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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 100, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/100\hfil Robin boundary-value problems]
{Robin boundary-value problems for quasilinear elliptic equations with 
subcritical and critical nonlinearities}

\author[D. A. Kandilakis, M. Magiropoulos \hfil EJDE-2016/100\hfilneg]
{Dimitrios A. Kandilakis, Manolis Magiropoulos}

\address{Dimitrios A. Kandilakis \newline
School of Architectural Engineering, Technical University of Crete,
73100 Chania, Greece}
\email{dkandylakis@isc.tuc.gr}

\address{Manolis Magiropoulos \newline
Department of Electrical Engineering,
Technological Educational Institute of Crete,
71410 Heraklion, Crete, Greece}
\email{mageir@staff.teicrete.gr}

\thanks{Submitted February 2, 2016. Published April 19, 2016.}
\subjclass[2010]{35J50, 35J65, 47J10}
\keywords{Quasilinear elliptic problems; Robin boundary condition;
\hfill\break\indent subcritical nonlinearities; critical nonlinearities; 
fibering method; mountain pass theorem}

\begin{abstract}
 By using variational methods we study the existence of positive
 solutions for a class of quasilinear elliptic problems with Robin 
 boundary conditions.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$
with a smooth boundary $\partial \Omega$. In this article we study the
nonlinear Robin problem:
\begin{gather*}
-\Delta_{p}u=\lambda|u|^{p-2}u+a(x)|u|^{q-2}u\quad \text{in }\Omega, \label{1a} \\
|\nabla u|^{p-2}\frac{\partial u}{\partial \eta}+b(x)|u|^{p-2}u=\mu
\rho(x)|u|^{r-2}u\quad \text{on }\partial \Omega, \label{1b}
\end{gather*}
where $\Delta_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$,
$1<p<N$, denotes the $p$-Laplace operator,
$\frac{\partial u}{\partial \eta}(x)$ denotes
the outward unit normal at $x\in \partial \Omega$, $\lambda$, $\mu$ are
parameters, $\mu>0$, $a:\Omega \to \mathbb{R}$,
$b,\rho:\partial \Omega \to \mathbb{R} $ are essentially bounded functions,
with $b(x)\geq0$ and $mx\in \partial \Omega:b(\cdot)>0\}>0$. Restrictions on
$q,r$ are given in the subsequent sections. With respect to the parameter $\mu$,
 we notice that its role is
crucial in the critical case examined in Section 3.

Quasilinear problems of the form $-\Delta_{p}u=f(x,u)$ with Dirichlet boundary
conditions have received considerable attention;
see \cite{Bar-Liu,Dra-Tac,M-M-NSP,Per-Sil,Tiw}. This equation with
Neumann boundary conditions (i.e. $b(\cdot)\equiv 0$ and $\rho(\cdot)\equiv0$) and
$a(\cdot)$ being a constant has been studied in \cite{B-D-Hu}, where existence of
solutions has been provided for $\lambda \in(0,\lambda^{\ast})$, for a suitable
$\lambda^{\ast}>0$. The same authors in \cite{B-D-H} provide positive
solutions to the aforementioned problem but with a critical term added to the
right hand side of \eqref{1a}. In \cite{Bon-Ro} the existence of solutions is
proved for \eqref{1a}-\eqref{1b} when $\lambda$ appears on the boundary
condition, $a(\cdot)\equiv0$, and $r$ can be subcritical, critical or
supercritical. Multiplicity of solutions is examined in \cite{NSP-Rad} where
the right hand side of \eqref{1a} is a real Carath\'{e}odory function
$f(x,u,\lambda)$ defined on $\Omega \times \mathbb{R}\times(0,+\infty)$
and the boundary condition is Neumann. Multiplicity of
solutions is also proved in \cite{NSP-Radu} for $\lambda>\lambda_2$, for
$\lambda_2$ being the second eigenvalue of the $p$-Laplacian operator with
Robin boundary conditions, while in \cite{NSP-Radul} existence of positive
solutions is shown for $\lambda<\lambda_1$. Existence of solutions depending
on the Fu\u{c}ik spectrum of the p-Laplace operator is examined in
\cite{Wink}. When $\Omega$ is an exterior domain, existence and nonexistence
of solutions is examined in \cite{Fil-Puc-Rad}. In case the potential is
nonsmooth we refer to \cite{Gas-NSP}. The fibering method, attributed to
Pohozaev, is useful when the right hand sides of the equation and the boundary
condition are power-like, see \cite{Dr-Po}, \cite{Poh}. For systems of
equations the interested reader may see \cite{Boz-Mit}.

Our aim in this work is to provide existence results concerning positive
solutions to \eqref{1a}-\eqref{1b} when $q$ is either subcritical or
critical, $r$ is subcritical and $\lambda \leq \lambda_1$, where $\lambda_1$
is the first eigenvalue of the associated eigenvalue problem. When the
exponents are subcritical, our proofs rely on the fibering method and the
mountain pass theorem developed in Ambrosetti-Rabinowitz \cite{Ambr-Rab},
while in the case of $q$ being critical we use the concentration-compactness
principle of Lions \cite{Lions1, Lions2}. A useful survey of results
concerning the mountain pass theorem is provided in \cite{Puc-Rad}.

