\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 15, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/15\hfil Lack of coercivity for N-Laplace equation]
{Lack of coercivity for N-Laplace equation \\ with critical
exponential nonlinearities \\ in a bounded domain}

\author[S. Goyal, K. Sreenadh \hfilneg]
{Sarika Goyal, Konijeti Sreenadh}  % in alphabetical order

\address{Sarika Goyal \newline
Department of Mathematics,
Indian Institute of Technology Delhi,
Hauz Khaz,  New Delhi-16, India}
\email{sarika1.iitd@gmail.com}

\address{Konijeti Sreenadh \newline
Department of Mathematics,
Indian Institute of Technology Delhi,
Hauz Khaz,  New Delhi-16, India}
\email{sreenadh@maths.iitd.ac.in}

\thanks{Submitted May 23, 2013. Published January 8, 2014.}
\subjclass[2000]{35J35, 35J60, 35J92}
\keywords{Quasilinear problem; critical exponent;
Trudinger-Moser embedding; \hfill\break\indent sign-changing weight function}

\begin{abstract}
 In this article, we study the existence and multiplicity of
 non-negative solutions of the  $N$-Laplacian equation
 \begin{gather*}
 -\Delta_N u+V(x)|u|^{N-2}u = \lambda h(x)|u|^{q-1}u+ u|u|^{p} e^{|u|^{\beta}} \quad
 \text{in }\Omega \\
 u \geq 0 \quad \text{in } \Omega,\quad u\in W^{1,N}_0(\Omega),\\
 u =0 \quad \text{on }  \partial \Omega
 \end{gather*}
 where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 2$,
 $0< q<N-1 <p+1$, $\beta\in (1,\frac{N}{N-1}]$ and $\lambda>0$. By
 minimization on a suitable subset of the Nehari manifold, and using 
 fiber maps, we find conditions on $V$, $h$ for the
 existence and multiplicity of solutions, when $V$ and $h$ are sign
 changing and unbounded functions.
\end{abstract}

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

\section{Introduction}

 We consider the following quasilinear
 elliptic equation in $\Omega\subset\mathbb{R}^N$, $N\geq 2$:
\begin{equation} \label{PlaVh}
\begin{gathered}
 -\Delta_N u+V(x)|u|^{N-2}u = \lambda h(x)|u|^{q-1}u+ u|u|^{p}  
 e^{|u|^{\beta}} \quad  \text{in }\Omega \\
 u \geq 0 \quad \text{in } \Omega,\quad u\in W^{1,N}_0(\Omega),\\
  u =0\quad \text{on } \partial \Omega
 \end{gathered}
\end{equation}
 where $ \Delta_N u= \nabla\cdot(|\nabla u|^{N-2}\nabla u)$, $0< q<N-1< p+1$,
$\beta\in (1,\frac{N}{N-1}]$ and $\lambda>0$.
 Let $\gamma=\frac{N}{N-q-1}$, $k=\frac{p+2+\beta}{q+1}>1$ and
$k'=\frac{k}{k-1}$. We assume the following:
 \begin{itemize}
 \item[(A1)] $V\in L^{s}(\Omega)$, $s>1$ be an indefinite and unbounded function;
\item[(A2)] $h^{+}\not\equiv 0$, $h$ can be indefinite and vanish in some
open subset of $\Omega$ and moreover
$ h\in L^{\gamma}(\Omega)$.
\end{itemize}
These conditions ensure that $ E_{V}(u):= \int_{\Omega} (|\nabla u|^{N} +
V(x)|u|^{N}) dx$ is weakly lower semi-continuous on
$W^{1,N}_0(\Omega)$, where as $ H(u):= \int_{\Omega} h(x)|u|^{q+1} dx $
 is weakly continuous on
$W^{1,N}_0(\Omega)$.  Problems of the type \eqref{PlaVh} are
motivated by the following Trudinger-Moser inequality.

 \begin{theorem}[\cite{moser}] \label{thmA}
 For $N\geq2$, $u\in W^{1,N}_0(\Omega)$
\begin{equation} \label{eq002}
\sup_{\| u\| \leq 1}\int_{\Omega} e^{\alpha |u|^{\frac{N}{N-1}}} dx <\infty
\end{equation}
if and only if ${\alpha}\leq {\alpha_N}$, where $\alpha_N = N
w_{N-1}^{\frac{1}{N-1}}$, $w_{N-1}=$ volume of $ S^{N-1}$. 
\end{theorem}

The embedding $W_0^{1,N}(\Omega) \ni u\longmapsto e^{|u|^{\beta}} \in
L^{1}(\Omega)$ is compact for all $\beta\in(1,\frac{N}{N-1})$
and is continuous for $\beta=\frac{N}{N-1}$. The non-compactness of
the embedding can be shown using a sequence of functions that are
truncations of fundamental solution of $-\Delta_N$ on
$W^{1,N}_0(\Omega)$. The existence results for quasilinear problems
with exponential terms on bounded domains was initiated and studied
by Adimurthi \cite{A}.

 Starting from the pioneering works of Tarantello \cite{TA} and
Ambrosetti-Brezis-Cerami \cite{ABC}, a lot of work has been done to
address the multiplicity of positive solutions for semilinear and
quasilinear elliptic problems with positive nonlinearities.
Recently, many works are devoted to the study of these multiplicity
results with polynomial type nonlinearity with sign-changing weight
functions using the Nehari manifold and fibering map analysis (see
\cite{GA, COA, WUFI, DP,AE, AEJ, TA, WU,WU6,WU9,WU10}).
Nonhomogeneous elliptic equation with exponential nonlinearity is
also dealt in \cite{PS}.  In \cite{qu}, Quoirin studied the
quasilinear equation
$$
  -\Delta_p u +V(x)u^{p-1} = \lambda a(x) u^{r-1}+b(x)u(x)^{q-1}  \text{ in }
 \Omega, \quad u(x)=0  \text{ on } \partial\Omega,
$$
where $1<r<p<q<p^*$, $V(x)$, $a(x)$ and $b(x)$ are indefinite
functions. Under suitable condition on $V$, $a$, $b$, he showed the
existence of four non-negative solutions when $\lambda_1(V)<0$, the
first eigenvalue of $-\Delta _{N}+V$ . Also In \cite{hqu}, Quoirin and
Ubilla showed the existence and multiplicity of non-negative
solutions of the following equation:
$$
  -\Delta u +V(x)u^{p-1} = \lambda a(x) u^{r-1}+b(x)u^{q-1}  \text{ in }  \Omega,
\quad u=0  \text{ on } \partial\Omega,
$$
where $1<r<2<q\leq \frac{2N}{N-2}$, $V(x)$, $a(x)$ and $b(x)$ are
sign changing. Our work in this paper is motivated by the work of
Quoirin \cite{qu}.

 The first and second eigenvalues of the
operator $-\Delta _{N}+V$ on $W^{1,N}_0(\Omega)$ are denoted by
$\lambda_1(V)<\lambda_2(V)$ respectively and $\phi_{V}>0$ be its
$L^{N}$-normalized eigenfunction corresponding to $\lambda_1(V)$. Also
$\lambda_1(V)$ is characterized as $\min\{E_{V}(u): \|u\|_{N}=1\}$, is
simple and principal so that $\phi_{V}$ is unique and positive.

 When $\lambda_1(V)\leq 0$, the effect of potential $V$ on
\eqref{PlaVh} is relevant as $E_{V}(u)$ becomes non-coercive. We
will see that in this case, \eqref{PlaVh} has existence and
multiplicity of non-negative solutions in critical and subcritical
case respectively which are distinguished by the sign of $V$ and
$h$.  As in \cite{qu} and \cite{cqu}, we define,
\begin{gather*}
\alpha(V,h):= \min\{E_{V}(u) : \|u\|_{N}=1, H(u)=0\},\\
\beta(V,h):= \min\{E_{V}(u) : H(u)=1\}.
\end{gather*}
Then $\alpha(V,h)$ is well-defined by assuming the convention that
$\min \emptyset =\infty$. It is clear that $\lambda_1(V)\leq \alpha(V,h)$
and $\lambda_1(V)= \alpha(V,h)$ if and only if $H(\phi_V)=0$.
Also one can easily see that $\beta{(V,h)}$ is well-defined if
$\alpha(V,h)>0$ (see \cite[Lemma 4.3]{qu}).
Also we introduce some symbols:
\begin{gather*}
E^{\pm}:=\{u\in W^{1,N}_0(\Omega): E_{V}(u)\gtrless 0\}, \quad
E_0:=\{u\in W^{1,N}_0(\Omega): E_{V}(u)= 0\},\\
H^{\pm}:=\{u\in W^{1,N}_0(\Omega): H(u)\gtrless 0\}, \quad
H_0:=\{u\in W^{1,N}_0(\Omega): H(u)= 0\},
\end{gather*}
and $H^{\pm}_0: =H^{\pm}\cup H_0$, $E^{\pm}_0: = E^{\pm}\cup E_0$.
We have the following existence result.

