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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 131, pp. 1--15.\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-2006/131\hfil A semilinear elliptic problem]
{A semilinear elliptic problem involving nonlinear boundary condition
and sign-changing potential}

\author[T.-F. Wu\hfil EJDE-2006/131\hfilneg]
{Tsung-Fang Wu}

\address{Tsung-Fang Wu \newline
Department of Applied Mathematics,
 National University of Kaohsiung,
Kaohsiung 811, Taiwan}
\email{tfwu@nuk.edu.tw}

\date{}
\thanks{Submitted July 6, 2006. Published October 17, 2006}
\thanks{Partially supported by the National Science Council of Taiwan (R.O.C.)}
\subjclass[2000]{35J65, 35J50, 35J55}
\keywords{Semilinear elliptic equations; Nehari manifold; \hfill\break\indent
Nonlinear boundary condition}

\begin{abstract}
 In this paper, we study the multiplicity of nontrivial nonnegative
 solutions for a semilinear elliptic equation involving nonlinear
 boundary condition and sign-changing potential. With the help of
 the Nehari manifold, we prove that the semilinear elliptic
 equation:
 \begin{gather*}
 -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\
 \frac{\partial u}{\partial \nu }=g(x)|u|
 ^{p-2}u \quad \text{on }\partial \Omega ,
 \end{gather*}
has at least two nontrivial nonnegative solutions for $\lambda $
is sufficiently small.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

In this paper, we consider the multiplicity of nontrivial nonnegative
solutions for the following semilinear elliptic equation
\begin{equation}
\begin{gathered}
-\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\
\frac{\partial u}{\partial \nu }=g(x)|u| ^{p-2}u \quad \text{on
}\partial \Omega ,
\end{gathered}  \label{Efg}
\end{equation}
where $1<q<2<p<\frac{2(N-1)}{N-2}$, $\lambda >0$, $\Omega $ is
a bounded domain in $\mathbb{R}^{N}$ with smooth boundary,
$\frac{\partial}{\partial \nu }$ is the outer normal derivative and
$f,g:\overline{\Omega }\to \mathbb{R}$ are continuous functions
which change sign in $\overline{\Omega }$. Associated with \eqref{Efg}, we
consider the energy functional $J_{\lambda }$ in $H^{1}(\Omega )$,
\[
J_{\lambda }(u)=\frac{1}{2}\| u\|_{H^{1}}^{2}- \frac{\lambda
}{q}\int_{\Omega }f|u|^{q}dx-\frac{1}{p} \int_{\partial \Omega
}g|u|^{p}ds.
\]
where $ds$ is the measure on the boundary and $\|
u\|_{H^{1}}^{2}=\int_{\Omega }|\nabla u|^{2}+u^{2}dx$. It is well
known that $J_{\lambda }$ is of $C^{1}$ in $H^{1}(\Omega)$ and the
solutions of equation \eqref{Efg} are the critical points of the
energy functional $J_{\lambda }$.

The fact that the number of solutions of equation \eqref{Efg}
is affected by the nonlinear boundary conditions has been the focus of a
great deal of research in recent years.
Garcia-Azorero, Peral and Rossi \cite{GPR} have investigated
\eqref{Efg} when $f\equiv g\equiv 1$.
They found that there exist positive numbers
$\Lambda _{1},\Lambda _{2}$ with $\Lambda _{1}\leq \Lambda _{2}$ such that
equation \eqref{Efg} admits at least two positive solutions for
$\lambda \in (0,\Lambda _{1})$ and no positive solution exists
for $\lambda >\Lambda _{2}$.
Also see Chipot-Chlebik-Fila-Shafrir \cite{CCF},
Chipot-Shafrir-Fila \cite{CSF}, Flores-del Pino \cite{FDP}, Hu \cite{H},
Pierrotti-Terracini \cite{PT} and Terraccini \cite{Te} where problems
similar to equation \eqref{Efg} have been studied.

The purpose of this paper is to consider the multiplicity of nontrivial
nonnegative solutions of equation \eqref{Efg} with
sign-changing potential. We prove that equation \eqref{Efg} has
at least two nontrivial nonnegative solutions for $\lambda $ is sufficiently
small.

\begin{theorem}\label{t1}
There exists $\lambda _{0}>0$ such that for $\lambda \in (0,\lambda _{0})$,
equation \eqref{Efg} has at least two
nontrivial nonnegative solutions.
\end{theorem}

Among the other interesting problems which are similar of equation
\eqref{Efg}, Ambro\-setti-Brezis-Cerami \cite{ABC} have
investigated the  equation
\begin{equation}
\begin{gathered}
-\Delta u=\lambda |u|^{q-2}u+|u| ^{p-2}u \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered} \label{Elambda}
\end{equation}
where $1<q<2<p\leq \frac{2N}{N-2}$. They proved that there exists
$\lambda_{0}>0$ such that \eqref{Elambda} admits at least two
positive solutions for $\lambda \in (0,\lambda _{0})$, has a
positive solution for $\lambda =\lambda _{0}$, and no positive
solution for $\lambda >\lambda _{0}$. Actually,
Adimurthi-Pacella-Yadava \cite{APY}, Damascelli-Grossi-Pacella
\cite{DGP}, Ouyang-Shi \cite{OS} and Tang \cite{Ta} proved that
there exists $\lambda _{0}>0$ such that equation \eqref{Elambda}
in the unit ball $B^{N}(0;1)$ has exactly two positive solutions
for $\lambda \in (0,\lambda _{0})$, has exactly one positive
solution for $\lambda =\lambda _{0}$ and no positive solution
exists for $\lambda >\lambda _{0}$. Generalizations of the result
of equation \eqref{Elambda} were done by Ambrosetti-Azorero-Peral
\cite{AAP}, de Figueiredo-Gossez-Ubilla \cite{FGU} and Wu
\cite{Wu}.

This paper is organized as follows. In section 2, we give some notation and
preliminaries. In section 3, we prove that \eqref{Efg}
has at least two nontrivial nonnegative solutions for $\lambda $ is
sufficiently small.

\section{Notation and Preliminaries}

Throughout this section, we denote by $S_{p}$, $C_{p}$ the best Sobolev
embedding and trace constant for the operators
$H^{1}(\Omega ) \hookrightarrow L^{p}(\Omega )$,
$H^{1}(\Omega ) \hookrightarrow L^{p}(\partial \Omega )$,
 respectively. Now, we
consider the Nehari minimization problem: For $\lambda >0$,
\[
\alpha _{\lambda }=\inf \{J_{\lambda }(u) : u\in
\mathbf{M}_{\lambda }\},
\]
where $\mathbf{M}_{\lambda }=\{u\in H^{1}(\Omega )
\backslash \{0\}: \langle J_{\lambda }'(
u),u\rangle =0\}$. Define
\[
\psi _{\lambda }(u)=\langle J_{\lambda }'( u),u\rangle =\|
u\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u|^{q}dx-\int_{\partial
\Omega }g|u|^{p}ds.
\]
Then for $u\in \mathbf{M}_{\lambda }$,
\[
\langle \psi _{\lambda }'(u),u\rangle =2\| u\|_{H^{1}}^{2}-\lambda
q\int_{\Omega }f| u|^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds.
\]
Similarly to the method used in Tarantello \cite{T}, we split
$\mathbf{M} _{\lambda }$ into three parts:
\begin{gather*}
\mathbf{M}_{\lambda }^{+} =\{u\in \mathbf{M}_{\lambda }:
\langle \psi _{\lambda }'(u)
,u\rangle >0\}, \\
\mathbf{M}_{\lambda }^{0} =\{u\in \mathbf{M}_{\lambda }:
\langle \psi _{\lambda }'(u)
,u\rangle =0\}, \\
\mathbf{M}_{\lambda }^{-} =\{u\in \mathbf{M}_{\lambda }:
\langle \psi _{\lambda }'(u) ,u\rangle <0\}.
\end{gather*}
Then, we have the following results.

\begin{lemma}\label{g3}
There exists $\lambda _{1}>0$ such that for each
$\lambda \in (0,\lambda _{1})$ we have $\mathbf{M}_{\lambda }^{0}=\phi $.
\end{lemma}

\begin{proof}
We consider the following two cases.