As usual $X:=W^{1,p}(\Omega)$ is equipped  with the norm
\[
\| u\| _{1,p}=\Big(  \int _{\Omega}| \nabla
u| ^{p}dx+\int _{\Omega}| u| ^{p}dx\Big)  ^{1/p}.
\]
The action functional $I(\cdot)$ corresponding to problem \eqref{1a}-\eqref{1b} is
defined on $X$  by
\[
I_{\lambda}(u)=\frac{1}{p}\Big[  \int _{\Omega}| \nabla
u| ^{p}dx-\lambda \int _{\Omega}| u|
^{p}dx+\int _{\partial \Omega}b(x)| u| ^{p}
d\sigma \Big]  -\frac{1}{q}A(u)-\frac{\mu}{r}P(u),
\]
where $P(u):=\int _{\partial \Omega}\rho(x)| u|^{r}d\sigma$ and
$A(u):=\int _{\Omega}a(x)| u|^{q}dx$.

Consider the eigenvalue problem
\begin{gather}
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\lambda|u|^{p-2}u\quad
\text{in }\Omega, \label{ei1} \\
|\nabla u|^{p-2}\frac{\partial u}{\partial \eta}+b(x)|u|^{p-2}u=0\quad
\text{on }\partial \Omega. \label{ei2}
\end{gather}
It is known that the smallest eigenvalue $\lambda_1$ is isolated and
positive with corresponding normalized eigenvector $u_1\in C^{1}(\Omega)$
(that is, $\| u_1\| =1$) which is positive in $\Omega$,
\cite[Lemma 5.3]{Le}. Furthermore,
\begin{equation}
\lambda_1=\inf \Big\{  \frac{\int _{\Omega}| \nabla
u| ^{p}dx+\int _{\partial \Omega}b(x)| u|
^{p}d\sigma}{\int _{\Omega}| u| ^{p}dx}:u\in
W^{1,p}(\Omega)\backslash \{0\} \Big\}  . \label{eigen}
\end{equation}

\section{Subcritical exponents}

 In what follows we assume that $1<q<p^{\ast}:=\frac{Np}{N-p}$ and
$1<r<\widehat{p}^{\ast}:=\frac{p(N-1)}{N-p}$.

\subsection{Existence of solutions when $\lambda<\lambda_1$}

\begin{lemma} \label{lem1}
The expression
\[
[ u]=\left[  \int _{\Omega}| \nabla u|
^{p}dx-\lambda \int _{\Omega}| u| ^{p}
dx+\int _{\partial \Omega}b(x)| u| ^{p}
d\sigma \right]  ^{1/p}
\]
is a norm on $X$ and is equivalent to $\|\cdot\| _{1,p}$.
\end{lemma}

 The proof of the above lemma follows from \cite[Proposition 2]{B-D-Hu}.

Depending on the relative ordering of the exponents $p,q,r$, we
distinguish the following four cases.
\smallskip

\noindent\textbf{Case 1.} $p<\min \{q,r\}$.
We assume
\begin{itemize}

\item[(H1)] $a(\cdot)\geq0$ and $m\{x\in \Omega:a(\cdot)>0\}>0$.

\item[(H2)] $\rho(\cdot)\geq0$ on $\partial \Omega$ and $m\{x\in \partial \Omega
:\rho(\cdot)>0\}>0$.
\end{itemize}
Let $Y$ be an Banach space and $\Sigma:=\{A\subseteq X\backslash \{0\}: A
\text{ is closed and }A=-A\}$. The genus of a set $A\in \Sigma$ is defined by
\[
\gamma(A):=\min \{n\in \mathbb{N} :\exists \varphi \in C(A,
\mathbb{R} ^{n}\backslash \{0\})\text{ with }\varphi(x)=-\varphi(-x)\}.
\]


\begin{theorem} \label{Ambr-Rab}
Suppose that $I:Y\to\mathbb{R} $ is an even $C^{1}(Y,\mathbb{R})$ function such
that:
\begin{itemize}
\item[(i)] I satisfies the Palais-Smale condition.

\item[(ii)] $I(u)>0$ if $0<\|u\|<r$ and $I(u)\geq c>0$ if $\|u\|=r$, for some $r>0$.