\begin{theorem}\label{th11}
Let $\beta\in(1,\frac{N}{N-1}]$, $V$ and $h$ satisfy
{\rm (A1), (A2)} respectively. Then there exists $\lambda_0>0$ such that
for $\lambda\in(0,\lambda_0)$, \eqref{PlaVh} admits a non-negative
solution $u_\lambda$ in each of the following cases:
\begin{enumerate}
\item  $\lambda_1(V)>0$,

\item  $\lambda_1(V)=0$ and $\phi_{V}\in H^{-}$,

\item $\lambda_1(V)<0<\lambda_2(V)$, $\phi_{V}\in H^{-}$ and  $\alpha(V,h)>0$.
\end{enumerate}
Moreover, $u_{\lambda}$ is a local minimum for $J_{\lambda}$ on
$W^{1,N}_0(\Omega)$.
\end{theorem}

 We have the following multiplicity result in the subcritical
case when $h(x)$ changes sign:


\begin{theorem}\label{th12}
 For $\beta\in (1,\frac{N}{N-1})$, $V$ and $h$ satisfy
{\rm (A1), (A2)} respectively. Then for $\lambda\in(0,\lambda_0)$, \eqref{PlaVh}
has a non-negative second solution  $v_\lambda$ in each the cases (1)-(3) above.
\end{theorem}

 Finally, in the critical case, we obtain the following
multiplicity result.

\begin{theorem}\label{th13}
For $\beta =\frac{N}{N-1}$, $h\geq 0$ and $\lambda_1(V)>0$ then for
$\lambda\in(0,\lambda_0)$, \eqref{PlaVh} has at least two non-negative
solutions.
\end{theorem}

Here $\lambda_0$ is the maximum of $\lambda$ such that for $\lambda<\lambda_0$, the
fibering map $t\mapsto J_\lambda(tu)$  has exactly two
critical points for each $u\in E^+\cap H^+$.


 The Euler functional associated with the problem \eqref{PlaVh} is
$J_{\lambda} :W^{1,N}_0(\Omega) \to \mathbb{R}$ defined as
\begin{equation}\label{eq1}
J_{\lambda}(u)= \frac{1}{N} E_{V}(u)- \frac{\lambda}{q+1} H(u)- \int_{\Omega}
G(u) dx,
\end{equation}
where $g(u)=u|u|^{p}e^{|u|^{\beta}}$ and $ G(u)=\int_0^{u} g(s)ds$.

\begin{definition} \rm
We say that $u\in W^{1,N}_0(\Omega)$ is a weak solution of
\eqref{PlaVh} if for all $\phi \in W^{1,N}_0(\Omega)$, we have
\begin{equation} \label{def1}
\int_{\Omega}\left(|\nabla u|^{N-2}\nabla u \nabla \phi +
V(x)|u|^{N-2}u\phi\right)dx = \int_{\Omega}g(u)\phi dx +\lambda \int_{\Omega}
h(x)|u|^{q-1}u\phi dx.
\end{equation}
\end{definition}

We remark that the similar existence results with some obvious
modification can be proved for critical exponent problem for
$p$-Laplacian with $p<N$ and $g(u)= |u|^{p^{*}-2}u$ where
$p^{*}=\frac{Np}{N-p}$, while subcritical case is studied by Quoirin
in \cite{qu}. The multiplicity in the critical case can be obtained
with some condition on $p$, $q$ as in \cite{fgu}.

 This paper is organized as follows: In section 2, we introduce
Nehari manifold and study the behavior of the Nehari manifold using
the fibering map analysis for \eqref{PlaVh}. Section 3 contains the
existence results for critical and subcritical non-linearities. In
section 4, we show the existence of a second solution. In
section 5 we study non-existence results.

 We shall throughout use the following notation: The norm on
$W^{1,N}_0(\Omega)$ and $L^{p}(\Omega)$ are denoted by $\|\cdot\|$,
$\|u\|_{p}$ respectively. The weak convergence is denoted by
$\rightharpoonup$ and $\to$ denotes strong convergence.

\section{Nehari manifold and fibering map analysis for \eqref{PlaVh}}

 The energy functional $J_{\lambda}$ is not
bounded below on the space $W^{1, N}_0(\Omega)$, but is bounded below
on an appropriate subset of $W^{1, N}_0(\Omega)$ and a minimizer on
subsets of this set gives rise to solutions of \eqref{PlaVh}.
To obtain the existence results, we introduce the Nehari manifold
\begin{equation*}
\mathcal{N}_{\lambda}= \{u\in W^{1, N}_0(\Omega): \langle
J_{\lambda}'(u),u\rangle=0\}=\{u\in W^{1, N}_0(\Omega) :
\phi_u'(1)=0\}
\end{equation*}
where $\langle\cdot,\cdot\rangle$ denotes the duality between $W^{1,N}_0(\Omega)$
and its dual space. Therefore $u\in \mathcal N_{\lambda}$ if and only if
\begin{equation}\label{eq2}
E_{V}(u)- \lambda H(u)- \int_{\Omega}{g(u) u}dx =0 .
\end{equation}
We note that $\mathcal N_{\lambda}$ contains every non zero solution of
\eqref{PlaVh}. Now as we know that the Nehari manifold is
closely related to the behaviour of the functions
$\phi_u: \mathbb{R}^+\to \mathbb{R}$  defined as $\phi_u(t)=J_{\lambda}(tu)$.
Such maps are called fiber maps and were introduced by Drabek and Pohozaev in
\cite{DP}. For $u\in W^{1, N}_0(\Omega)$, we have
\begin{gather*}
\phi_u(t) = \frac{t^N}{N} E_{V}(u)- \frac{\lambda
 t^{q+1}}{q+1} H(u) -\int_{\Omega} {G(tu)} dx ,\\
\phi_u'(t) = t^{N-1} E_{V}(u)- {\lambda
 t^{q}} H(u)- \int_{\Omega}{g(tu) u} dx,\\
\phi_u''(t) = (N-1) t^{N-2} E_{V}(u)- q \lambda
 t^{q-1} H(u) - \int_{\Omega}{g'(tu)u^2} dx.
\end{gather*}
Then it is easy to see that $tu\in \mathcal N_{\lambda}$ if and only if
$\phi_u'(t)=0$ and in particular, $u\in \mathcal{N}_{\lambda}$ if
and only if $\phi_u'(1)=0$. Thus it is natural to split
$\mathcal N_{\lambda}$ into three parts corresponding to local minima,
local maxima and points of inflection. For this we set
\begin{gather*}
\mathcal N_{\lambda}^{\pm}:= \{u\in \mathcal{N}_{\lambda}:
\phi_u''(1) \gtrless0\}
=\{tu\in W^{1, N}_0(\Omega) : \phi_u'(t)=0,\; \phi_u''(t)\gtrless  0\},\\
\mathcal N_{\lambda}^{0}:= \{u\in \mathcal{N}_{\lambda}:
\phi_u''(1) = 0\}=\{tu\in W^{1,
N}_0(\Omega) : \phi_u'(t)=0,\; \phi_u''(t)= 0\}.
\end{gather*}

Now we describe the behavior of the fibering map $\phi_u$
according to the sign of $H(u)$ and $E_{V}(u)$.

\noindent Case 1: $u\in H^{+}\cap E^{-}$.
In this case $\phi_u(0)=0$, $\phi_u'(t)<0$ for all  $t>0$
which implies that $\phi_u$ is strictly decreasing and hence no
critical point.

\noindent Case 2: $u\in H^{-}\cap E^{-}$.
In this case, firstly we define $m_u: \mathbb{R}^{+} \to \mathbb{R}$ by
\begin{equation*}
m_u(t)= t^{N-1-q} E_{V}(u)-  t^{-q}\int_{\Omega}g(tu)u dx.
\end{equation*}
Clearly, for $t>0$, $tu\in \mathcal{N}_{\lambda}$ if and only if $t$ is a
solution of
 $m_u(t)={\lambda} H(u)$.
\begin{align*}%\label{eq01}
m_u' (t) &= (N- 1-q) t^{(N-2-q)} E_{V}(u) - t^{-q}
\int_{\Omega}g'(tu)u^2 dx+ qt^{-q-1} \int_{\Omega}{g(tu) u} dx\\
&= (N- 1-q) t^{N-2-q}E_{V}(u) - (1+p-q)t^{-1-q}\int_{\Omega}g(tu)u \\
&\quad - \beta t^{-q-1+\beta}\int_{\Omega}|u|^{\beta}g(tu)u.
\end{align*}
Therefore $m_u'(t)<0$ for all $t>0$, since $u\in E^{-}$.
As $u\in H^{-}$ so there exists $t_*(u)$ such that $m_u(t_*)={\lambda}
H(u)$. Thus for $0<t<t_*$, $\phi_u'(t)= t^q(m_u(t)-{\lambda}
H(u))>0$ and for $t>t_*$, $\phi_u'(t)<0$. Hence $\phi_u$ is
increasing on $(0,t_*)$, decreasing on $(t_*, \infty)$. Since
$\phi_u(t)>0$ for $t$ close to $0$ and $\phi_u(t)\to -\infty$ as
$t\to \infty$, we obtain $\phi_u$ has exactly one critical point
$t_1(u)$, which is a global maximum point. Hence $t_1(u)u \in
\mathcal{N}^{-}_{\lambda}$.