\noindent Case (I): $u\in \mathbf{M}_{\lambda }$ and
$\int_{\partial \Omega }g|u|^{p}ds\leq 0$. We have
\[
\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial
\Omega }g|u|^{p}ds.
\]
Thus,
\begin{align*}
\langle \psi _{\lambda }'(u),u\rangle &=2\|
u\|_{H^{1}}^{2}-\lambda q\int_{\Omega }f|
u|^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds \\
&=(2-q)\| u\|_{H^{1}}^{2}+(q-p) \int_{\partial \Omega
}g|u|^{p}ds>0
\end{align*}
and so $u\in \mathbf{M}_{\lambda }^{+}$.

\noindent Case (II): $u\in \mathbf{M}_{\lambda }$ and
$\int_{\partial \Omega }g|u|^{p}ds>0$. Suppose that
$\mathbf{M}_{\lambda }^{0}\neq \phi $ for all $\lambda >0$. If
$u\in \mathbf{M}_{\lambda }^{0}$, then we have
\begin{align*}
0 &=\langle \psi _{\lambda }'(u),u\rangle =2\|
u\|_{H^{1}}^{2}-\lambda q\int_{\Omega }f|u|^{q}dx-p\int_{\partial
\Omega }g|u|^{p}ds
\\
&=(2-q)\| u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|u|^{p}ds.
\end{align*}
Thus,
\begin{equation}
\| u\|_{H^{1}}^{2}=\frac{p-q}{2-q}\int_{\partial \Omega
}g|u|^{p}ds  \label{1}
\end{equation}
and
\begin{equation}
\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial
\Omega }g|u| ^{p}ds=\frac{p-2}{2-q}\int_{\partial \Omega
}g|u|^{p}ds. \label{2}
\end{equation}
Moreover,
\begin{align*}
(\frac{p-2}{p-q})\| u\|_{H^{1}}^{2} &= \|
u\|_{H^{1}}^{2}-\int_{\partial \Omega }g|u|^{p}ds\\
&=\lambda \int_{\Omega }f|u|^{q}dx \\
&\leq \lambda \| f\|_{L^{p^{\ast }}}\|
u\|_{L^{p}}^{q} \\
&\leq \lambda \| f\|_{L^{p^{\ast }}}S_{p}^{q}\|
u\|_{H^{1}}^{q},
\end{align*}
where $p^{\ast }=\frac{p}{p-q}$. This implies
\begin{equation}
\| u\|_{H^{1}}\leq \Big[ \lambda (\frac{p-q}{p-2}
)\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{1/(2-q)}.  \label{4}
\end{equation}
Let $I_{\lambda }:\mathbf{M}_{\lambda }\to \mathbb{R}$ be given by
\[
I_{\lambda }(u)=K(p,q)\Big(\frac{\|
u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega
}g|u|^{p}ds}\Big)^{1/(p-2)}-\lambda \int_{\Omega }f|u|^{q}dx,
\]
where $K(p,q)=(\frac{2-q}{p-q})^{(p-1)/(p-2)}
(\frac{p-2}{2-q})$. Then $I_{\lambda }(u)=0$ for
all $u\in \mathbf{M}_{\lambda }^{0}$. Indeed, from $(\ref{1})$
and $(\ref{2})$ it follows that for $u\in \mathbf{M}_{\lambda
}^{0}$ we have
\begin{equation}
\begin{aligned}
I_{\lambda }(u)
&= K(p,q)\Big(\frac{\| u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega
  }g|u|^{p}ds}\Big)^{1/(p-1)}-\lambda
\int_{\Omega }f|u|^{q}dx   \\
&= \big(\frac{2-q}{p-q}\big)^{\frac{p}{p-1}}(\frac{p-2}{2-q})
\Big(\frac{(\frac{p-q}{2-q})^{p-1} \big(\int_{\partial \Omega
}g|u|^{p}ds\big)^{p-1}}{
\int_{\partial \Omega }g|u|^{p}ds}\Big)^{\frac{1}{p-2}}\\
&\quad -\frac{p-2}{2-q}\int_{\partial \Omega }g|u|^{p}ds = 0.
\end{aligned} \label{5}
\end{equation}
However, by \eqref{4}, the H\"{o}lder and Sobolev trace
inequality, for $u\in \mathbf{M}_{\lambda }^{0}$
\begin{align*}
I_{\lambda }(u) &\geq K(p,q)\Big(\frac{ \|
u\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega
}g|u|^{p}ds}\Big)^{1/(p-2)}-\lambda
S_{p}^{q}\| f\|_{L^{p^{\ast }}}\| u\|_{H^{1}}^{q} \\
&\geq \| u\|_{H^{1}}^{q}\Big(K(p,q)\Big(
\frac{\| u\|_{H^{1}}^{2(p-1)}}{
C_{p}^{p}\| g\|_{\infty }\| u\|
_{H^{1}}^{p+q(p-2)}}\Big)^{1/(p-2)}-\lambda
S_{p}^{q}\| f\|_{L^{p^{\ast }}}\Big)\\
&\geq \| u\|_{H^{1}}^{q}\big\{K(p,q)C_{p}^{
\frac{p}{2-p}}\lambda ^{\frac{1-q}{2-q}}\Big[ \big(\frac{p-q}{p-2}\big)
\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{\frac{1-q}{2-q}}
 -\lambda S_{p}^{q}\| f\|_{L^{p^{\ast }}}\big\}.
\end{align*}
This implies that for $\lambda $ sufficiently small we have
$I_{\lambda}(u)>0$ for all $u\in \mathbf{M}_{\lambda }^{0}$, this
contradicts \eqref{5}. Thus, we can conclude that there
exists $\lambda _{1}>0$ such that for $\lambda \in (0,\lambda _{1})$, we
have $\mathbf{M}_{\lambda }^{0}=\phi $.
\end{proof}

By Lemma \ref{g3}, for $\lambda \in (0,\lambda _{1})$ we write
$\mathbf{M}_{\lambda }=\mathbf{M}_{\lambda }^{+}\cup \mathbf{M}_{\lambda }^{-}$
and define
\[
\alpha _{\lambda }^{+}=\inf_{u\in \mathbf{M}_{\lambda }^{+}}J_{\lambda
}(u);\quad\text{}\alpha _{\lambda }^{-}(\Omega )
=\inf_{u\in \mathbf{M}_{\lambda }^{-}}J_{\lambda }(u).
\]
The following lemma shows that the minimizers on $\mathbf{M}_{\lambda }$ are
``usually'' critical points for $J_{\lambda }$.

\begin{lemma} \label{g2}
For $\lambda \in (0,\lambda _{1})$. If $u_{0}$ is a local
minimizer for $J_{\lambda }$ on $\mathbf{M}_{\lambda }$, then $J_{\lambda
}'(u_{0})=0$ in $H^{\ast }(\Omega )$.
\end{lemma}

\begin{proof}
If $u_{0}$ is a local minimizer for $J_{\lambda }$ on
$\mathbf{M}_{\lambda}$, then $u_{0}$ is a solution of
the optimization problem
\[
\text{minimize }J_{\lambda }(u)\quad\text{subject to }\psi_{\lambda }(u)=0.
\]
Hence, by the theory of Lagrange multipliers, there exists $\theta \in
\mathbb{R}$ such that
\[
J_{\lambda }'(u_{0})=\theta \psi _{\lambda }'(u_{0})\quad
\text{in }H^{\ast }(\Omega ).
\]
Thus,
\begin{equation}
\langle J_{\lambda }'(u_{0}),u_{0}\rangle
_{H^{1}}=\theta \langle \psi _{\lambda }'(u_{0})
,u_{0}\rangle _{H^{1}}.  \label{6}
\end{equation}
By Lemma \ref{g3}, $u_{0}\in \mathbf{M}_{\lambda }^{+}\cup \mathbf{M}
_{\lambda }^{-}$, we have
$\langle \psi _{\lambda }'(u_{0}),u_{0}\rangle _{H^{1}}\neq 0$ and so by
\eqref{6} $\theta =0$. This completes the proof.
\end{proof}