\item[(iii)] There exists a subspace $Y_{m}\subseteq E$ of dimension $m$ and a
compact subset $A_{m}\subseteq Y_{m}$ with $I<0$ on $A_{m}$ such that $0$ lies
in a bounded component (in $Y_{m}$) of $Y_{m}\backslash A$.
\end{itemize}
Let
$\Gamma:=\{h\in C(Y,Y):h(0)=0, h \text{ is an odd homeomorhism, }
I(h(B_1))\geq0\}$,
$K_{m}:=\{K\subseteq Y: K \text{ is compact,}K=-K, \gamma(K\cap h(\partial
B_1))\geq m\text{ for every }h\in \Gamma \}$,
 where $B_1$ denotes the unit ball of $Y$.
Then
\[
c_{m}:=\underset{K\in K_{m}}{\inf}\underset{u\in K}{\max}I(u)
\]
is a critical value of $I$ with $0<c<c_{m}\leq c_{m+1}<+\infty$. Furthermore,
if $c_{m}=c_{m+1}=\dots =c_{m+n}$, then $\gamma(K_{c_{m}})\geq n+1$, where
$K_{c_{m}}:=\{u\in X:I'(u)=0$, $I(u)=c_{m}\}$.
\end{theorem}

For the proof of the above  Theorem we refer the reader to \cite{Ambr-Rab}.

\begin{theorem} \label{thm3}
Assume that {\rm (H1)} and {\rm (H2)} hold. Then  \eqref{1a}-\eqref{1b} admits
infinitely many solutions.
\end{theorem}

\begin{proof}
We will show first that $I$ satisfies the Palais-Smale condition. So let
$\{u_n\}_{n\in \mathbb{N} }$ be a sequence in $X$ such that
$|I(u_n)|\leq M$ and $I' (u_n)\to0$. For $k\in(p$, $\min \{q,r\})$ we have
\[
-M+o_n(1)[u_n]\leq I(u_n)-\frac{1}{k}I'(u_n)u_n\leq
M+o_n(1)[u_n],
\]
and so
\begin{equation}
\begin{aligned}
-M+o_n(1)[u_n]
&\leq \big(\frac{1}{p}-\frac{1}{k}\big)  [u_n
]^{p}+\big(  \frac{1}{k}-\frac{1}{q}\big)  A(u_n)+\mu \big(  \frac{1}
{k}-\frac{1}{r}\big)  P(u_n) \\
&\leq M+o_n(1)[u_n],
\end{aligned} \label{bound}
\end{equation}
which implies $\{u_n\}_{n\in \mathbb{N}}$ is bounded in $X$.
Without loss of generality we may assume that
$u_n\to u$ weakly in $X$ and strongly in $L^{p}(\Omega)$,
$L^{q}(\Omega)$, $L^{p}(\partial \Omega)$ and
$L^{r}(\partial \Omega)$.
Therefore,
\begin{gather}
\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla(u_n-u)dx\to0,  \\
\int_{\Omega}a(|u_n|^{q-2}u_n-|u|^{q-2}u)(u_n-u)dx\to0,
\label{PS1} \\
\int_{\partial \Omega}b(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u)d\sigma
\to0, \label{PS2} \\
\int_{\partial \Omega}\rho(|u_n|^{r-2}u_n-|u|^{r-2}u)(u_n
-u)d\sigma \to 0 \label{PS3}
\end{gather}
as $n\to+\infty$. Since $I'(u_n)\to0$,
\eqref{PS1}-\eqref{PS3} imply that
\begin{equation}
\langle I'(u_n)-I'(u),u_n-u\rangle
\to 0 \quad \text{as }n\to+\infty. \label{conv}
\end{equation}
Thus,
\begin{align*}
&\int_{\Omega}\big[|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla u\big]
 (\nabla u_n-\nabla u) \,dx\\
&-\lambda \int_{\Omega}(|u_n|^{p-2}u_n -|u|^{p-2}u)(u_n-u)dx \\
&+\int_{\partial \Omega}b(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u)d\sigma\\
&-\int_{\Omega}a(|u_n|^{q-2}u_n-|u|^{q-2}u)(u_n-u)dx \\
&-\mu \int_{\partial \Omega}\rho(|u_n|^{r-2}u_n-|u|^{r-2}u)(u_n
-u)d\sigma \to0 \quad \text{as }n\to+\infty.
\end{align*}
Consequently,
\[
\int_{\Omega}[|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla
u](\nabla u_n-\nabla u)dx\to0\quad \text{as }n\to+\infty.
\]
As a consequence of Holder's inequality we have
\begin{equation}
\begin{aligned}
&\int_{\Omega}[|\nabla u_n|^{p-2}\nabla u_n-|\nabla u|^{p-2}\nabla
u](\nabla u_n-\nabla u)dx \\
&\geq \Big[  \Big(  \int_{\Omega}|\nabla u_n|^{p}dx\Big)  ^{(p-1)/p}
-\Big(  \int_{\Omega}|\nabla u|^{p}dx\Big)  ^{(p-1)/p}\Big] \\
&\quad \times \Big[  \Big(  \int_{\Omega}|\nabla u_n|^{p}dx\Big)  ^{1/p}
-\Big(\int_{\Omega}|\nabla u|^{p}dx\Big)  ^{1/p}\Big]  .
\end{aligned} \label{Holder}
\end{equation}
Therefore, $\| u_n\| _{1,p}\to \|u\| _{1,p}$. The uniform convexity of
 $X$ implies that $u_n \to u$ in $X$. Note that
\[
I(u)=\frac{1}{p}[u]^{p}-\frac{1}{q}A(u)-\frac{\mu}{r}P(u)\geq \frac{1}
{p}[u]^{p}-c_1[u]^{q}-c_2[u]^{r},
\]
by the Sobolev embedding, and so $I(u)>0$ for $\|u\|=\rho$ and
$I(u)\geq c_3>0$ for $\|u\|<\rho$, provided $\rho$ is sufficiently small.
 Suppose that
$\{  X_n\}  _{n\in\mathbb{N}}$ is a sequence of subspaces of $X$
with dimension $\dim(X_n)=n$ such that
$\frac{\partial u}{\partial \eta}\neq0$ if $u\in X_n\backslash \{0\}$. Then,
for $u\in B_1^{n}:=\{v\in X_n:[v]=1\}$ and $\zeta$ sufficiently large
\[
I(\zeta u)=\frac{\zeta^{p}}{p}[u]^{p}-\frac{\zeta^{q}}{q}A(u)-\frac{\mu
\zeta^{r}}{r}P(u)<\frac{\zeta^{p}}{p}-\frac{\zeta^{q}}{q}\underset{u\in
B_1^{n}}{\min}A(u)-\frac{\mu \zeta^{r}}{r}\underset{u\in B_1^{n}}{\min
}P(u)<0.
\]
We can now apply Theorem \ref{Ambr-Rab} to complete the proof.
\end{proof}