\noindent Case 3: $u\in E^{+}\cap H^{-}$.
In this case, we have
\begin{equation}\label{eq011}
\begin{aligned}
&m_u'(t)\\
&= (N- 1-q) t^{(N-2-q)} E_{V}(u) - t^{-q}
\int_{\Omega}g'(tu)u^2 dx+ qt^{-q-1} \int_{\Omega}g(tu) u   \\
&= t^{N-2-q}\Big[(N-1-q)E_{V}(u)-(1+p-q)t^{1-N}\\
&\quad\times \int_{\Omega}g(tu)u-\beta
t^{1-N+\beta}\int_{\Omega}|u|^{\beta}g(tu)u \Big].
\end{aligned}
\end{equation}
It is easy to see that $ \lim_{t\to 0^{+}}m_u'(t)>0$
and sum of second and third term in \eqref{eq011} is a monotone
function in $t$. Therefore there exists a unique $t_*=t_{*}(u)>0$
such that $m_u(t)$ is increasing on $(0,t_*)$, decreasing on $(t_*,
\infty)$ and $m_u'(t_*)=0$. As $m_u(t)\to -\infty$ as
$t\to \infty$, $u\in H^{-}$ so $\exists$ $t_1(u)$ such that
$m_u(t_1)={\lambda} H(u)$. Thus for $0<t<t_1$,
$\phi_u'(t)>0$ and for $t>t_1$, $\phi_u'(t)<0$.
Thus $\phi_u$ has exactly one critical point $t_1(u)$, which is
a global maximum point. Hence $t_1(u)u \in \mathcal{N}^{-}_{\lambda}$

\noindent Case 4: $u\in E^{+}\cap H^+$.
 In this case, we claim that there exists $\lambda_0>0$ and a unique
$t_*$ such that for $\lambda\in(0,\lambda_0)$, $\phi_u$ has exactly two
critical points $t_1(u)$ and $t_2(u)$ such that
$t_1(u)<t_*(u)<t_2(u)$, and moreover $t_1(u)$ is a local minimum
point and $t_2(u)$ is a local maximum point. Thus $t_1(u)u\in\mathcal{N}_{\lambda}^{+}$ and $t_2(u)u\in\mathcal{N}_{\lambda}^{-}$.

 To show this we need following Lemmas:

\begin{lemma}\label{le2}
If $\alpha(V,h)>0$, $\beta(V,h)>0$, then $H_0^{+}\subseteq E^{+}$ and
moreover, there exists a constant $K>0$ such that $E_{V}(u)\geq K
\|u\|^{N}$ for all $u\in H^{+}_0$.
\end{lemma}

\begin{proof}
Let $u\in H^{+}_0$ then $H(u)\geq 0$ and $\alpha(V,h)>0$,
$\beta(V,h)>0$ implies that $E_{V}(u)>0$. Next we show that there
exists a constant $K>0$ such that $E_{V}(u)\geq K \|u\|^{N}$ for all
$u\in H^{+}_0$. Suppose this is not true, then for each $n$, there
exist $u_n \in H_0^{+}$ such that
$E_{V}(u_n)<\frac{\|u_n\|^N}{n}$. Let $v_n= \frac{u_n}{\|u_n\|}$.
Then $v_n$ is bounded. So there exists a subsequence $v_n$ such that
$v_n\rightharpoonup v_0$ weakly in $W^{1,N}_0(\Omega)$. Also
$H(v_0)\geq 0$ and $0\leq E_{V}(v_n)<\frac{1}{n}$ implies
$E_{V}(v_0)\leq 0$. Moreover $v_0\ne 0$ because if $v_0=0$ then we obtain $\|v_n\|\leq E_V(v_n)+\int_{\Omega}|V(x)||v_n|^N dx\to 0$, which is
a contradiction as $\|v_n\|=1$. Thus $E_{V}(v_0)\leq 0 \leq H(v_0)$
imply $\alpha(V,h)\leq 0$, $\beta(V,h)\leq 0$ which contradict the given
assumptions. 
\end{proof}

Next, we define $ \Lambda= \{u\in W^{1,N}_0(\Omega) \;|\;
E_V(u) \leq \frac{N-q}{(N-1-q)}\int_{\Omega}{g'(u){u}^2}dx\}$.
Then, we prove the following Lemma.

\begin{lemma}\label{le3}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ then there exists $\lambda_0>0$ such that
for every $\lambda\in(0,\lambda_0)$,
\begin{equation}\label{eq8}
\Lambda_m:=\inf_{u\in \Lambda\setminus\{0\} \cap H^{+}_0}\Big\{\int_{\Omega}
\left(p+2-N +\beta{|u|^{\beta}}\right)|u|^{p+2} e^{|u|^{\beta}}- (N-1-q)
{\lambda} H(u) \Big\}>0.
\end{equation}
\end{lemma}

\begin{proof}
 Step 1: $\inf_{u\in \Lambda\setminus\{0\}\cap H^{+}_0} E_{V}(u)>0$.
In view of Lemma \ref{le2} it is sufficient to show that
\[
\inf_{u\in \Lambda\setminus\{0\}\cap H^{+}_0} \|u\|>0.
\]
Suppose this is not true. Then we find a sequence $\{u_n\}
\subset \Lambda\setminus\{0\}\cap H^{+}_0$ such that $\|u_n\| \to 0$
and we have
\begin{align}\label{eq9}
 E_{V}(u_n)\leq  \big(\frac{N-q}{N-1-q}\big)
\int_{\Omega}{g'(u_n)u_{n}^2}\;dx\quad \forall n .
\end{align}
From $g(u)=u|u|^pe^{|u|^{\beta}}$, H\"{o}lders inequality and Sobolev
inequality, we have
\begin{align*}
\int_{\Omega} {g'(u_n)u_{n}^2} dx 
&= \int_{\Omega} {\left(p+1+\beta|u_n|^{\beta}\right)|{u_n}|^{p+2}e^{|u_n|^{\beta}}} dx\\
&\leq C\int_{\Omega}|{u_n}|^{p+2} e^{(1+\delta)|u_n|^{\beta}} dx\\
&\leq C\Big(\int_{\Omega}{|{u_n}|^{(p+2)t'}} dx
\Big)^{1/t'}\Big( \int_{\Omega} e^{t(1+\delta)|u_n|^{\beta}} \, dx\Big)^{1/t}\\
&\leq C' \|u_n\|^{p+2} \Big(\sup_{\|w_n\|\leq 1}
\int_{\Omega}e^{t(1+\delta)\|u_n\|^{\beta}|w_n|^{\beta}}
dx\Big)^{1/t},
\end{align*}
since  $\|u_n\|\to 0$ as $n\to\infty$, we can choose 
$\alpha=t(1+\delta){\|u_n\|}^{\beta}$ such that $\alpha \leq {\alpha_N}$.
Hence by this, \eqref{eq9} and Lemma \ref{le2}, we obtain $1\leq
K'\|u_n\|^{p+2-N}\to 0$ as $n\to \infty$, since $p+2 > N$,
which gives a contradiction.

 Step 2: Let $ C_1=\inf_{u\in \Lambda\setminus\{0\} \cap
H^{+}_0}\int_{\Omega} {\left(p+2-N+\beta{|u|^{\beta}}\right)|u|^{p+2}
e^{|u|^{\beta}}} dx$. Then $C_1>0$.
From Step 1 and the definition of $\Lambda$, we obtain
\[
0< \inf_{u\in \Lambda\setminus\{0\} \cap H^{+}_0}\int_{\Omega}{g'(u)u^2}dx 
=\inf_{u\in \Lambda\setminus\{0\}\cap H^{+}_0}
\int_{\Omega}{\left(p+1+\beta{|u|^{\beta}}\right)|u|^{p+2}e^{|u|^{\beta}}} dx.
\]
Using this it is easy to check that
\[
 \inf_{u\in \Lambda \setminus\{0\}\cap H^{+}_0}\int_{\Omega}
{\left(p+2-N+\beta{|u|^{\beta}}\right)|u|^{p+2} e^{|u|^{\beta}}} dx>0.
\]
This completes step 2.