\begin{lemma} \label{g1} \begin{itemize}
\item[(i)] If $u\in \mathbf{M}_{\lambda }^{+}$, then
$\int_{\Omega }f|u|^{q}dx>0$;
\item[(ii)] If $u\in \mathbf{M}_{\lambda }^{-}$, then
$\int_{\partial \Omega }g|u|^{p}ds>0$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) Case (I): $\int_{\partial \Omega}g|u|^{p}ds\leq 0$. We have
\[
\lambda \int_{\Omega }f|u|^{q}dx =\|u\|_{H^{1}}^{2}-\int_{\partial
\Omega }g|u|^{p}ds>0.
\]
Case (II): $\int_{\partial \Omega }g|u|^{p}ds>0$. We have
\[
\| u\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|u|^{q}dx-\int_{\partial
\Omega }g|u|^{p}ds=0
\]
and
\[
\| u\|_{H^{1}}^{2}>\frac{p-q}{2-q}\int_{\partial \Omega
}g|u|^{p}ds.
\]
Thus,
\[
\lambda \int_{\Omega }f|u|^{q}dx=\| u\|_{H^{1}}^{2}-\int_{\partial
\Omega }g|u| ^{p}ds>\frac{p-2}{2-q}\int_{\partial \Omega }g|u|
^{p}ds>0.
\]
(ii) Since
\[
(2-q)\| u\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega
}g|u|^{p}ds=\langle \psi _{\lambda }'(u) ,u\rangle <0.
\]
It follows that $\int_{\partial \Omega }g|u| ^{p}ds>0$. This
completes the proof.
\end{proof}

For each $u\in \mathbf{M}_{\lambda }^{-}$, we write
\[
t_{\rm max}=\Big(\frac{(2-q)\| u\|_{H^{1}}^{2}
}{(p-q)\int_{\partial \Omega }g|u|^{p}ds} \Big)^{1/(p-2)}<1.
\]
Then we have the following lemma.

\begin{lemma} \label{g4}
Let $p^{\ast }=\frac{p}{p-q}$ and
$\lambda _{2}=(\frac{p-2}{p-q})(\frac{2-q}{p-q})^{\frac{2-q}{p-2}}
C_{p}^{\frac{p(2-q)}{2-p}}S_{p}^{-q}\| f\|_{L^{p^{\ast}}}^{-1}$.
Then for each $u\in \mathbf{M}_{\lambda }^{-}$ and
$\lambda \in(0,\lambda _{2})$, we have
\begin{itemize}
\item[(i)] if $\int_{\Omega }f|u|^{q}dx\leq 0$, then
$J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(tu)>0;$

\item[(ii)] if $\int_{\Omega }f|u|^{q}dx>0$, then there is a
unique $0<t^{+}=t^{+}(u)<t_{\rm max}$ such that
$t^{+}u\in \mathbf{M}_{\lambda }^{+}$ and
\[
J_{\lambda }(t^{+}u)=\inf_{0\leq t\leq t_{\rm max}}J_{\lambda
}(tu),J_{\lambda }(u)=\sup_{t\geq t_{\rm max}}J_{\lambda }(tu).
\]
\end{itemize}
\end{lemma}

\begin{proof}
Fix $u\in \mathbf{M}_{\lambda }^{-}$. Let
\[
h(t)=t^{2-q}\| u\| _{H^{1}}^{2}-t^{p-q}\int_{\partial \Omega
}g|u|^{p}ds \quad \text{for }t\geq 0.
\]
We have $h(0)=0$, $h(t)\to -\infty $ as $t\to \infty $,
$h(t)$ achieves its maximum at $t_{\rm max}$, increasing for
$t\in [ 0,t_{\rm max})$ and decreasing for
$t\in (t_{\rm max},\infty)$. Moreover,
\begin{align*}
&h(t_{\rm max})\\
&= \Big(\frac{(2-q)\| u\|_{H^{1}}^{2}}{ (p-q)\int_{\partial \Omega
}g|u|^{p}ds} \Big)^{\frac{2-q}{p-2}}\| u\|_{H^{1}}^{2}
-\Big(\frac{(2-q)\| u\|_{H^{1}}^{2}}{(p-q) \int_{\partial \Omega
}g|u|^{p}ds}\Big)^{\frac{p-q}{
p-2}}\int_{\partial \Omega }g|u|^{p}ds \\
&= \| u\|_{H^{1}}^{q}\Big[ (\frac{2-q}{p-q})
^{\frac{2-q}{p-2}}-(\frac{2-q}{p-q})^{\frac{p-q}{p-2}}\Big]
\Big(\frac{\| u\|_{H^{1}}^{p}}{\int_{\partial \Omega
}g|u|^{p}ds}\Big)^{\frac{2-q}{p-2}} \\
&\geq \| u\|_{H^{1}}^{q}(\frac{p-2}{p-q})
(\frac{2-q}{p-q})^{\frac{2-q}{p-2}}C_{p}^{\frac{p(2-q)}{2-p}}
\end{align*}
or
\begin{equation}
h(t_{\rm max})\geq \| u\|_{H^{1}}^{q}(
\frac{p-2}{p-q})(\frac{2-q}{p-q})^{\frac{2-q}{p-2}
}C_{p}^{\frac{p(2-q)}{2-p}}.  \label{12}
\end{equation}
(i): $\int_{\Omega }f|u|^{q}dx\leq 0$. There is a unique
$t^{-}>t_{\rm max}$ such that $h(t^{-})=\lambda\int_{\Omega
}f|u|^{q}dx$ and $h'(t^{-})<0$. Now,
\begin{align*}
&(2-q)\| t^{-}u\|_{H^{1}}^{2}-(
p-q)\int_{\partial \Omega }|t^{-}u|^{p}ds \\
&= (t^{-})^{1+q}\Big[ (2-q)(t^{-})
^{1-q}\| u\|_{H^{1}}^{2}-(p-q)(
t^{-})^{p-q-1}\int_{\partial \Omega }g|u|^{p}ds\Big]  \\
&= (t^{-})^{1+q}h'(t^{-})<0,
\end{align*}
and
\begin{align*}
&\langle J_{\lambda }'(t^{-}u),t^{-}u\rangle  \\
&= (t^{-})^{2}\| u\|_{H^{1}}^{2}-( t^{-})^{q}\lambda \int_{\Omega
}f|u| ^{q}dx-(t^{-})^{p}\int_{\partial \Omega }g|u|^{p}ds \\
&= (t^{-})^{q}\Big[ h(t^{-})-\lambda \int_{\Omega }f|u|^{q}dx\Big]
=0.
\end{align*}
Thus, $t^{-}u\in \mathbf{M}_{\lambda }^{-}$ or $t^{-}=1$. Since for
$t>t_{\rm max}$, we have
\begin{gather*}
(2-q)\| tu\|_{H^{1}}^{2}-(p-q)
\int_{\partial \Omega }g|tu|^{p}ds < 0, \\
\frac{d^{2}}{dt^{2}}J_{\lambda }(tu) < 0, \\
\frac{d}{dt}J_{\lambda }(tu)=t\| u\| _{H^{1}}^{2}-\lambda
t^{q-1}\int_{\Omega }f|u| ^{q}dx-t^{p-1}\int_{\partial \Omega
}g|u|^{p}ds=0 \quad \text{for }t=t^{-}.
\end{gather*}
Thus, $J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(
tu)$. Moreover,
\[
J_{\lambda }(u)\geq J_{\lambda }(tu)\geq \frac{ t^{2}}{2}\|
u\|_{H^{1}}^{2}-\frac{t^{p}}{p}\int_{\partial \Omega
}g|u|^{p}ds\quad\text{for all }t\geq 0.
\]
By routine computations, $g(t)=\frac{t^{2}}{2}\|
u\|_{H^{1}}^{2}-\frac{t^{p}}{p}\int_{\partial \Omega }g|u|^{p}ds$
achieves its maximum at $t_{0}=(\| u\|_{H^{1}}^{2}/\int_{\partial
\Omega }g|u| ^{p}ds)^{1/(p-2)}$. Thus,
\[
J_{\lambda }(u)\geq \frac{p-2}{2p}\Big(\frac{\|
u\|_{H^{1}}^{p}}{\int_{\partial \Omega }g|u|
^{p}ds}\Big)^{\frac{2}{p-2}}>0.
\]
(ii): $\int_{\Omega }f|u|^{q}dx>0$. By (\ref{12}) and
\begin{align*}
h(0)&= 0<\lambda \int_{\Omega }f|u| ^{q}dx\leq \lambda \|
f\|_{L^{p^{\ast
}}}S_{p}^{q}\| u\|_{H^{1}}^{q} \\
&< \| u\|_{H^{1}}^{q}(\frac{p-2}{p-q})(
\frac{2-q}{p-q})^{\frac{2-q}{p-2}}C_{p}^{\frac{p(2-q)}{
2-p}}\\
&\leq h(t_{\rm max})\quad \text{for }\lambda \in (0,\lambda _{2}),
\end{align*}
there are unique $t^{+}$ and $t^{-}$ such that $0<t^{+}<t_{\rm max}<t^{-}$,
\begin{gather*}
h(t^{+})=\lambda \int_{\Omega }f|u|^{q}dx=h(t^{-}),\\
h'(t^{+})>0>h'(t^{-}).
\end{gather*}
We have $t^{+}u\in \mathbf{M}_{\lambda }^{+}$,
$t^{-}u\in \mathbf{M}_{\lambda }^{-}$, and
$J_{\lambda }(t^{-}u)\geq J_{\lambda }(tu)\geq J_{\lambda }(t^{+}u)$
for each $t\in [ t^{+},t^{-}] $ and
$J_{\lambda }(t^{+}u)\leq J_{\lambda }(tu)$ for each $t\in [ 0,t^{+}]$.
 Thus, $t^{-}=1$ and
\[
J_{\lambda }(u)=\sup_{t\geq 0}J_{\lambda }(tu)
,J_{\lambda }(t^{+}u)=\inf_{0\leq t\leq t_{\rm max}}J_{\lambda
}(tu).
\]
This completes the proof.\end{proof}