\noindent\textbf{Case 2.} $1<r<q<p$
We assume
\begin{itemize}
\item[(H1')] $a(\cdot)\geq0$ or $a(\cdot)\leq0$ in $\Omega$ and
$m\{x\in \Omega:a(\cdot)\neq 0\}>0$.
\end{itemize}

\begin{theorem} \label{thm4}
If $1<r<q<p$ and {\rm (H1'), H(2)} hold, then \eqref{1a}-\eqref{1b}
admits a positive solution.
\end{theorem}

\begin{proof}
Assume first that $a(\cdot)\geq0$. We consider the open set
$Z:=\{u\in X:A(u)>0\text{ or }P(u)>0\}$.
\end{proof}

For $u\in Z$, $t\geq0$, one forms
\[
I(tu)=\frac{t^{p}}{p}H_{\lambda}(u)-\frac{t^{q}}{q}A(u)-\frac{\mu t^{r}}
{r}P(u)\text{,}
\]
where $H_{\lambda}(u):=[ u]  ^{p}$.

For $t>0$, let
\[
I_t(tu)=t^{p-1}H_{\lambda}(u)-t^{q-1}A(u)-\mu t^{r-1}P(u).
\]
For critical points, we obtain
\begin{equation}
t^{p}H_{\lambda}(u)-t^{q}A(u)-\mu t^{r}P(u)=0, \label{r(u)}
\end{equation}
that has always a unique solution $t=t(u)$. Let
$S_{\lambda}=Z\cap \{u\in X:H_{\lambda}(u)=1\}$. We notice that
$\{t(u):u\in S_{\lambda}\}$ is bounded.

For $u\in Z$, we define $\widehat{I}(u):=I(t(u)u)$. In view of \eqref{r(u)},
\begin{equation}
\widehat{I}(u)=\big(  \frac{1}{p}-\frac{1}{q}\big)  t(u)^{p}H_{\lambda
}(u)+\big(  \frac{1}{q}-\frac{1}{r}\big)  \mu t(u)^{r}P(u)<0.
\label{a2}
\end{equation}
Notice that $\widehat{I}(\cdot)$ is bounded below in $S_{\lambda}$. Let
$M=\inf_{u\in S_{\lambda}} \widehat{I}(u)$.
 Let $\{u_n\}  _{n\in\mathbb{N}}\subseteq S_{\lambda}$
be a minimizing sequence for
$\widehat{I}/S_{\lambda}$. Since $\{  u_n\}  _{n\in\mathbb{N}}$ is bounded in
 $X$, we may assume that $u_n\rightharpoonup u$ in $X$. At
the same time, $t(u_n)\to \widehat{t}$ in $\mathbb{R}$. Thus
$t(u_n)u_n\rightharpoonup \widehat{t}u$ in $X$. By weak lower
semicontinuity of $I(\cdot)$, we have
\[
I(\widehat{t}u)\leq \liminf_{n\to+\infty} I(t(u_n)u_n)=M.
\]
Thus $\widehat{t}u\neq0$. Because of the corresponding compact Sobolev
embeddings, $A(u_n)\to A(u)$ and $P(u_n)\to P(u)$.
Exploiting \eqref{r(u)} for each $n$, one has
\[
t(u_n)^{p-r}=t(u_n)^{q-r}A(u_n)+\mu P(u_n).
\]
Letting $n\to+\infty$, we obtain
\begin{equation}
\widehat{t}^{p-r}=\widehat{t}^{q-r}A(u)+\mu P(u). \label{a3}
\end{equation}
Since $\widehat{t}>0$, either $A(u)>0$ or $P(u)>0$, thus $u\in Z$, and $t(u)$
is well defined. The weak lower semicontinuity of the norm applies to give
\begin{equation}
\widehat{t}^{p-r}H_{\lambda}(u)\leq \widehat{t}^{q-r}A(u)+\mu P(u) \label{a4}
\end{equation}
or
\[
H_{\lambda}(u)\leq \widehat{t}^{q-p}A(u)+\widehat{t}^{r-p}\mu P(u)\text{.}
\]
At the same time,
\[
H_{\lambda}(u)=t(u)^{q-p}A(u)+t(u)^{r-p}\mu P(u)\text{.}
\]
Since the map $f(t)=t^{q-p}A(u)+t^{r-p}\mu P(u),t>0$ is strictly decreasing,
the last two relations imply $\widehat{t}\leq$ $t(u)$. Let us assume that
$\widehat{t}<$ $t(u)$. We set $F(y):=I(yu),y\geq0$.
For $y\in [\widehat{t}, t(u)]$, one has
$F'(y)=y^{p-1}H_{\lambda} (u)-y^{q-1}A(u)-y^{r-1}\mu P(u)=y^{r-1}
[  y^{p-r}H_{\lambda} (u)-y^{q-r}A(u)-\mu P(u)]  $,
which is negative everywhere but at
$y=t(u)$, since \eqref{r(u)} has a unique solution. Thus $F(y)$ is strictly
decreasing in $[\widehat{t}$, $t(u)]$, so
\[
I(t(u)u)<I(\widehat{t}u)\leq M.
\]
We take $k>0$ such that $ku\in S_{\lambda}$ (actually, combining \eqref{a3}
and \eqref{a4} one sees that $k\geq1$). We have
\[
t(ku)^{p}=t(ku)^{q}A(ku)+t(ku)^{r}\mu P(ku)
\]
or
\[
\big(  kt(ku)\big)  ^{p}H_{\lambda}(u)=\big(  kt(ku)\big)
^{q}A(u)+\big(  kt(ku)\big)  ^{r}\mu P(u),
\]
thus $kt(ku)=t(u)$. Then
\[
I(t(ku)ku)=I(t(u)u)<I(\widehat{t}u)\leq M,
\]
which is a contradiction. Thus $\widehat{t}=t(u)$,
$H_{\lambda}(u)=1$, and
$t(u)u$ is a nontrivial solution of our problem. Since
$|t(u)u| $ will also be a minimizer, by Harnack's inequality we may
assume that $t(u)u$ is positive.