Step 3: Let
$\lambda<\frac{1}{(N-q-1)}(\frac{C_1}{l})^{\frac{(k-1)}{k}}$, where
$l=\int_{\Omega}|h(x)|^{\frac{k}{k-1}}dx$. Then \eqref{eq8} holds.
 Using H\"{o}lder's inequality and (A2) we have,
\begin{align*}
H(u)
&\leq  \Big(\int_{\Omega}|h(x)|^{\frac{k}{k-1}}dx\Big)^{\frac{k-1}{k}} 
\Big(\int_{\Omega} |u|^{(q+1)k} dx \Big)^{1/k}\\
&= l^{\frac{k-1}{k}} \Big(\int_{\Omega} |u|^{p+2+\beta} dx \Big)^{1/k}\\
&\leq l^{\frac{k-1}{k}}
\Big(\int_{\Omega}\big(p+2-N+\beta{|u|^{\beta}}\big)
|u|^{p+2}e^{|u|^{\beta}} dx \Big)^{1/k}\\
&\leq\big(\frac{l}{C_1}\big)^{\frac{k-1}{k}}
\int_{\Omega}\left(p+2-N+\beta{|u|^{\beta}}\right) |u|^{p+2}e^{|u|^{\beta}}\,dx .
\end{align*}
The above inequality combined with step 2 proves the Lemma.
\end{proof}

The following Lemma completes the proof of claim made in case (4) above.

\begin{lemma}\label{le5}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ and $\lambda$ be such that
\eqref{eq8} holds. Then for every $u\in H^{+}\setminus\{0\}$, there
is a unique $t_*=t_{*}(u)>0$ and unique 
$t_1= t_1(u)<t_{*}<t_2= t_2(u)$ such that
$t_1u\in \mathcal{N}_{\lambda}^{+}$, $t_2u\in\mathcal{N}_{\lambda}^{-}$  and
 $ J_{\lambda}(t_1u)= \min_{0\leq t\leq t_2}J_{\lambda}(tu)$,
$ J_{\lambda}(t_2u)= \max_{t\geq t_{*}}J_{\lambda}(tu)$.
\end{lemma}

\begin{proof}
 Fix $0\ne u\in H^{+}$, Then by Lemma \ref{le2}, $u\in E^{+}$. 
Define $m_u: \mathbb{R}^{+} \to \mathbb{R}$ by
\[
m_u(t)= t^{N-1-q} E_{V}(u)-  t^{-q}\int_{\Omega}{g(tu) u} dx.
\]
We note that $m_u(t)\to -\infty$ as $t \to\infty $ and
\begin{align}
m_u'(t)& = (N- 1-q) t^{(N-2-q)} E_{V}(u) - t^{-q}
\int_{\Omega}g'(tu)u^2 dx+ qt^{-q-1} \int_{\Omega}g(tu) u \,dx \label{eq03}\\
&= t^{N-2-q}\Big[(N- 1-q)E_{V}(u)-(1+p-q)t^{1-N}\int_{\Omega} g(tu)u \nonumber\\
&\quad -\beta t^{1-N+\beta} \int_{\Omega}|u|^{\beta}g(tu)u dx \Big].\label{eq04}
\end{align}
It is easy to see that $ \lim_{t\to 0^{+}}m_u'(t)>0$
and sum of second and third term in \eqref{eq03} is a monotone
function in $t$. So there exists a unique $t_*=t_{*}(u)>0$ such that
$m_u(t)$ is increasing on $(0,t_*)$, decreasing on $(t_*, \infty)$
and $m_u'(t_*)=0$. Using this and \eqref{eq03}, we obtain
$t_*{u}\in \Lambda\setminus\{0\} \cap H^{+}$. From
$t_{*}^{q+2}m_u'(t_*)= 0$ and by definition of $m_u$, we obtain
\[
m_u(t_*)=\frac{1}{t_{*}^{q+1}(N-1-q)}\Big[ \int_{\Omega}{g'(t_{*}u)
(t_*u)^2}  dx -(N-1) \int_{\Omega}{g(t_*u) t_*u}  dx \Big].
\]
Using Lemma \ref{le3} and  that
$g'(s)s^2-(N-1)g(s)s=(p+2-N+\beta|s|^{\beta})|s|^{p+2}
e^{|s|^{\beta}}$, we have
\begin{align*}
m_u(t_*)- {\lambda} H(u) 
&=\frac{1}{ t_{*}^{q+1} (N-1-q)}\Big[\int_{\Omega} \big( g'(t_{*}u)
(t_*u)^2 -(N-1)g(t_*u) t_*u  \big)dx  \\
&\quad -(N-1-q) {\lambda} H(t_*(u)) \Big]\\
&> \frac{\Lambda_m}{t_{*}^{q+1} (N-1-q)} >0.
\end{align*}
Since $m_u(0)=0$, $m_u$ is increasing in $(0,t_*)$ and
strictly decreasing in $(t_*,\infty)$, $ \lim_{t\to
\infty}m_u(t)=-\infty$ and $u\in H^+$. Then there exists a unique
$t_1=t_1(u)<t_*$ and $t_2=t_2(u)>t_*$  such that
$m_u(t_1)= {\lambda} H(u) = m_u(t_2)$ implies $t_1{u}$,
$t_2{u}\in \mathcal{N}_{\lambda}$. Also $m_u'(t_1)>0$ and
$m_u'(t_2)<0$ give $t_1u\in \mathcal{N}_{\lambda}^{+}$ and
$t_2u\in \mathcal{N}_{\lambda}^{-}$. Since
$\phi_u'(t)=t^q(m_u(t)- \lambda H(u))$.
Then $\phi_u'(t)<0$ for all  $t\in [0,t_1)$ and
$\phi_u'(t)>0$ for all  $t\in (t_1,t_2)$ so
$\phi_u(t_1)=\min_{0\leq t\leq t_2} \phi_u(t)$. Also
$\phi_u'(t)>0$ for all  $t\in [t_*,t_2)$,
$\phi_u'(t_2)=0$ and $\phi_u'(t)<0$ for all
 $t\in (t_2,\infty)$ implies that $ \phi_u(t_2)= \max_{t\geq t_*}
\phi_u(t)$.
\end{proof}

\begin{lemma}\label{le4}
If  $\alpha(V,h)>0$, $\beta(V,h)>0$ and $\lambda$ be such that
\eqref{eq8} holds. Then $\mathcal{N}_{\lambda}^{0}= \{0\}$.
\end{lemma}

\begin{proof}
Suppose $u\in \mathcal{N}_{\lambda}^{0}$, $u\not\equiv 0$. Then by
definition of $\mathcal{N}_{\lambda}^{0}$, we have the following two equations
\begin{gather}\label{eq10}
{(N-1)}E_{V}(u)=\int_{\Omega}{g'(u)u^2} dx+ \lambda q H(u), \\
\label{eq11}
E_{V}(u) =\int_{\Omega}{g(u)u} dx+ \lambda H(u).
\end{gather}
Let $u\in H^+\cap \mathcal{N}_{\lambda}^{0}$ and $\lambda\in(0,\lambda_0)$. Then from
above equations, we can easily deduce that
\[
{(N-1-q)}E_{V}(u)\leq \int_{\Omega}{g'(u)u^2} dx ,
\]
which shows $u\in\Lambda\setminus\{0\}$. Noting that
$g'(s)s^2-(N-1)g(s)s= (p+2-N+\beta|s|^{\beta})|s|^{p+2}
e^{|s|^{\beta}}$, from \eqref{eq10} and \eqref{eq11}, we obtain
\[
(N-1-q)\lambda H(u) =
\int_{\Omega}\left(p+2-N+\beta|u|^{\beta}\right)|u|^{p+2}e^{|u|^{\beta}} dx,
\]
which violates Lemma \ref{le3}. Hence $\mathcal{N}_{\lambda}^{0}= \{0\}$. In
other cases, $u\in H^-\cap E^-\cap\mathcal{N}_\lambda^{0}$ and
$u\in H^-\cap E^+\cap\mathcal{N}_\lambda^{0}$, we have $1$ is critical point of
$\phi_u$ and $\phi_u''(1)=0$ but $u\in H^-\cap E^-$ and $u\in H^-\cap E^+$
implies that $\phi_u$ has exactly one critical point corresponding
to global maxima i.e $\phi_u''(1)\ne0$ which is a
contradiction. Hence $\mathcal{N}_{\lambda}^{0}= \{0\}$.
\end{proof}

\section{Existence of solutions}

 In this section we show that $J_\lambda$ is bounded below on $\mathcal{N}_{\lambda}$.
Also we show that under suitable condition on $V$ and $h$, $J_{\lambda}$
attains its minimizer on $E^{+}\cap H^{+}\cap \mathcal{N}_{\lambda}^{+}$.\\
We define $\theta_{\lambda} := \inf \{J_{\lambda}(u)\mid u\in
\mathcal{N}_{\lambda}\}$ and prove the following lower bound.