Next, we establish the existence of nontrivial nonnegative solutions for the
equation
\begin{equation}
\begin{gathered}
-\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \text{in }\Omega , \\
u=0 \quad \text{on }\partial \Omega ,
\end{gathered}  \label{13}
\end{equation}
Associated with equation \eqref{13}, we consider the energy
functional
\[
K_{\lambda }(u)=\frac{1}{2}\| u\|_{H^{1}}^{2}- \frac{\lambda
}{q}\int_{\Omega }f|u|^{q}dx
\]
and the minimization problem
\[
\beta _{\lambda }=\inf \{K_{\lambda }(u) : u\in \mathbf{N}_{\lambda }\},
\]
where $\mathbf{N}_{\lambda }=\{u\in H_{0}^{1}(\Omega )
\backslash \{0\}: \langle K_{\lambda }'(u),u\rangle =0\}$.
Then we have the following result.

\begin{theorem} \label{m4}
Suppose that $\lambda >0$. Then equation \eqref{13}
has a nontrivial nonnegative solution $v_{\lambda }$ with $K_{\lambda
}(v_{\lambda })=\beta _{\lambda }<0$.
\end{theorem}

\begin{proof}
First, we need to show that $K_{\lambda }$ is bounded below on
$\mathbf{N}_{\lambda }$ and $\beta _{\lambda }<0$. Then for
$u\in \mathbf{N}_{\lambda}$,
\[
\| u\|_{H^{1}}^{2}=\lambda \int_{\Omega }f|u|^{q}dx\leq \lambda \|
f\|_{L^{q^{\ast }}}S_{p}^{-\frac{q}{2}}\| u\|_{H^{1}}^{q}.
\]
where $p^{\ast }=\frac{p}{p-q}$. This implies
\begin{equation}
\| u\|_{H^{1}}\leq (\lambda \| f\|
_{L^{p^{\ast }}}S_{p}^{-\frac{q}{2}})^{\frac{1}{2-q}}.  \label{11}
\end{equation}
Hence,
\begin{align*}
K_{\lambda }(u)&= \frac{1}{2}\| u\|_{H^{1}}-
\frac{\lambda }{q}\int_{\Omega }f|u|^{q}dx \\
&= \big(\frac{1}{2}-\frac{1}{q}\big)\| u\|_{H^{1}}^{2}
\\
&\leq \big(\frac{1}{2}-\frac{1}{q}\big)\big(\lambda \|
f\|_{L^{p^{\ast }}}S_{p}^{-\frac{q}{2}}\big)^{\frac{1}{2-q}}
\end{align*}
for all $u\in \mathbf{N}_{\lambda }$ and $\beta _{\lambda }<0$. Let $\{
v_{n}\}$ be a minimizing sequence for $K_{\lambda }$ on $\mathbf{N}
_{\lambda }$. Then by $(\ref{11})$ and the compact imbedding
theorem, there exist a subsequence $\{v_{n}\}$ and $v_{\lambda
} $ in $H_{0}^{1}(\Omega )$ such that
\[
v_{n}\rightharpoonup v_{\lambda }\quad\text{weakly in }H_{0}^{1}(\Omega )
\]
and
\begin{equation}
v_{n}\to v_{\lambda }\quad\text{strongly in }L^{q}(\Omega ).  \label{10}
\end{equation}
First, we claim that $\int_{\Omega }f|v_{\lambda }| ^{q}dx>0$. If
not,
\[
K_{\lambda }(v_{n})\geq \frac{1}{2}\| v_{\lambda
}\|_{H^{1}}^{2}-\frac{\lambda }{q}\int_{\Omega }f|v_{\lambda
}|^{q}dx+o(1) \geq \frac{1}{2}\| v_{\lambda }\|_{H^{1}}^{2}+o(1),
\]
this contradicts $K_{\lambda }(v_{n})\to \beta_{\lambda }(\Omega
)<0$ as $n\to \infty $. Thus, $\int_{\Omega }f|v_{\lambda
}|^{q}dx>0$. In particular, $v_{\lambda }\not\equiv 0$. Now, we
prove that $v_{n}\to v_{\lambda }$ strongly in $H_{0}^{1}(\Omega
)$. Suppose otherwise, then $\|v_{\lambda
}\|_{H^{1}}<\liminf_{n\to \infty }\| v_{n}\|_{H^{1}}$ and so
\[
\| v_{\lambda }\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|v_{\lambda
}|^{q}dx<\liminf_{n\to \infty } \Big(\|
v_{n}\|_{H^{1}}^{2}-\lambda \int_{\Omega }f|v_{n}|^{q}dx\Big)=0.
\]
Since $\int_{\Omega }f|v_{\lambda }|^{q}dx>0$, there is a unique
$t_{0}\neq 1$ such that $t_{0}v_{\lambda }\in \mathbf{N}_{\lambda
}$. Thus,
\[
t_{0}v_{n}\rightharpoonup t_{0}v_{\lambda }\quad\text{weakly in }
H_{0}^{1}(\Omega ).
\]
Moreover,
\[
K_{\lambda }(t_{0}v_{\lambda })<K_{\lambda }(v_{\lambda
})<\lim_{n\to \infty }K_{\lambda }(v_{n})=\beta
_{\lambda },
\]
which is a contradiction. Hence $v_{n}\to v_{\lambda }$ strongly in
$H_{0}^{1}(\Omega )$. This implies $v_{\lambda }\in \mathbf{N}_{\lambda }$
and
\[
K_{\lambda }(v_{n})\to K_{\lambda }(v_{\lambda
})=\beta _{\lambda }\quad \quad\text{as }n\to \infty .
\]
Since $K_{\lambda }(v_{\lambda })=K_{\lambda }(\|
v_{\lambda }\|)$ and $\|v_{\lambda }\|
\in \mathbf{N}_{\lambda }$, without loss of generality, we may assume that
$v_{\lambda }$ is a nontrivial nonnegative solution of equation
\eqref{13}.
\end{proof}