If $a(\cdot)\leq0$ in $\Omega$, we define $\hat{Z}:=\{u\in X:$ $P(u)>0\}$.
It is clear that \eqref{r(u)} has a unique positive solution and $M<0$. Furthermore,
since the limit $u$ of a minimizing sequence satisfies \eqref{a3}, we have
that $P(u)>0$. Thus $u\in \hat{Z}\ $and $| t(u)u| $ is a
positive solution of \eqref{1a},\ref{1b}.
\smallskip

\noindent\textbf{Case 3.} $1<q<r<p$.

\begin{theorem} \label{thm5}
Suppose that $1<q<r<p$ and {\rm (H1), (H2)} hold. Then 
 \eqref{1a}-\eqref{1b} admits a positive solution.
\end{theorem}

\begin{proof}
Note that for every $u\in Z$ \eqref{r(u)} has a unique positive solution
$t:=t(u)$. Furthermore, the set $\{t(u):u\in Z\}$ is bounded. Let $\widehat
{I}(u):=I(t(u)u)$. Then, in view of \eqref{r(u)},
\begin{equation}
\begin{aligned}
\widehat{I}(u)
&=t^{p}\big(  \frac{1}{p}-\frac{1}{q}\big)  
+\mu t^{r}\big(\frac{1}{q}-\frac{1}{r}\big)  P(u)\\
&\leq t^{p}\big(  \frac{1}{p}-\frac{1}{q}\big) 
 +t^{p}\big(  \frac{1}{q}-\frac{1}{r}\big)  
=t^{p}\big(  \frac{1}{p}-\frac{1}{r}\big)  <0.
\end{aligned} \label{neg}
\end{equation}
We can now proceed as in case 2.
\end{proof}
\smallskip

\noindent\textbf{Case 4.} $1<r<p<q$.
We assumpe
\begin{itemize}
\item[(H3)] $a(\cdot)\leq0$ and $m\{x\in \Omega:a(\cdot)<0\}>0$.
\end{itemize}

\begin{theorem} \label{thm6}
If $1<r<p<q$ and {\rm (H2), (H3)} hold, then \eqref{1a}-\eqref{1b}
admits a positive solution.
\end{theorem}

\begin{proof}
Once more, \eqref{r(u)} has a unique positive solution $t:=t(u)$ for every
$u\in \hat{Z}$. Furthermore, the set $\{t(u):u\in \hat{Z}\}$ is bounded and
$\widehat{I}(u)<0\ $in $\hat{Z}$. We proceed as in case 2.
\end{proof}

\subsubsection{Existence of solutions when $\lambda=\lambda_1$}

In this section we  assume that (H2) and (H3) hold.
\smallskip

\noindent\textbf{Case 5.} $1<r<q<p$.