\begin{theorem}\label{th1}
$J_{\lambda}$ is bounded below on $\mathcal{N}_{\lambda}$. Moreover, there exists
a constant $C=C(p,q,N)>0$ such that 
$\theta_{\lambda} \geq - C \lambda^{\frac{k}{k-1}}$.
\end{theorem}

\begin{proof} 
Let $u \in \mathcal{N}_{\lambda}$. Then
\begin{equation} \label{eq3}
J_{\lambda}(u) = \frac{1}{N} \int_{\Omega}{g(u)u}\, dx- \int_{\Omega} {G(u)}\, dx
-{\lambda}\big(\frac{1}{q+1}-\frac{1}{N}\big) H(u).
\end{equation}
If $u\in H^{-}_0$, then $J_\lambda(u)$ is bounded below by $0$. If
$u\in H^+$ then by using H\"{o}lder's inequality, we have
\[
H(u) \leq  l^{\frac{k-1}{k}} \Big(\int_{\Omega} |u|^{(q+1)k}\,dx\Big)^{1/k},
\]
where $l=\int_{\Omega} |h(x)|^{k/k-1} dx$. Also, It is easy to see that
\begin{equation}
\frac{1}{N}g(u)u- G(u)\geq
\big(\frac{1}{N}-\frac{1}{p+2}\big)|u|^{p+2+\beta} ,
\end{equation}
From the above inequalities, we obtain
\[
J_{\lambda}(u)\geq \big(\frac{1}{N}-\frac{1}{p+2}\big)\int_{\Omega}
|u|^{(q+1)k} dx -
\frac{\lambda(N-q-1)l^{\frac{k-1}{k}}}{N(q+1)}\Big(\int_{\Omega}
|u|^{(q+1)k} dx\Big)^{1/k},
\]
where $k=\frac{p+2+\beta}{q+1}$. By considering the global minimum of
the function $\rho(x): \mathbb{R}^+ \to \mathbb{R}$ defines as
\[
\rho(x)= \big(\frac{1}{N}-\frac{1}{p+2}\big)x^k -
\Big(\frac{\lambda(N-q-1)l^{\frac{k-1}{k}}}{N(q+1)}\Big) x,
\]
it can
be shown that
\begin{equation*}
\inf_{u\in \mathcal{N}_{\lambda}}J_{\lambda}(u) \geq
\rho\Big[\Big(\frac{\lambda(N-q-1)(p+2)l^{\frac{k-1}{k}}}{k(q+1)(p+2-N)}
\Big)^{\frac{1}{k-1}}\Big].
\end{equation*}
From this, it follows that
\begin{equation}\label{eq5}
\theta_{\lambda} \geq -C(p,q,N)\lambda^{\frac{k}{k-1}},
\end{equation}
where
\[
C(p,q,N)= \Big(\frac{1}{k^{\frac{1}{k-1}}}-\frac{1}{k^{\frac{k}{k-1}}}\Big)
\frac{l(p+2)^{\frac{1}{k-1}}(N-q-1)^{\frac{k}{k-1}}}{N(p+2-N)
^{\frac{1}{k-1}}(q+1)^{\frac{k}{k-1}}}>0.
\]
Hence $J_{\lambda}$ is bounded below on $\mathcal{N}_{\lambda}$.
\end{proof}

 The following lemma shows that minimizers for $J_{\lambda}$ on any
subset of $\mathcal{N}_{\lambda}$ are usually critical points for $J_{\lambda}$.

\begin{lemma}\label{le1}
Let $u$ be a local minimizer for $J_{\lambda}$ in any of the subsets
$\mathcal{N}_{\lambda}^{\pm}\cap E^+\cap H^+$ of $\mathcal{N}_{\lambda}$ such that
$u\notin \mathcal{N}_{\lambda}^{0}$, then $u$ is a non-negative critical point
for $J_{\lambda}$.
\end{lemma}

\begin{proof} 
Let $u$ be a local minimizer for $J_{\lambda}$ in any of the
subsets of $\mathcal{N}_{\lambda}$. We can take $u\geq 0$ as
$J_{\lambda}(|u|)=J_{\lambda}(u)$ for every $u$. Then, in any case $u$ is a
minimizer for $J_{\lambda}$ under the constraint $I_{\lambda}(u):=\langle
J_{\lambda}'(u),u\rangle =0$. Hence, by the theory of Lagrange
multipliers, there exists $\mu \in \mathbb{R}$ such that 
$ J_{\lambda}'(u)= \mu I_{\lambda}'(u)$. 
Thus $\langle J_{\lambda}'(u),u\rangle= \mu \langle I_{\lambda}'(u),u\rangle = \mu\phi_u''(1)$=0,
but $u\notin \mathcal{N}_{\lambda}^{0}$ and so
$\phi_u''(1) \ne 0$. Hence $\mu=0$ completes the
proof.
\end{proof}

\begin{lemma}\label{le01}
If $\alpha(V,h)>0$, $\beta(V,h)>0$ and
$\{u_n\}\in \mathcal{N}_{\lambda}\cap H^{+}_0$ be a sequence such that 
$J_{\lambda}(u_n)$ is bounded. Then the sequence $\{u_n\}$ is bounded.
\end{lemma}

\begin{proof} On $\mathcal{N}_{\lambda}$,
\begin{equation} \label{eq05}
\begin{aligned}
J_{\lambda}(u_n) &=\big(\frac{1}{N}-\frac{1}{p+2}\big)E_{V}(u_n)
-\frac{\lambda(p+1-q)}{(q+1)(p+2)} H(u_n)\\
&\quad +\int_{\Omega}\Big(\frac{1}{p+2}g(u_n)u_n - G(u_n)\Big) \,dx.
\end{aligned}
\end{equation}
 As $u_n\in H^{+}_0$, we have
\[
\frac{(p+2-N)}{N(p+2)}E_{V}(u_n)\leq J_{\lambda}(u_n)+
\frac{\lambda(p+1-q)}{(q+1)(p+2)} H(u_n).
\]
Then by using Lemma \ref{le2}
and H\"{o}lders inequality, we obtain
 \[
K\|u_n\|^N \leq \frac{(p+2-N)}{N(p+2)}E_{V}(u_n)\leq J_{\lambda}(u_n) +
\frac{\lambda(p+1-q)}{(q+1)(p+2)}\|h\|_{(\frac{N}{q+1})'}{\|u_n\|^{q+1}},
\]
and hence $\{u_n\}$ is bounded.
\end{proof}

\begin{lemma}\label{le31}
Let  $\alpha(V,h)>0$, $\beta(V,h)>0$, and let $\lambda$ satisfy \eqref{eq8}.
Then given $u\in \mathcal{N}_{\lambda}\setminus \{0\}$, there exist $\epsilon>0$
and a differentiable function 
$\xi :  B(0,\epsilon)\subset W^{1,N}_0(\Omega) \to \mathbb{R}$ such that
$\xi(0)=1$, the function
$\xi(w)(u-w)\in\mathcal{N} _{\lambda}$ and for all $w\in W^{1,N}_0(\Omega)$,
\begin{equation} \label{eq:3.1}
\langle\xi'(0),w\rangle =\frac{ N
R(u,w)-\int_{\Omega}\big({g(u)+g'(u)u}\big)w\,dx - \lambda
(q+1)\int_{\Omega} h(x)|u|^{q-1}uw\, dx}
{(N-q-1)E_{V}(u)- \int_{\Omega}{g'(u) u^2}dx+ q\int_{\Omega} g(u) u \,dx},
\end{equation}
 where $R(u,w)={\int_{\Omega}(|\nabla u|^{N-2}\nabla u
\nabla w + V(x)|u|^{N-2}uw)dx}$.
\end{lemma}

\begin{proof} 
Fix $u\in \mathcal{N}_{\lambda}\setminus \{0\}$, define a function
$G_u: \mathbb{R}\times W^{1,N}_0(\Omega) \to \mathbb{R}$ as follows:
\[
G_u(t,v)= t^{N-1-q}E_{V}(u-v)-  t^{-q}\int_{\Omega} g(t(u-v)) (u-v)
dx -\lambda H(u-v).
\]
 Then $G_u\in C^{1}(\mathbb{R}\times W^{1,N}_0(\Omega); \mathbb{R})$, 
$G_u(1,0)= \langle J_{\lambda}'(u),u \rangle= 0$ and
\[
\frac{\partial}{\partial t}G_u(1,0)= (N-1-q)E_{V}(u)- \int_{\Omega}
 g'(u) u^2 dx +q \int_{\Omega} g(u) u dx \ne 0,
\]
since $\mathcal{N}_{\lambda}^{0}=\{0\}$. By the Implicit function
theorem, there exist $\epsilon>0$ and a differentiable function 
$\xi :  B(0,\epsilon)\subset W^{1,N}_0(\Omega) \to \mathbb{R}$ such that
$\xi(0)=1$, and $G_u(\xi(w),w)=0$ for all $w\in  B(0,\epsilon)$
which is equivalent to $\langle J_{\lambda}'(\xi(w)(u-w)),\xi(w)(u-w) \rangle=0$ 
for all $w\in  B(0,\epsilon)$ and hence
$\xi(w)(u-w)\in \mathcal{N} _{\lambda}$. Now
differentiating $G_u(\xi(w),w)=0$ with respect to $w$ we obtain
\eqref{eq:3.1}. 
\end{proof}

\begin{lemma}\label{le33}
 Let $\alpha(V,h)>0$, $\beta(V,h)>0$ then exists a constant $C_2>0$ such 
that $\theta_{\lambda}\leq -\frac{(p+1-q)}{(q+1)(p+2)N}C_2$.
\end{lemma}