Then we have the following results.

\begin{lemma} \label{g5}
\begin{itemize}
\item[(i)] $ \alpha _{\lambda }\leq \alpha _{\lambda }^{+}\leq \beta
_{\lambda }<0$;
\item[(ii)] $J_{\lambda }$ is coercive and bounded below on $\mathbf{
M}_{\lambda }$ for all $\lambda \in (0,\frac{p-2}{p-q}]$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) Let $v_{\lambda }$ be a positive solution of equation
\eqref{13} such that $K(v_{\lambda })=\beta _{\lambda }$. Since
$v_{\lambda }\in C^{2}(\overline{\Omega })$. Then we have
$\int_{\partial \Omega }g|v_{\lambda }| ^{p}ds=0$ and $v_{\lambda
}\in \mathbf{M}_{\lambda }^{+}$. This implies
\[
J_{\lambda }(v_{\lambda })=\frac{1}{2}\| v_{\lambda
}\|_{H^{1}}^{2}-\frac{\lambda }{q}\int_{\Omega }f| v_{\lambda
}|^{q}dx=\beta _{\lambda }<0
\]
and so
$\alpha _{\lambda }\leq \alpha _{\lambda }^{+}\leq \beta _{\lambda }<0$.

\noindent (ii) For $u\in \mathbf{M}_{\lambda }$, we have
$\|u\|_{H^{1}}^{2}=\lambda \int_{\Omega }f|u|^{q}dx
+\int_{\partial \Omega }g|u|^{p}ds$. Then by the H\"{o}lder and
Young inequalities,
\begin{align*}
J_{\lambda }(u)&= \frac{p-2}{2p}\| u\| _{H^{1}}^{2}-\lambda
\big(\frac{p-q}{pq}\big)\int_{\Omega }f|u|^{q}dx \\
&\geq \frac{p-2}{2p}\| u\|_{H^{1}}^{2}-\lambda
\big(\frac{p-q}{pq}\big)\| f\|_{L^{p^{\ast
}}}S_{p}^{q}\| u\|_{H^{1}}^{q} \\
&\geq \big[ \frac{p-2}{2p}-\lambda \big(\frac{p-q}{2p}\big)\big]
\| u\|_{H^{1}}^{2}-\lambda \big(\frac{(p-q)
(2-q)}{2pq}\big)\big(\| f\|_{L^{p^{\ast
}}}S_{p}^{q}\big)^{\frac{2}{2-q}} \\
&= \frac{1}{2p}\big[ (p-2)-\lambda (p-q)\big]
\| u\|_{H^{1}}^{2}-\lambda \big(\frac{(p-q)
(2-q)}{2pq}\big)\big(\| f\|_{L^{p^{\ast
}}}S_{p}^{q}\big)^{\frac{2}{2-q}}.
\end{align*}
Thus, $J_{\lambda }$ is coercive on $\mathbf{M}_{\lambda }$ and
\[
J_{\lambda }(u)\geq -\lambda \big(\frac{(p-q)
(2-q)}{2pq}\big)\big(\| f\|_{L^{p^{\ast
}}}S_{p}^{q}\big)^{\frac{2}{2-q}}
\]
for all $\lambda \in (0,\frac{p-2}{p-q}]$.
\end{proof}

\section{Proof of Theorem \ref{t1}}

First, we will use the idea of Ni-Takagi \cite{NT} to get the following
results.

\begin{lemma} \label{m6}
For each $u\in \mathbf{M}_{\lambda }$, there exist $\epsilon >0$
and a differentiable function $\xi :B(0;\epsilon )\subset
H^{1}(\Omega )\to \mathbb{R}^{+}$ such that $\xi (0)=1$, the function
$\xi (v)(u-v)\in \mathbf{M}_{\lambda }$ and
\begin{equation}
\langle \xi '(0),v\rangle =\frac{ 2\int_{\Omega }\nabla u\nabla
vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial \Omega
}g|u| ^{p-2}uvds}{(2-q)\| u\|_{H^{1}}^{2}-( p-q)\int_{\partial
\Omega }g|u|^{p}ds} \label{21}
\end{equation}
for all $v\in H^{1}(\Omega )$.
\end{lemma}

\begin{proof}
For $u\in \mathbf{M}_{\lambda }$, define a function
$F:\mathbb{R}\times H^{1}(\Omega )\to \mathbb{R}$ by
\begin{align*}
F_{u}(\xi ,w)
&= \langle J_{\lambda }'(\xi (u-w)),\xi (u-w)\rangle \\
&= \xi ^{2}\int_{\Omega }|\nabla (u-w)|^{2}+(u-w)^{2}dx
 -\xi ^{q}\lambda \int_{\Omega }f|u-w|^{q}dx \\
&\quad -\xi ^{p}\int_{\partial \Omega }g|u-w|^{p}ds.
\end{align*}
Then $F_{u}(1,0)=\langle J_{\lambda }'(u),u\rangle =0$ and
\begin{align*}
\frac{d}{d\xi }F_{u}(1,0) &= 2\| u\|_{H^{1}}^{2}-\lambda
q\int_{\partial \Omega }f|u|
^{q}dx-p\int_{\partial \Omega }g|u|^{p}ds \\
&= (2-q)\| u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega
}g|u|^{p}ds\neq 0.
\end{align*}
According to the implicit function theorem, there exist $\epsilon >0$ and a
differentiable function
$\xi :B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R}$ such that
$\xi (0)=1$,
\[
\langle \xi '(0),v\rangle =\frac{ 2\int_{\Omega }\nabla u\nabla
vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial \Omega
}g|u| ^{p-2}uvds}{(2-q)\| u\|_{H^{1}}^{2}-( p-q)\int_{\partial
\Omega }g|u|^{p}ds}
\]
and
\[
F_{u}(\xi (v),v)=0\quad\text{for all }v\in B(
0;\epsilon )
\]
which is equivalent to
\[
\langle J_{\lambda }'(\xi (v)(u-v)),\xi (v)(u-v)\rangle =0 \quad
\quad\text{for all }v\in B(0;\epsilon ),
\]
that is $\xi (v)(u-v)\in \mathbf{M}_{\lambda }$.
\end{proof}

\begin{lemma} \label{m7}
For each $u\in \mathbf{M}_{\lambda }^{-}$, there exist
$\epsilon>0 $ and a differentiable function
$\xi ^{-}:B(0;\epsilon )\subset H^{1}(\Omega )\to \mathbb{R}^{+}$ such that
$\xi ^{-}(0)=1$, the function
$\xi ^{-}(v)(u-v)\in \mathbf{M}_{\lambda }^{-}$ and
\begin{equation}
\langle (\xi ^{-})'(0) ,v\rangle =\frac{2\int_{\Omega }\nabla
u\nabla vdx-\lambda q\int_{\Omega }f|u|^{q-2}uvdx-p\int_{\partial
\Omega }g|u|^{p-2}uvds}{(2-q)\|
u\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega }g|u|^{p}ds} \label{22}
\end{equation}
for all $v\in H^{1}(\Omega )$.
\end{lemma}