\begin{theorem} \label{thm7}
Assume that $1<r<q<p$ and {\rm (H2), (H3)} hold. then  \eqref{1a}-\eqref{1b} 
admits a positive solution.
\end{theorem}

\begin{proof}
Let $H_{\lambda}^{P}(u):=[u]^{p}-A(u)$ and $S_{\lambda}^{P}:=\{u\in X:P(u)>0$
and $H_{\lambda}^{P}(u)=1\}$. If $u\in S_{\lambda}^{P}$, then \eqref{r(u)} has
a unique solution $t(u)\ $with $\widehat{I}(u)<0$. 
Define $M=\inf_{u\in S_{\lambda}^{P}} \widehat{I}(u)$ and assume 
that $u_n\in S_{\lambda }^{P}$ is such that $\widehat{I}(u_n)\to M$. We claim that
$\|u_n\|_{1,p}$, $n\in \mathbb{N}$, is bounded. Indeed, let us assume that 
it is not, that is, $\|u_n\|_{1,p}\to+\infty$. Define $z_n:=\frac{u_n}{d_n}$, where
$d_n=\|u_n\|_{1,p}$. Then
\[
d_n^{p}[z_n]^{p}-d_n^{q}A(z_n)=1.
\]
Consequently,
\begin{equation}
[ z_n]^{p}\leq \frac{1}{d_n^{p}}\to0,\quad 
0\leq-A(z_n)\leq \frac{1}{d_n^{q}}\to0. \label{eigen0}
\end{equation}
Thus
\begin{equation}
\lambda_1\int _{\Omega}| z_n| ^{p} dx\to1. \label{nonzero}
\end{equation}
Since $\|z_n\|_{1,p}=1$, we may assume that $z_n\to z$ weakly in
$X$. Therefore, \eqref{nonzero} implies that
\begin{equation}
\lambda_1\int _{\Omega}| z| ^{p}dx=1, \label{eigen1}
\end{equation}
and so $z\neq0$. By \eqref{nonzero} and \eqref{eigen1} we see that $[z]=0$,
that is, $z$ is an eigenvector corresponding to $\lambda_1$. On the other
hand, since $A(z_n)\to A(z)$, \eqref{nonzero} yields $A(z)=0$,
contradicting the fact $z>0$ in $\Omega$. Thus, $\|u_n\|_{1,p}$, 
$n\in\mathbb{N}$, is indeed bounded. So we may assume that $u_n\to u$ weakly in
$X$. Note that, for an infinite number of $n's$, either 
$[u_n]^{p}\geq \frac{1}{2}$, or $-A(u_n)\geq \frac{1}{2}$. In either case,
\eqref{r(u)} implies that $r(u_n)$ is bounded. Since \eqref{r(u)} implies
that $P(u)>0$, we see that $u\in S_{\lambda}^{P}$. We can now proceed as in
case 1 to get a solution.
\end{proof}

\noindent\textbf{Case 6.} $1<r<p<q$.

\begin{theorem} \label{thm8}
Assume that $1<r<p<q$ and {\rm (H2), (H3)} hold. Then  \eqref{1a}-\eqref{1b} 
admits a positive solution.
\end{theorem}

\begin{proof}
We use the  inequality
\begin{align*}
\widehat{I}(u)
&=t^{p}\big(  \frac{1}{p}-\frac{1}{q}\big)  
+\mu t^{r}\big( \frac{1}{q}-\frac{1}{r}\big)  P(u) \\
&\leq \mu t^{r}\big(  \frac{1}{p}-\frac{1}{q}\big)  P(u)
+\mu t^{r}\big(\frac{1}{q}-\frac{1}{r}\big)  P(u)\\
&=\mu t^{r}\big(  \frac{1}{p}-\frac{1}{r}\big)  P(u)<0,
\end{align*}
to show that $\underset{u\in S_{\lambda}^{P}}{\inf}\widehat{I}(u)<0$ and by
following the same steps as in case 5 we obtain a positive solution.
\end{proof}

\section{The critical case $q=p^{\ast}$}

In this section we study the critical problem $q=p^{\ast}:=\frac{Np}{N-p}$.
with $p<r<\frac{p(N-1)}{N-p.}$ and $\lambda<\lambda_1$. The proof follows
closely the lines of \cite[Theorem 1.4]{F-R}. Since the embedding
$X\hookrightarrow L^{p^{\ast}}(\Omega)$ is no longer compact we do not expect
that the Palais-Smale condition holds. So we prove a local Palais-Smale
condition which is true if $I(\cdot)$ lies below a certain energy value.