\begin{proof} 
$\alpha(V,h)>0$, $\beta(V,h)>0$ implies that $E^+\cap H^{+}_0
\neq \emptyset$. Let $v\in E^+\cap H^+$. Then by the fibering map
analysis, we can find $t_1=t_1(v)>0$ such that $t_1v\in \mathcal{N}_{\lambda}^+$. 
Thus
\begin{equation} \label{eq31}
\begin{aligned}
J_{\lambda}(t_1v)
&=\big(\frac{1}{N}-\frac{1}{q+1}\big)E_{V}(t_1v)-
\int_{\Omega}G(t_1v)dx +\frac{1}{q+1} \int_{\Omega} g(t_1v) t_1v\,dx \\
&\leq \frac{q+N}{N(q+1)}\int_{\Omega}g(t_1v) t_1vdx -\int_{\Omega}
G(t_1v)dx -\frac{1}{N(q+1)} \int_{\Omega} g'(t_1v) (t_1v)^2dx,
\end{aligned}
\end{equation}
since $t_1v\in \mathcal{N}_{\lambda}^+\cap E^+$. We now consider the
function
\begin{equation*}
\rho(s)= \frac{q+N}{N(q+1)}g(s)s- G(s)-\frac{1}{N(q+1)}g'(s)s^2.
\end{equation*}
Then
\begin{align*}
\rho'(s)
&= \frac{(q+N-2)}{N(q+1)}g'(s)s-\frac{q(N-1)}
 {N(q+1)}g(s)-\frac{1}{(q+1)N}g''(s)s^2\\
&= \Big(\frac{(q+N-2-p)(p+1)-(N-1)q}{N(q+1)} \Big) g(s)\\
& \quad  + \beta \Big( \frac{q-p+N-2-\beta - p-1}{N(q+1)} \Big)
g(s)|s|^{\beta} - \frac{\beta^2}{N(q+1)} g(s)|s|^{2\beta}.
\end{align*}
Now it is not difficult to see that coefficients in the first and
second term are negative, since $p> N-2$. As $\rho(0)=0$, it follows
that $\rho(s)\leq 0$ for all  $s\in\mathbb{R}^{+}$. Also it can be easily
verified that
\begin{gather*}
\lim_{s\to 0}\frac{\rho(s)}{|s|^{p+2}}= -\frac{(p+1-q)(p+2-N)}{N(q+1)(p+2)};\\
\lim_{s\to \infty}\frac{\rho(s)}{|s|^{p+2+\beta} e^{|s|^{\beta}}}=
-\frac{\beta}{N(q+1)}.
\end{gather*}
From these two estimates, we obtain
\begin{equation}\label{eq30}
\rho(s)\leq-\frac{(p+1-q)}{N(q+1)(p+2)}\left(p+2-N+\beta|s|^{\beta}\right)
|s|^{p+2}e^{|s|^{\beta}}.
\end{equation}
Therefore, using \eqref{eq31} and \eqref{eq30}, we obtain
\begin{align*}
J_{\lambda}(t_1v)
&\leq - \frac{(p+1-q)}{N(q+1)(p+2)}\int_{\Omega}
\left(p+2-N+\beta|t_1v|^{\beta}\right) |t_1v|^{p+2}  e^{|t_1v|^{\beta}} dx  \\
& \leq - \frac{(p+1-q)}{N(q+1)(p+2)}\int_{\Omega}|t_1v|^{p+2+\beta} dx
\end{align*}
Hence $\theta_{\lambda}\leq\inf_{u\in \mathcal{N}_{\lambda}^{+}\cap
H^{+}}J_{\lambda}(u)\leq  -\frac{(p+1-q)}{N(q+1)(p+2)}\; C_2$, where
$C_2=\int_{\Omega} |t_1v|^{p+2+\beta}dx$.
\end{proof}

 By Lemma \ref{th1}, $J_{\lambda}$ is bounded below on $\mathcal{N}_{\lambda}$. 
So, by Ekeland's Variational principle, we can find a
sequence $\{u_n\}\in \mathcal{N}_{\lambda}\setminus \{0\}$ such that
\begin{gather}\label{eq32}
J_{\lambda}(u_n)\leq \theta_{\lambda}+\frac{1}{n}, \\
\label{eq33}
J_{\lambda}(v)\geq J_{\lambda}(u_n)- \frac{1}{n}\|v-u_n\|\quad \forall\, 
v\in \mathcal{N}_{\lambda}.
\end{gather}
We claim that if $\alpha(V,h)>0$, $\beta(V,h)>0$ then 
$u_n\in E^{+}\cap H^{+}$.  Now from \eqref{eq32} and Lemma \ref{le33},
we have
\begin{align}\label{eq34}
 J_{\lambda}(u_n)&\leq -\frac{(p+1-q)}{N(q+1)(p+2)}\; C_3.
\end{align}
 As $u_n\in \mathcal{N}_{\lambda}$, we have
\begin{equation} \label{eq38}
J_{\lambda}(u_n) =\big(\frac{1}{N}-\frac{1}{q+1}\big)E_{V}(u_n)
+\int_{\Omega}\big(\frac{1}{q+1}g(u_n)u_n - G(u_n)\big) dx.
\end{equation}
 From \eqref{eq34} and \eqref{eq38}, we obtain
\begin{equation} \label{eq39}
E_{V}(u_n)\geq \frac{(p+1-q)}{(N-q+1)(p+2)}\; C_3>0
\end{equation}
 Also as $u_n\in \mathcal{N}_{\lambda}$, we have
\begin{align*}
&J_{\lambda}(u_n) \\
&=\big(\frac{1}{N}-\frac{1}{p+2}\big)E_{V}(u_n)
-\frac{\lambda(p+1-q)}{(q+1)(p+2)} H(u_n)
+\int_{\Omega}\big(\frac{1}{p+2}g(u_n)u_n - G(u_n)\big) dx.
\end{align*}
By this equality,  \eqref{eq39} and \eqref{eq34}, we obtain
\begin{equation} \label{eq35}
H(u_n) \geq \frac{C_3}{\lambda N}>0\quad \forall n.
\end{equation}
Thus we have $u_n\in \mathcal{N}_{\lambda}\cap E^{+}\cap H^{+}$. Now we
prove the following result.

\begin{proposition}\label{pro1}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ and $\lambda$ satisfies \eqref{eq8}.
Then $\|J_{\lambda}'(u_n)\|_{*}\to 0$ as $n\to \infty$.
\end{proposition}

\begin{proof}
 Step 1: $\liminf_{n\to \infty} E_{V}(u_n)>0$.
 Applying H\"{o}lders inequality in \eqref{eq35}, we have
$K'\|u_n\|^{q+1}\geq H(u_n) \geq \frac{C_3}{\lambda N}>0$ which implies
that $ \liminf_{n\to \infty} \|u_n\|>0$. Using this and Lemma
\ref{le2} we obtain $\liminf_{n\to \infty} E_{V}(u_n)>0$.

Step 2: We claim that
\begin{equation}\label{liminf3.4}
 K:=\liminf_{n \to \infty}\Big\{(N-1-q) E_{V}(u_n) - \int_{\Omega}
g'(u_n) {u^2_n} dx + q \int_{\Omega} g(u_n) u_n dx\Big\}>0.
\end{equation}
Assume by contradiction that for some subsequence of $\{u_n\}$,
still denoted by $\{u_n\}$ we have
\[
(N- 1-q) E_{V}(u_n) - \int_{\Omega} g'(u_n) u_n^2 dx + q
\int_{\Omega}g(u_n) u_n dx= o_{n}(1).
\]
From this and the fact that $E_{V}(u_n)$ is bounded away from $0$,
we obtain that
$ \liminf_{n\to \infty}\int_{\Omega}{g'(u_n) u_n^2}dx >0$.
 Hence, we obtain $u_n \in \Lambda\setminus \{0\}$ for all  $n$ large.
Using this and the fact that $u_n \in \mathcal{N}_{\lambda}\setminus \{0\}$,
we have
\begin{align*}
o_{n}(1)= \lambda(N-q-1) H(u_n)-\int_{\Omega}(g'(u_n) u_n^2
-(N-1)g(u_n)u_n) dx < -\Lambda_m
\end{align*}
by \eqref{eq8}, which is a contradiction.

 Finally, we show that $\|J_{\lambda}'(u_n)\|_{*}\to 0$ as 
$n\to \infty$. By Lemma \ref{le31}, we obtain a sequence of functions
$\xi_n : B(0,\epsilon_n)\to \mathbb{R}$ for some $\epsilon_n>0$ such that
$\xi_n(0)=1$ and $\xi_n(w)(u_n-w)\in \mathcal{N}_{\lambda}$ for all
$w\in B(0,\epsilon_n)$. Choose $0<\rho <\epsilon_n$ and $f\in
W^{1,N}_0(\Omega) $ such that $\|f\|=1$. Let $w_\rho=\rho f$. Then
$\|w_{\rho}\|=\rho<\epsilon_n$ and
$\eta_{\rho}=\xi_n(w_\rho)(u_n-w_\rho)\in\mathcal{N}_{\lambda}$ for all  $n$.
Since $\eta_{\rho} \in\mathcal{N}_{\lambda}$, we deduce from \eqref{eq33} and
Taylor's expansion,
\begin{equation} \label{eq:3.7}
\begin{aligned}
\frac{1}{n}\|\eta_{\rho}-u_n\|&\geq J_{\lambda}(u_n)-J_{\lambda}(\eta_\rho)
=\langle J_{\lambda}'(\eta_\rho),u_n-\eta_\rho\rangle + o(\|u_n-\eta_\rho\|) \\
&= (1-\xi_n(w_{\rho}))\langle J_{\lambda}'(\eta_\rho),u_n\rangle + \rho
\xi_n(w_{\rho})\langle J_{\lambda}'(\eta_\rho),f\rangle +
o(\|u_n-\eta_\rho\|).
\end{aligned}
\end{equation}
We note that as $\rho \to 0$, we have
$\frac{1}{\rho}\|\eta_\rho -u_n\|= \|u_n\langle\xi_{n}'(0),f\rangle -f\|$.
Now dividing \eqref{eq:3.7} by $\rho$ and taking the limit $\rho \to 0$,
and using $u_n\in \mathcal{N}_{\lambda}$, we obtain
\begin{equation}\label{eq36}
\langle J_{\lambda}'(u_n),f \rangle\leq
\frac{1}{n}\left(\|u_n\|\|\xi'_{n}(0)\|_{*}+1\right)\le
\frac{1}{n}\frac{C_4\|f\|}{K},
\end{equation}
by Lemma \ref{le31} and \eqref{liminf3.4}. This completes the proof.
\end{proof}
 We can now prove the following result.