\begin{proof}
Similar to the argument in Lemma \ref{m6}, there exist $\epsilon >0$ and a
differentiable function $\xi ^{-}:B(0;\epsilon )\subset
H^{1}(\Omega )\to \mathbb{R}$ such that $\xi ^{-}(
0)=1$ and $\xi ^{-}(v)(u-v)\in \mathbf{M}
_{\lambda }$ for all $v\in B(0;\epsilon )$. Since
\[
\langle \psi _{\lambda }'(u),u\rangle =(2-q)\|
u\|_{H^{1}}^{2}-(p-q) \int_{\partial \Omega }g|u|^{p}ds<0.
\]
Thus, by the continuity of the function $\xi ^{-}$, we have
\begin{align*}
&\langle \psi _{\lambda }'(\xi ^{-}(v)
(u-v)),\xi ^{-}(v)(u-v) \rangle \\
&= (2-q)\| \xi ^{-}(v)(u-v) \|_{H^{1}}^{2}-(p-q)\int_{\partial
\Omega }g|\xi ^{-}(v)(u-v)|^{p}ds <0
\end{align*}
if $\epsilon $ sufficiently small, this implies that
$\xi ^{-}(v)(u-v)\in \mathbf{M}_{\lambda }^{-}$.
\end{proof}

\begin{proposition} \label{m8}
Let $\lambda _{0}=\min \{\lambda _{1},\lambda _{2},\frac{p-1
}{p-q}\}$, Then for $\lambda \in (0,\lambda _{0})$:
\begin{itemize}
\item[(i)] There exists a minimizing sequence $\{u_{n}\}
\subset \mathbf{M}_{\lambda }$ such that
\begin{align*}
J_{\lambda }(u_{n})&= \alpha _{\lambda }+o(1), \\
J_{\lambda }'(u_{n})&= o(1)\quad \text{in } H^{\ast }(\Omega );
\end{align*}
\item[(ii)] there exists a minimizing sequence
$\{u_{n}\}\subset \mathbf{M}_{\lambda }^{-}$ such that
\begin{align*}
J_{\lambda }(u_{n})&= \alpha _{\lambda }^{-}+o(1), \\
J_{\lambda }'(u_{n})&= o(1)\quad \text{in }H^{\ast }(\Omega ).
\end{align*}
\end{itemize}
\end{proposition}

\begin{proof}
(i) By Lemma \ref{g5} (ii) and the Ekeland
variational principle \cite{E}, there exists a minimizing sequence
$\{u_{n}\}\subset \mathbf{M}_{\lambda }$ such that
\begin{gather}
J_{\lambda }(u_{n})<\alpha _{\lambda }+\frac{1}{n},  \label{16} \\
J_{\lambda }(u_{n})<J_{\lambda }(w)+\frac{1}{n}
\| w-u_{n}\|_{H^{1}}\quad\text{for each }w\in \mathbf{M}
_{\lambda }.  \label{17}
\end{gather}
By taking $n$ large, from Lemma \ref{g5} (i), we have
\begin{equation} \label{23}
\begin{aligned}
J_{\lambda }(u_{n})
&= (\frac{1}{2}-\frac{1}{p})
\| u_{n}\|_{H^{1}}^{2}-(\frac{1}{q}-\frac{1}{p}
)\lambda \int_{\Omega }f|u_{n}|^{q}dx \\
&< \alpha _{\lambda }+\frac{1}{n}<\frac{\beta _{\lambda }}{2}.
\end{aligned}
\end{equation}
This implies
\begin{equation}
\| f\|_{L^{p^{\ast }}}S_{p}^{q}\| u_{n}\| _{H^{1}}^{q}\geq
\int_{\Omega }f|u_{n}|^{q}dx>\frac{-pq }{2\lambda (p-q)}\beta
_{\lambda }>0.  \label{24}
\end{equation}
Consequently, $u_{n}\neq 0$ and putting together \eqref{23},
\eqref{24} and the H\"{o}lder inequality, we obtain
\begin{gather}
\| u_{n}\|_{H^{1}}>\Big[ \frac{-pq}{2\lambda (
p-q)}\beta _{\lambda }S_{p}^{-q}\| f\|_{L^{p^{\ast
}}}^{-1}\Big] ^{1/q}  \label{25-1} \\
\| u_{n}\|_{H^{1}}<\Big[ \frac{2(p-q)}{
(p-2)q}\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big]
^{1/(2-q)}  \label{25-2}
\end{gather}
Now, we show that
\[
\| J_{\lambda }'(u_{n})\|
_{H^{-1}}\to 0\quad \text{as }n\to \infty .
\]
Applying Lemma \ref{m6} with $u_{n}$ to obtain the functions
$\xi_{n}:B(0;\epsilon _{n})\to \mathbb{R}^{+}$ for some
$\epsilon _{n}>0$, such that $\xi _{n}(w)(u_{n}-w)
\in \mathbf{M}_{\lambda }$. Choose $0<\rho <\epsilon _{n}$.
Let $u\in H^{1}(\Omega )$ with $u\not\equiv 0$ and let
$w_{\rho }=\frac{\rho u}{\| u\|_{H^{1}}}$. We set
$\eta _{\rho }=\xi_{n}(w_{\rho })(u_{n}-w_{\rho })$.
Since $\eta_{\rho }\in \mathbf{M}_{\lambda }$, we deduce from
\eqref{17} that
\[
J_{\lambda }(\eta _{\rho })-J_{\lambda }(u_{n})
\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\|_{H^{1}}
\]
and by the mean value theorem, we have
\[
\langle J_{\lambda }'(u_{n}),\eta _{\rho}-u_{n}\rangle
+o(\| \eta _{\rho }-u_{n}\|_{H^{1}})\geq
-\frac{1}{n}\| \eta _{\rho }-u_{n}\|_{H^{1}}.
\]
Thus,
\begin{equation} \label{18}
\begin{aligned}
&\langle J_{\lambda }'(u_{n}),-w_{\rho
}\rangle +(\xi _{n}(w_{\rho })-1)
\langle J_{\lambda }'(u_{n}),(u_{n}-w_{\rho})\rangle   \\
&\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\|
_{H^{1}}+o(\| \eta _{\rho }-u_{n}\|_{H^{1}}).
\end{aligned}
\end{equation}
 Since $\xi _{n}(w_{\rho })(u_{n}-w_{\rho })\in
\mathbf{M}_{\lambda }$ and $(\ref{18})$ it follows that
\begin{align*}
&-\rho \langle J_{\lambda }'(u_{n}),\frac{u}{
\| u\|_{H^{1}}}\rangle +(\xi _{n}(
w_{\rho })-1)\langle J_{\lambda }'(
u_{n})-J_{\lambda }'(\eta _{\rho }),(
u_{n}-w_{\rho })\rangle \\
&\geq -\frac{1}{n}\| \eta _{\rho }-u_{n}\|
_{H^{1}}+o(\| \eta _{\rho }-u_{n}\|_{H^{1}}).
\end{align*}
Thus,
\begin{equation} \label{19}
\begin{aligned}
\langle J_{\lambda }'(u_{n}),\frac{u}{\|
u\|_{H^{1}}}\rangle
&\leq \frac{\| \eta _{\rho}-u_{n}\|_{H^{1}}}{n\rho }
+\frac{o(\| \eta _{\rho}-u_{n}\|_{H^{1}})}{\rho }   \\
&\quad +\frac{(\xi _{n}(w_{\rho })-1)}{\rho }
\langle J_{\lambda }'(u_{n})-J_{\lambda }'(\eta _{\rho }),
(u_{n}-w_{\rho })\rangle .
\end{aligned}
\end{equation}
Since
$\| \eta _{\rho }-u_{n}\|_{H^{1}}\leq \rho \|\xi
_{n}(w_{\rho })\|+\|\xi _{n}(w_{\rho})-1\|\| u_{n}\|_{H^{1}}$
and
\[
\lim_{\rho \to 0}\frac{\|\xi _{n}(w_{\rho })
-1\|}{\rho }\leq \| \xi _{n}'(0)
\|,
\]
if we let $\rho \to 0$ in $(\ref{19})$ for a fixed $n$,
then by $(\ref{25-2})$ we can find a constant $C>0$,
independent of $\rho $, such that
\[
\langle J_{\lambda }'(u_{n}),\frac{u}{\|
u\|_{H^{1}}}\rangle \leq \frac{C}{n}(1+\| \xi
_{n}'(0)\|).
\]
 The proof will be complete once we show that
$\| \xi _{n}'(0)\|$ is uniformly bounded in $n$. By
\eqref{21}, \eqref{25-2} and the H\"{o}lder inequality, we have
\[
\langle \xi _{n}'(0),v\rangle \leq \frac{ b\|
v\|_{H^{1}}}{|(2-q)\| u_{n}\|_{H^{1}}-(p-q)\int_{\partial \Omega
}g|u_{n}|^{p}ds|}\quad \text{for some }b>0.
\]
We only need to show that
\begin{equation}
|(2-q)\| u_{n}\|_{H^{1}}-( p-q)\int_{\partial \Omega }g|u_{n}|
^{p}ds|>c  \label{26}
\end{equation}
for some $c>0$ and $n$ large enough. We argue by contradiction. Assume that
there exists a subsequence $\{u_{n}\}$, we have
\begin{equation}
(2-q)\| u_{n}\|_{H^{1}}-(p-q) \int_{\partial \Omega
}g|u_{n}|^{p}ds=o(1). \label{27}
\end{equation}
Combining $(\ref{27})$ with $(\ref{25-1})$, we can
find a suitable constant $d>0$ such that
\begin{equation}
\int_{\partial \Omega }g|u_{n}|^{p}ds\geq d\quad \text{for $n$
 sufficiently large.}  \label{28}
\end{equation}
In addition $(\ref{27})$, and the fact that
$u_{n}\in \mathbf{M}_{\lambda }$ also give
\[
\lambda \int_{\Omega }f|u_{n}|^{q}dx=\|
u_{n}\|_{H^{1}}^{2}-\int_{\partial \Omega }g|
u_{n}|^{p}ds=\frac{p-2}{2-q}\int_{\partial \Omega }g|
u_{n}|^{p}ds+o(1)
\]
and
\begin{equation}
\| u_{n}\|_{H^{1}}\leq \Big[ \lambda (\frac{p-q}{p-2
})\| f\|_{L^{p^{\ast }}}S_{p}^{q}\Big] ^{\frac{1}{
2-q}}+o(1).  \label{37}
\end{equation}
This implies
\begin{equation}
I_{\lambda }(u_{n})=K(p,q) \Big(\frac{\|
u_{n}\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega
}g|u_{n}|^{p}ds}\Big)^{1/(p-2)}-\lambda \int_{\Omega
}f|u_{n}|^{q}dx=o(1). \label{29}
\end{equation}
However, by \eqref{28}, \eqref{37} and $\lambda \in (0,\lambda _{0})$,
\begin{align*}
I_{\lambda }(u_{n}) &\geq K(p,q)\Big(\frac{\|
u_{n}\|_{H^{1}}^{2(p-1)}}{\int_{\partial \Omega
}g|u_{n}|^{p}ds}\Big)^{1/(p-2)}-\lambda S_{p}^{q}\|
f\|_{L^{p^{\ast }}}\| u_{n}\|
_{H^{1}}^{q} \\
&\geq \| u_{n}\|_{H^{1}}^{q}\Big(K(p,q)
\Big(\frac{\| u_{n}\|_{H^{1}}^{2(p-1)}}{
C_{p}^{p}\| u_{n}\|_{H^{1}}^{p+q(p-2)}}
\Big)^{1/(p-2)}-\lambda S_{p}^{q}\| f\|
_{L^{p^{\ast }}}\Big)\\
&\geq \| u_{n}\|_{H^{1}}^{q}\Big\{K(p,q)
C_{p}^{\frac{p}{2-p}}\lambda ^{\frac{1-q}{2-q}}\big[ (\frac{p-q}{p-2}
)\| f\|_{L^{p^{\ast }}}S_{p}^{q}\big] ^{\frac{1-q}{2-q}}
 -\lambda \| f\|_{L^{p^{\ast }}}\Big\}.
\end{align*}
this contradicts \eqref{29}. We get
\[
\langle J_{\lambda }'(u_{n}),\frac{u}{\|
u\|_{H^{1}}}\rangle \leq \frac{C}{n}.
\]
This completes the proof of $(i)$.