In what follows we assume that $a(\cdot)\equiv1$ and (H2) holds.
Consider the problem
\begin{gather}
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\lambda|u|^{p-2}u+|u|^{q-2}
u\quad \text{in }\Omega, \label{crit1} \\
|\nabla u|^{p-2}\frac{\partial u}{\partial \eta}+b(x)|u|^{p-2}u=\mu
\rho(x)h(u)\quad \text{on }\partial \Omega. \label{crit2}
\end{gather}
Let
\[
S=\inf_{u\in D^{1,p}(\mathbb{R}^{N})\backslash \{0\}} 
\frac{\int_{\Omega}|\nabla u|^{p}dx}{\int_{\Omega}| u| ^{p^{\ast}}dx},
\]
be the best Sobolev constant, where 
$u\in D^{1,p}(\mathbb{R}^{N})$ is the completion of $C_0^{\infty}(\mathbb{R}^{N})$
under the gradient norm.

\begin{lemma} \label{lem2}
Suppose that $\{u_n\}_{n\in\mathbb{N}}$ is a sequence in $X$ satisfying 
the Palais-Smale condition with energy level $c<\frac{1}{N}S^{\frac{N}{p}}$, that is
\[
I(u_n)\to c\quad\text{and}\quad I'(u_n)\to0.
\]
Then $\{u_n\}_{n\in\mathbb{N}}$ has a convergent subsequence in $X$.
\end{lemma}

\begin{proof}
The boundedness of $\{u_n\}_{n\in \mathbb{N}}$ is a consequence of \eqref{bound}. 
Thus, $\{u_n\}_{n\in\mathbb{N}}$ has a subsequence, still denoted by 
$\{u_n\}_{n\in\mathbb{N}}$, which converges weakly to $u\in X$. 
By \cite[Lemma 3.6]{Medeir} there
exists a set of points $\{x_j\}_{j\in J}\subseteq \overline{\Omega}$, $J$ at
most countable, and nonnegative numbers $\mu_j,\nu_j$ satisfying
\begin{gather*}
|\nabla u_n|^{p}\to \mu \geq|\nabla u|^{p}
+\sum _{j\in J} \mu_j\delta_{x_j}\,, \\
|u_n|^{p^{\ast}}\to \nu=|u|^{p^{\ast}}
+\sum _{j\in J} \nu_j\delta_{x_j}\,,\\
S\nu_j^{\frac{p}{p^{\ast}}}\leq \mu_j\quad \text{if }x_j\in \Omega, \\
S\nu_j^{\frac{p}{p^{\ast}}}\leq2^{\frac{p}{N}}\mu_j\text{ if }x_j
\in \partial \Omega.
\end{gather*}
Let $k\in\mathbb{N}$, $\varepsilon>0$ and take $\varphi \in C^{\infty}(\Omega)$ 
such that
\[
\varphi \equiv1\text{ in }B(x_k,\varepsilon),\quad \varphi \equiv0\quad
\text{in }X\backslash B(x_k,2\varepsilon),\quad|\nabla \varphi|\leq \frac
{2}{\varepsilon}.
\]
Since $I'(u_n)(\varphi u_n)\to0$ an $n\to+\infty$,
we obtain
\begin{align*}
&\lim_{n\to+\infty} \Big[  \int_{\Omega}|\nabla u_n
|^{p-2}\nabla u_n\nabla \varphi u_ndx+\int_{\Omega}|\nabla u_n
|^{p}\varphi dx\Big]  \\
&=\lambda \int_{\Omega}|u|^{p}\varphi dx
-\int_{\partial \Omega}b(x)|u|^{p}\varphi d\sigma
+\lim_{n\to +\infty} \int_{\Omega}|u_n|^{p^{\ast}}\varphi dx
+\mu \int_{\partial \Omega}\rho(x)|u|^{r}\varphi d\sigma \\
&=\lambda \int_{\Omega}|u|^{p}\varphi dx-\int_{\partial \Omega}|u|^{p}\varphi
d\sigma+\int_{\Omega}\varphi d\nu+\mu \int_{\partial \Omega}\rho(x)|u|^{r}
\varphi d\sigma.
\end{align*}
Note that, by the Holder inequality,
\begin{align*}
&\lim_{n\to+\infty} \big| \int_{\Omega}|\nabla
u_n|^{p-2}\nabla u_n\nabla \varphi u_n\,dx\big| \\
& \leq \lim_{n\to+\infty} \Big(  \int_{\Omega}|u_n
|^{p}\varphi dx\Big)  ^{\frac{p-1}{p}}
\lim_{n\to+\infty} \Big(  \int_{\Omega}|\nabla \varphi|^{p}|u_n|^{p}dx\Big)  ^{1/p}\\
& \leq C\Big(\int_{B(x_k,2\varepsilon)\cap \Omega}|\nabla \varphi|^{p}
|u|^{p}dx\Big)  ^{1/p} \\
&\leq C\Big(  \int_{B(x_k,2\varepsilon)\cap \Omega}|\nabla \varphi
|^{N}dx\Big)  ^{1/N}\Big(  \int_{B(x_k,2\varepsilon)\cap \Omega
}|u|^{p^{\ast}}dx\Big)  ^{1/p^{\ast}} \\
&\leq C'\int_{B(x_k,2\varepsilon)\cap \Omega}|u|^{p^{\ast}
}dx\to0 \quad \text{as }\varepsilon \to0,
\end{align*}
and so
\begin{align*}
&\lim_{\varepsilon \to0} \Big[  \int_{\Omega}\varphi d\mu
-\lambda \int_{\Omega}|u|^{p}\varphi dx
+\int_{\partial \Omega} b(x)|u|^{p}\varphi d\sigma
-\int_{\Omega}\varphi d\nu
 -\mu \int_{\partial \Omega}\rho(x)|u|^{r}\varphi d\sigma \Big] \\
&=\mu_k-\nu_k=0.
\end{align*}
Consequently, $S\nu_k^{\frac{p}{p^{\ast}}}\leq \nu_k$ if
$x_k\in \Omega$ or $2^{-\frac{p}{N}}S\nu_k^{\frac{p}{p^{\ast}}}\leq \nu_k$
if $x_k \in \partial \Omega$, implying that $S^{\frac{N}{p}}\leq \nu_k$ if
$x_k \in \Omega$ or $\frac{1}{2}S^{\frac{N}{p}}\leq \nu_k$ if 
$x_k\in \partial \Omega$. On the other hand,
\begin{align*}
c&=\lim_{n\to+\infty} I(u_n)
=\lim_{n\to+\infty } I(u_n)-\underset{n\to+\infty}{\lim}\frac{1}{p}I'(u_n)(u_n)\\
&=\big(  \frac{1}{p}-\frac{1}{p^{\ast}}\big) \int_{\Omega}|u|^{p^{\ast}}
+\big(  \frac{1}{p}-\frac{1}{p^{\ast}}\big)  \int_{\Omega}
\sum _{j\in J} \nu_j\delta_{x_j}
+\mu \big(  \frac{1}{p}-\frac{1}{r}\big)
\int_{\partial \Omega}\rho(x)|u|^{r}d\sigma \\
&\geq \big(  \frac{1}{p}-\frac{1}{p^{\ast}}\big)  \nu_k
=\frac{1}{N} S^{N/p}.
\end{align*}
Thus $\nu_k=0$ for every $k\in J$, implying that 
$\int_{\Omega} |u_n|^{p^{\ast}}dx\to \int_{\Omega}|u|^{p^{\ast}}dx$. The result
follows by exploiting the continuity of the inverse $p$-Laplace operator.
\end{proof}