\begin{lemma}\label{le34}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ and let $\lambda$ satisfy \eqref{eq8}.
Then there exists a function $u_{\lambda}\in \mathcal{N}_{\lambda}^{+}\cap H^+\cap
E^+$ such that $ J_{\lambda}(u_{\lambda})=\inf_{u\in \mathcal{N}_{\lambda}\setminus
\{0\}}J_{\lambda}(u)$.
\end{lemma}

\begin{proof} 
Let $u_n$ be a minimizing sequence for $J_{\lambda}$ on 
$\mathcal{N}_{\lambda}\setminus \{0\}$ satisfying \eqref{eq32} and \eqref{eq33}.
Then $\{u_n\}$ is bounded in $W_0^{1,N}(\Omega)$ by Lemma \ref{le01}.
Also there exists a subsequence of $\{u_n\}$ (still denoted by
$\{u_n\}$) and a function $u_\lambda$ such that 
$u_n \rightharpoonup u_\lambda$ weakly in $W_0^{1,N}(\Omega)$, 
$u_n\to u_\lambda$ strongly in $L^{\alpha}(\Omega)$ for all  $\alpha\geq 1$ and 
$u_n(x)\to u_{\lambda}(x)$ a.e in $\Omega$. Also $H(u_n) \to H(u_\lambda)$. 
By Proposition \ref{pro1}, $\|J_{\lambda}'(u_n)\|_{*}\to 0$. Then we have
\begin{gather*}
\nabla u_n(x)\to \nabla u_{\lambda}(x)\quad \text{a.e. in }\Omega,\\
{g(u_n)} \to {g(u_\lambda)}\quad \text{strongly in }L^{1}({\Omega}),\\
|\nabla u_n|^{N-2}\nabla u_n\rightharpoonup |\nabla u_{\lambda}|^{N-2}\nabla u_\lambda
\quad \text{weakly in }(L^{\frac{N}{N-1}}(\Omega))^{N}. 
\end{gather*}
In particular, it follows that $u_{\lambda}$
solves \eqref{PlaVh} and hence $u_\lambda \in \mathcal{N}_{\lambda}$. Moreover,
$\theta_\lambda \leq J_{\lambda}(u_\lambda)\leq \liminf_{n\to
\infty}J_{\lambda}(u_n)=\theta_\lambda$. 
Hence $u_\lambda$ is a minimizer for $J_{\lambda}$ on $\mathcal{N}_{\lambda}$.

 Using \eqref{eq35}, we have $H(u_\lambda)>0$ and $E_{V}(u_\lambda)>0$,
since $\beta(V,h)>0$. Therefore there exists $t_1(u_\lambda)$ such that
$t_1(u_\lambda)u_\lambda \in \mathcal{N}_{\lambda}^{+}$. We now claim that
$t_1(u_\lambda)=1$ $(i.e$. $u_\lambda \in \mathcal{N}_{\lambda}^{+})$. Suppose
$t_1(u_\lambda)<1$. Then $t_2(u_\lambda)=1$ and hence $u_\lambda \in \mathcal{N}_{\lambda}^{-}$. 
Now $J_{\lambda}(t_1(u_\lambda)u_\lambda)\leq J_{\lambda}(u_\lambda)=
\theta_{\lambda}$ which is impossible, as $t_1(u_\lambda) u_\lambda \in \mathcal{N}_{\lambda}$.
\end{proof}

\begin{theorem}\label{th3.6}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ and $\lambda$ be such that \eqref{eq8} holds.
Then $u_\lambda \in \mathcal{N}_{\lambda}^{+}\cap E^+\cap H^+$ is also a
non-negative local minimum for $J_{\lambda}$ in $W^{1,N}_0(\Omega)$.
\end{theorem}

\begin{proof} 
Since $u_\lambda \in \mathcal{N}^{+}_{\lambda}$, we have 
$t_1(u_{\lambda})=1 <t_*(u_{\lambda})$. Hence by continuity of $u\mapsto t_*(u)$, given
$\epsilon>0$, there exists $\delta=\delta(\epsilon)>0$ such that $1+\epsilon< t_*(u_\lambda-w)$
for all  $\|w\|<\delta$. Also, from Lemma \ref{le33} we have, for $\delta>0$
small enough, we obtain a $C^1$ map $t: \text{B}(0,\delta)\to \mathbb{R}^+$
such that $t(w)(u_\lambda-w)\in \mathcal{N}_{\lambda}$, $t(0)=1$. Therefore, for
$\delta>0$ small enough we have 
$t_1(u_\lambda-w)= t(w)<1+\epsilon<t_*(u_\lambda-w)$ for all  $\|w\|<\delta$. 
Since $t_*(u_\lambda-w)>1$, we obtain
$J_{\lambda}(u_\lambda)<J_{\lambda}(t_1(u_\lambda-w)(u_\lambda-w))<J_{\lambda}(u_\lambda-w)$
for all  $\|w\|<\delta$. This shows that $u_\lambda$ is a local minimizer for
$J_{\lambda}$. We can take $u_{\lambda}\geq 0$ as $J_{\lambda}(|u_\lambda|)=
J_{\lambda}(u_\lambda)$.
\end{proof}

The following Lemma is taken from \cite{qu}.

\begin{lemma}\label{le12}
(i) If either $\lambda_1(V)>0$ or $\lambda_1(V)=0$ and $\phi_{V}\in
H^-$ then $\beta(V,h)>0$.

(ii)  If $\phi_{V}\in H^{-}$, $\lambda_1(V)<0<\lambda_2(V)$ and
$\alpha(V,h)>0$ then $\beta(V,h)>0$.
\end{lemma}

\begin{proof}
(i) follows from $\beta(V,h)\geq \lambda_1(V)$ and
$\beta(V,h)=\lambda_1(V)$ if and only if $\phi_{V}\in H^+$.

 To prove (ii), define ${\tilde{\beta}}(V,h):= \min\{E_V(u);
H(u)\geq 0, \|u\|_{N}=1\}$. This minimum is achieved say $u_0$ and
positive. Also ${\tilde{\beta}}(V,h)>0$ implies $\beta(V,h)>0$. If
$H(u_0)=0$ then ${\tilde{\beta}}(V,h)\geq \alpha(V,h)>0$. If $H(u_0)> 0$
then $\tilde{\beta}(V,h)$ is actually an eigenvalue of $-\Delta_N +V$.
Since $\phi_{V}\in H^{-}$ we see that $\tilde{\beta}(V,h)\geq
\lambda_2(V)>0$.
\end{proof}

Now the proof of Theorem \ref{th11} follows from Lemma
\ref{le12} and Theorem \ref{th3.6}.

\section{Multiplicity results}

\subsection{Existence of a second solution in the subcritical case 
$(1<\beta<\frac{N}{N-1})$}

\begin{lemma}\label{le001}
For $\beta\in(1,\frac{N}{N-1})$, $u_n\rightharpoonup u$ implies
$G(tu_n)\to G(tu)$ in $L^{1}(\Omega)$ for all $t\in \mathbb{R}$.
\end{lemma}

\begin{proof}
Let $u_n\rightharpoonup u$ then $tu_n\rightharpoonup tu$ for every
$t\in \mathbb{R}$. By compactness of embedding $u\longmapsto
\int_{\Omega}e^{|u|^{\beta}}$, we have 
$\int_{\Omega}e^{|tu_n|^{\beta}}\leq C$, for some $C>0$. From this one 
can easily show that $g(tu_n)\to g(tu)$ in $L^{1}(\Omega)$. 
Also there exists $M>0$ such that $G(s)\leq (1+g(s))M$ for every 
$s\in \mathbb{R}$. Then using this and by applying
Lebesgue dominated convergence theorem, we obtain $G(tu_n)\to G(tu)$ in
$L^{1}(\Omega)$.
\end{proof}

\begin{lemma}\label{le03}
If $\alpha(V,h)>0$, $\beta(V,h)>0$. Then $J_{\lambda}$ achieve its minimizers
on $\mathcal{N}_{\lambda}^{-}\cap H^{+}_0\cap E^+$.
\end{lemma}