\noindent (ii) Similarly, by using Lemma \ref{m7}, we can prove
(ii). We will omit detailed proof here.
\end{proof}

Now, we establish the existence of a local minimum for $J_{\lambda }$ on
$\mathbf{M}_{\lambda }^{+}$.

\begin{theorem} \label{t2}
Let $\lambda _{0}>0$ as in Proposition \ref{m8}, then for
$\lambda \in (0,\lambda _{0})$ the functional $J_{\lambda }$
has a minimizer
$u_{0}^{+}$ in $\mathbf{M}_{\lambda }^{+}$ and it satisfies
\begin{itemize}
\item[(i)] $J_{\lambda }(u_{0}^{+})=\alpha _{\lambda }=\alpha
_{\lambda }^{+}$;
\item[(ii)] $u_{0}^{+}$ is a nontrivial nonnegative solution of
equation \eqref{Efg};

\item[(iii)] $J_{\lambda }(u_{0}^{+})\to 0$ as $\lambda
\to 0$.
\end{itemize}
\end{theorem}

\begin{proof}
Let $\{u_{n}\}\subset \mathbf{M}_{\lambda }$ be a minimizing
sequence for $J_{\lambda }$ on $\mathbf{M}_{\lambda }$ such that
\[
J_{\lambda }(u_{n})=\alpha _{\lambda }+o(1)\quad
\text{and}\quad  J_{\lambda }'(u_{n})=o(1)\quad \text{in }H^{\ast }(\Omega ).
\]
Then by Lemma \ref{g5} and the compact imbedding theorem, there exist a
subsequence $\{u_{n}\}$ and $u_{0}^{+}\in H^{1}(\Omega )$ such
that
\[
u_{n}\rightharpoonup u_{0}^{+}\quad\text{weakly in }H^{1}(\Omega ),
u_{n}\to u_{0}^{+}\quad\text{strongly in }L^{p}(\partial \Omega )
\]
and
\begin{equation}
u_{n}\to u_{0}^{+}\quad\text{strongly in }L^{q}(\Omega ).  \label{7}
\end{equation}
First, we claim that $\int_{\Omega }f(x)\|u_{0}^{+}\|^{q}dx\neq 0$.
Suppose otherwise, by \eqref{7} we can conclude that
\[
\int_{\Omega }f|u_{n}|^{q}dx\to \int_{\Omega
}f|u_{0}^{+}|^{q}dx=0\quad \text{as }n\to \infty
\]
and so
\[
\| u_{n}\|_{H^{1}}^{2}=\int_{\partial \Omega }g|
u_{n}|^{p}ds+o(1).
\]
Thus,
\begin{align*}
J_{\lambda }(u_{n}) &= \frac{1}{2}\|
u_{n}\|_{H^{1}}^{2}-\frac{\lambda }{q} \int_{\Omega
}f|u_{n}|^{q}dx-\frac{1}{p}\int_{\partial \Omega } g|u_{n}|^{p}ds
\\
&= (\frac{1}{2}-\frac{1}{p})\int_{\partial \Omega }g|
u_{n}|^{p}ds+o(1)\\
&= (\frac{1}{2}-\frac{1}{p})\int_{\partial \Omega }g|
u_{0}^{+}|^{p}ds\quad\text{as }n\to \infty ,
\end{align*}
this contradicts $J_{\lambda }(u_{n})\to \alpha
_{\lambda }<0$ as $n\to \infty $. Moreover,
\[
o(1)=\langle J_{\lambda }'(u_{n})
,\phi \rangle =\langle J_{\lambda }'(u_{0})
,\phi \rangle +o(1)\quad\text{for all }\phi \in H^{1}(
\Omega ).
\]
Thus, $u_{0}^{+}\in \mathbf{M}_{\lambda }$ is a nonzero solution of equation
\eqref{Efg} and $J_{\lambda }(u_{0}^{+})\geq
\alpha _{\lambda }$. Now we prove that $u_{n}\to u_{0}^{+}$ strongly
in $H^{1}(\Omega )$. Suppose otherwise, then $\|
u_{0}^{+}\|_{H^{1}}<\liminf_{n\to \infty }
\| u_{n}\|_{H^{1}}$ and so
\begin{align*}
&\| u_{0}^{+}\|_{H^{1}}^{2}-\lambda \int_{\Omega
}f|u_{0}^{+}|^{q}dx-\int_{\partial \Omega }g|
u_{0}^{+}|^{p}ds \\
&< \liminf_{n\to \infty } \Big(\| u_{n}\|_{H^{1}}^{2}-\lambda
\int_{\Omega }f| u_{n}|^{q}dx-\int_{\partial \Omega }g|u_{n}|
^{p}ds\Big)=0,
\end{align*}
this contradicts $u_{0}^{+}\in \mathbf{M}_{\lambda }$. Hence
$u_{n}\to u_{0}^{+}$ strongly in $H^{1}(\Omega )$ and
\[
J_{\lambda }(u_{n})\to J_{\lambda }(
u_{0}^{+})=\alpha _{\lambda }\quad\text{as }n\to \infty .
\]
Moreover, we have $u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$. If not, then
$u_{0}^{+}\in \mathbf{M}_{\lambda }^{-}$ and by Lemma \ref{g4}, there are
unique $t_{0}^{+}$ and $t_{0}^{-}$ such that
$t_{0}^{+}u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$ and
$t_{0}^{-}u_{0}^{+}\in \mathbf{M}_{\lambda }^{-}$. In
particular, we have $t_{0}^{+}<t_{0}^{-}=1$. Since
\[
\frac{d}{dt}J_{\lambda }(t_{0}^{+}u_{0}^{+})=0\quad \text{and}\quad
\frac{d^{2}}{dt^{2}}J_{\lambda }(t_{0}^{+}u_{0}^{+})>0,
\]
there exists $t_{0}^{+}<\bar{t}\leq t_{0}^{-}$ such that
$J_{\lambda }(t_{0}^{+}u_{0}^{+})<J_{\lambda }(\bar{t}u_{0}^{+})$. By
Lemma \ref{g4},
\[
J_{\lambda }(t_{0}^{+}u_{0}^{+})<J_{\lambda }(\bar{t}
u_{0}^{+})\leq J_{\lambda }(t_{0}^{-}u_{0}^{+})
=J_{\lambda }(u_{0}^{+}),
\]
which is a contradiction. Since $J_{\lambda
}(u_{0}^{+})=J_{\lambda }(|u_{0}^{+}|)$ and $|u_{0}^{+}|\in
\mathbf{M}_{\lambda }^{+}$, by Lemma \ref{g2} we may assume that
$u_{0}^{+}$ is a nontrivial nonnegative solution of equation
\eqref{Efg}. From Lemma \ref{g5} it follows that
\[
0>J_{\lambda }(u_{0}^{+})\geq -\lambda \Big(\frac{(
p-q)(2-q)}{2pq}\Big)(\| f\|_{L^{p^{\ast }}}S_{p}^{q})^{\frac{2}{2-q}}
\]
and so $J_{\lambda }(u_{0}^{+})\to 0$ as $\lambda \to 0$.
\end{proof}

