph{Isolation and simplicity for the first eigenvalue of
the p-Laplacian with a nonlinear boundary condition},
Abstr. Appl. Anal. 7, no. 5 (2002) 287-293.

\bibitem{Mihail} M. Mihailescu;
\emph{Existence and multiplicity of weak solutions
for a class of degenerate nonlinear elliptic equations}, Boundary Value
Problems, Article ID 41295 (2006) 1-17.

\bibitem{Poh} S. I. Pohozaev;
\emph{The fibering method in nonlinear variational
problems}, in \emph{Topological and Variational Methods for Nonlinear
Boundary Value Problems}, Pitman Res. Notes Math. Ser.,
vol. 365, Longman, Harlow, 1997, pp. 35-88.

\bibitem{So-Wa-Zh} X. Song, W. Wang and P. Zhao;
\emph{Positive solutions of elliptic equations with nonlinear
boundary conditions}, Nonlinear Analysis 70 (2009) 328-334.

\bibitem{Ter} S. Terraccini;
\emph{Symmetry properties of positive solutions to some
elliptic equations with nonlinear boundary conditions},
Differential Integral Equations 8, no. 8 (1995) 1911--1922.

\bibitem{Zeid} E. Zeidler;
\emph{Nonlinear Functional Analysis and its
Applications, Vol. I.} (New York: Springer-Verlag, 1986).

\end{thebibliography}

\end{document}