\begin{theorem} \label{thm10}
There exists $\mu_0>0$ such that for $\mu \geq \mu_0$ problem \eqref{1a}-\eqref{1b}
admits a solution.
\end{theorem}

\begin{proof}
We will first verify the requirements for the mountain pass theorem. By the
Sobolev embedding and trace theorems we see that
\begin{align*}
I(u)&=\frac{1}{p}[u]^{p}-\frac{1}{p^{\ast}}A(u)-\frac{\mu}{r}P(u) \\
&\geq \frac{1}{p}[u]^{p}-C_1[u]^{p^{\ast}}-C_2[u]^{r},
\end{align*}
for some $C_1,C_2>0,$and so for a sufficiently small positive number
$\beta$ there exists $a>0$ such that $I(u)>a>0$ for $[u]=\beta$. We now take
$v\in X\backslash \{0\}$. It is easy to see that 
$\lim_{s\to +\infty} I(sv)=-\infty$. Thus, $I(s_0v)<0$ for sufficiently large
$s_0$.

Let $c:=\inf_{\gamma \in \Gamma} \sup_{t\in [0,1]} I(\gamma(t))$, where 
$\Gamma:=\{ \gamma \in C([0,1],X):\gamma(0)=0,\gamma (1)=s_0v\}$. 
We will show that $c<\frac{1}{N}S^{\frac{N}{p}}$ for large
enough $\mu$. Take $z\in X$ such that $\|z\|_{p^{\ast}}=1$. The maximum value
of $\eta \to I(\eta z)$, $\eta>0$, is assumed at the point $\eta_{\mu}$
satisfying $\frac{d}{d\eta}I(\eta_{\mu}z)=0$, that is
\begin{equation}
\eta_{\mu}^{p}[z]^{p}=\eta_{\mu}^{p^{\ast}}\|z\|_{p^{\ast}}^{p^{\ast}}+\mu
\eta_{\mu}^{r}P(z)=\eta_{\mu}^{p^{\ast}}+\mu \eta_{\mu}^{r}P(z). \label{infin}
\end{equation}
Therefore,
\[
\eta_{\mu}\leq [ z]^{\frac{p}{p^{\ast}-p}},
\]
which, in view of \eqref{infin}, yields $\lim_{\mu \to+\infty} \eta_{\mu}=0$. 
On the other hand,
\[
I(\eta_{\mu}z)=\eta_{\mu}^{p}\big(\frac{1}{p}-\frac{1}{p^{\ast}}\big)
[z]^{p}+\mu \eta_{\mu}^{r}\big(  \frac{1}{p^{\ast}}-\frac{1}{r}\big)
P(z)\leq \eta_{\mu}^{p}\big(  \frac{1}{p}-\frac{1}{p^{\ast}}\big)  [z]^{p},
\]
implying that $\lim_{\mu \to+\infty} I(\eta_{\mu}z)=0$. Thus,
for large enough $\mu$, say $\mu \geq \mu_0$, 
$I(\eta_{\mu}z)<\frac{1} {N}S^{\frac{N}{p}}$. By Lemma \ref{lem2}, 
$I(\cdot)$ satisfies the Palais-Smale
condition and the mountain pass theorem provides a solution.
\end{proof}

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\end{document}