Next, we establish the existence of a local minimum for $J_{\lambda }$ on
$\mathbf{M}_{\lambda }^{-}$.

\begin{theorem} \label{t3}
Let $\lambda _{0}>0$ as in Proposition \ref{m8}. Then for
$\lambda \in (0,\lambda _{0})$ the functional $J_{\lambda }$ has a minimizer
$u_{0}^{-}$ in $\mathbf{M}_{\lambda }^{-}$ and satisfies
\begin{itemize}
\item[(i)] $J_{\lambda }(u_{0}^{-})=\alpha _{\lambda }^{-}$;

\item[(ii)] $u_{0}^{-}$ is a nontrivial nonnegative solution of
equation \eqref{Efg}.
\end{itemize}
\end{theorem}

\begin{proof}
By Proposition \ref{m8} (ii), there exists a minimizing
sequence $\{u_{n}\}$ for $J_{\lambda }$ on $\mathbf{M}_{\lambda
}^{-}$ such that
\[
J_{\lambda }(u_{n})=\alpha _{\lambda }^{-}+o(1)
\quad\text{and}\quad J_{\lambda }'(u_{n})=o(1)
\quad\text{in }H^{\ast }(\Omega ).
\]
By Lemma \ref{g5} and the compact imbedding theorem, there exist a
subsequence $\{u_{n}\}$ and $u_{0}^{-}\in H^{1}(\Omega)$
such that
\begin{gather*}
u_{n}\rightharpoonup u_{0}^{-}\quad \text{weakly in }H^{1}(\Omega ),\\
u_{n}\to u_{0}^{-}\quad \text{strongly in }L^{p}(\partial \Omega ),\\
u_{n}\to u_{0}^{-}\quad\text{strongly in }L^{q}(\Omega ).
\end{gather*}
Since $(2-q)\| u_{n}\|_{H^{1}}^{2}-( p-q)\int_{\partial \Omega
}g|u_{n}|^{p}ds<0$, by the Sobolev trace inequality there exists
$C>0$ such that $\int_{\partial \Omega }g|u_{n}|^{p}ds>C$.
Moreover,
\[
o(1)=\langle J_{\lambda }'(u_{n})
,\phi \rangle =\langle J_{\lambda }'(u_{0})
,\phi \rangle +o(1)\quad\text{for all }\phi \in H^{1}(
\Omega )
\]
and
\begin{align*}
&(2-q)\| u_{0}\|_{H^{1}}^{2}-(
p-q)\int_{\partial \Omega }g|u_{0}|^{p}ds \\
&\leq \liminf_{n\to \infty } \Big((2-q) \|
u_{n}\|_{H^{1}}^{2}-(p-q)\int_{\partial \Omega
}g|u_{n}|^{p}ds\Big) \leq 0.
\end{align*}
Thus, $u_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$ is a nonzero solution of
equation \eqref{Efg}. Now we prove that $u_{n}\to
u_{0}^{-}$ strongly in $H^{1}(\Omega )$. Suppose otherwise, then $\|
u_{0}^{-}\|_{H^{1}}<\liminf_{n\to \infty }
\| u_{n}\|_{H^{1}}$ and so
\begin{align*}
&\| u_{0}^{-}\|_{H^{1}}^{2}-\lambda \int_{\Omega
}f|u_{0}^{-}|^{q}dx-\int_{\partial \Omega }g|
u_{0}^{-}|^{p}ds \\
&< \liminf_{n\to \infty } \Big(\| u_{n}\|_{H^{1}}^{2}-\lambda
\int_{\Omega }f| u_{n}|^{q}dx-\int_{\partial \Omega }g|u_{n}|
^{p}ds\Big)=0,
\end{align*}
this contradicts $u_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$. Hence
$u_{n}\to u_{0}^{-}$ strongly in $H^{1}(\Omega )$. This implies
\[
J_{\lambda }(u_{n})\to J_{\lambda }(
u_{0}^{-})=\alpha _{\lambda }^{-}\quad\text{as }n\to \infty .
\]
Since $J_{\lambda }(u_{0}^{-})=J_{\lambda }(|u_{0}^{-}|)$ and
$|u_{0}^{-}|\in \mathbf{M}_{\lambda }^{-}$, by Lemma \ref{g2} we
 may assume that $u_{0}^{-}$
is a nontrivial nonnegative solution of equation \eqref{Efg}.
\end{proof}

 Now, we complete the proof of Theorem \ref{t1}. By
Theorems \ref{t2}, \ref{t3}, we obtain equation \eqref{Efg} has
two nontrivial nonnegative solutions $u_{0}^{+}$ and $u_{0}^{-}$ such that
$u_{0}^{+}\in \mathbf{M}_{\lambda }^{+}$ and
$u_{0}^{-}\in \mathbf{M}_{\lambda }^{-}$. Since
$\mathbf{M}_{\lambda }^{+}\cap \mathbf{M}_{\lambda }^{-}=\phi $,
this implies that $u_{0}^{+}$ and $u_{0}^{-}$ are different.

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\end{document}