\begin{proof}
Note that $\alpha(V,h)>0$ and $\beta(V,h)>0$ imply
$H^{+}_0\cap E^+ \ne \emptyset$.
 Let $u_n\in \mathcal{N}_{\lambda}^{-}\cap H^{+}_0\cap E^+$
be a minimizing sequence for $J_{\lambda}$. Then $J_{\lambda}(u_n)$ is
bounded. By Lemma \ref{le01}, $\{u_n\}$ is a bounded sequence.
Therefore $u_n\rightharpoonup u_0$ weakly in $W^{1,N}_0(\Omega)$ and
$H(u_n)\to H(u_0)$. Also $H(u_0)\geq 0$ as $u_n\in H^{+}_0$. We
claim that $u_0\not\equiv 0$. Let us assume this for a moment, if
$H(u_0)=0$, $\alpha(V,h)>0$ then $E_{V}(u_0)>0$ and if $H(u_0)>0$,
$\beta(V,h)>0$ then $E_{V}(u_0)>0$. Thus $u_0\in H^{+}_0\cap E^+$
and we have $\phi_{u_0}$ has a global maximum at some $t_0$ so
that $t_0 u_0\in \mathcal{N}_{\lambda}^{-}\cap H_0^{+}\cap E^+$. Next we
claim that $u_n \to u_0$. Suppose this is not true then by using
Lemma \ref{le001}, we have
\[
J_{\lambda}(t_0 u_0)< \lim_{n\to \infty} J_{\lambda}(t_0 u_n).
\]
On the other hand, $u_n\in \mathcal{N}_{\lambda}^{-}$ implies that $1$ is a
global maximum point for $\phi_{u_n}$; i.e., $\phi_{u_n}(t_0)\leq
\phi_{u_n}(1)$. Thus we have
\[
\lim_{n\to\infty}J_{\lambda}(t_0 u_n)\leq  
\lim_{n\to \infty} J_{\lambda}(u_n)
= \inf_{u\in \mathcal{N}^{-}_{\lambda}\cap H_0^{+}\cap E^+} J_{\lambda}(u),
\]
which is a contradiction. Hence $u_n \to u_0$ and moreover 
$u_0\in \mathcal{N}^{+}_{\lambda}\cap H_0^+\cap E^+$, since 
$\mathcal{N}^{0}_{\lambda}=\emptyset$.

 Now we show that $u_0\not\equiv 0$. Suppose $u_0 \equiv 0$.
Then $E_{V}(u_n)= \lambda H(u_n)+\int_{\Omega}g(u_n)u_n dx\to \lambda
H(u_0)+\int_{\Omega}g(u_0)u_0 dx =0$ as $n\to\infty$. Therefore we have
$u_n\to 0$ strongly in $W^{1,N}_0(\Omega)$ as $n\to \infty$, since $\|u_n\|^N\leq
E_{V}(u_n)+\int_{\Omega}|V(x)||u_n|^N dx \to 0$ as $n\to \infty$. Now let
$v_n= u_n/|u_n\|$. Then $v_n\rightharpoonup v_0$ weakly
in $W^{1,N}_0(\Omega)$ and $H(v_n)\to H(v_0)$. Also $
E_{V}(u_n)\leq \frac{1}{N-1-q}\int_{\Omega}g'(u_n)u_{n}^2 dx$,
since $u_n\in \mathcal{N}_{\lambda}^{-}$. Thus
\begin{align*}
E_{V}(v_n)&\leq\frac{(p+1)\|u_n\|^{p+2-N}}{(N-1-q)}
 \int_{\Omega}|v_n|^{p+2} e^{\|u_n\|^{\beta}|v_n|^{\beta}}dx\\
&\quad+ \frac{\beta \|u_n\|^{p+2-N+\beta}}{(N-1-q)}
\int_{\Omega}|v_n|^{p+2+\beta} e^{\|u_n\|^{\beta}|v_n|^{\beta}}dx
\end{align*}
By using H\"{o}lders inequality, Sobolev embeddings, Moser Trudinger
inequality and $u_n\to 0$ strongly in $W^{1,N}_0(\Omega)$, one can easily show that
$\int_{\Omega}|v_n|^{p+2} e^{\|u_n\|^{\beta}|v_n|^{\beta}}$ and
$\int_{\Omega}|v_n|^{p+2+\beta}e^{\|u_n\|^{\beta}|v_n|^{\beta}}$ are
bounded. Thus $\limsup E_{V}(v_n)\to 0$ as $n\to \infty$. So
$E_{V}(v_0)\leq 0$. Also
\begin{align*}
&\lambda H(v_n)\\
&=\|u_n\|^{N-1-q}E_{V}(v_n)-\|u_n\|^{-1-q} \int_{\Omega}g(\|u_n\|v_n)\|u_{n}\|v_n\,dx \\
&\leq \|u_n\|^{N-1-q}E_{V}(v_n)+ \|u_n\|^{p+1-q}\int_{\Omega}|v_n|^{p+2}
  e^{\|u_n\|^{\beta}|v_n|^{\beta}}\,dx\\
&\leq K \|u_n\|^{N-1-q}+ \|u_n\|^{p+1-q}
\Big(\int_{\Omega}|v_n|^{(p+2)t'}\Big)^{1/t'}
\Big(\sup_{\|w_n\|\leq 1}
\int_{\Omega}e^{t\|u_n\|^{\beta}|w_n|^{\beta}}\Big)^{1/t}\\
&\leq K_1 \|u_n\|^{N-1-q} + K_2\|u_n\|^{p+1-q}
\|v_n\|^{p+2}\Big(\sup_{\|w_n\|\leq 1}
\int_{\Omega}e^{\alpha|w_n|^{\beta}}\Big)^{1/t}
 \to 0 \quad\text{as } n\to \infty,
\end{align*}
since $\|u_n\|\to 0$ as $n\to\infty$ we can choose $\alpha\leq \alpha_N$.
Thus $H(v_0)=0$. It is easy to see that $v_0\ne 0$. Hence
$\alpha(V,h)\leq 0$, which is a contradiction because $\alpha(V,h)>0$.
Hence $u_0\not\equiv0$.
\end{proof}


The proof of Theorem \ref{th12}  follows from Lemmas
\ref{le12} and  \ref{le03}.


\subsection{Existence of a second solution in the critical case 
$(\beta=\frac{N}{N-1})$}

For showing the existence of a second solutions we assume
$h\geq0$. Then $u\in H^{+}_0$ for every $u\in W^{1,N}_0(\Omega)$.
In this subsection, We show that the minimizing sequence in 
$\mathcal{N}_{\lambda}^{-}$ is a Palais-Smale sequence below the critical level. 
We analyze the critical level and show that the weak limit of
minimizing sequence is the required second solution of
\eqref{PlaVh}. To proceed further, we cut-off the nonlinearity from
$u_\lambda$. For this we define
\[
\tilde{g}(x,s) =
\begin{cases}
g(x,u_\lambda)& s\leq u_\lambda(x)\\
g(x,s)& s\geq u_\lambda(x)
\end{cases} \quad\text{and}\quad
\tilde{k}(x,s) =
\begin{cases}
h(x)u_{\lambda}^{q}(x)& s\leq u_\lambda(x)\\
h(x)s^q &s\geq u_\lambda(x),
\end{cases}
\]
where $\tilde {G}(x,s)=\int_0^{s}\tilde g(x,t)dt$ and 
$\tilde {K}(x,s)=\int_0^{s}\tilde k(x,t)dt$. Define 
$\tilde{J_{\lambda}}: W^{1,N}_0(\Omega)\to \mathbb{R}$ as
\begin{equation}
\tilde{J_{\lambda}}(u)= \frac{1}{N} E_{V}(u)-
\int_{\Omega}{\tilde{G}(x,u)}dx - {\lambda}\int_{\Omega} \tilde K(x,u)dx,
\end{equation}
where $\tilde {G}(x,u)=G(u)+ g(u_\lambda)u_\lambda -G(u_\lambda)$ and
$\tilde{K}(x,u)= K(u)+ \frac{q}{q+1}h(x)u_{\lambda}^{q+1}$. So
\begin{equation}\label{eq4.2}
\tilde{J_{\lambda}}(u)= J_{\lambda}(u)- \int_{\Omega} ( g(u_\lambda)u_\lambda-
G(u_\lambda))dx  -\frac{\lambda q}{q+1} H(u_\lambda).
\end{equation}
Then we have $\tilde{J}'_{\lambda}(u)=J'_{\lambda}(u)$,
$\tilde{\mathcal{N}}_{\lambda}=\mathcal{N}_{\lambda}$, 
$\tilde{\mathcal{N}}_{\lambda}^{+}=\mathcal{N}_{\lambda}^{+}$, 
$\tilde{\mathcal{N}}_{\lambda}^{-}=\mathcal{N}_{\lambda}^{-}$,
 $\tilde{\mathcal{N}}_{\lambda}^{0}=\mathcal{N}_{\lambda}^{0}$. Define
\[
\tilde{c}_1=\inf_{u\in\mathcal{N}_{\lambda}^{-}}\tilde{J}_{\lambda}(u),\quad
 {c}_1=\inf_{u\in \mathcal{N}_{\lambda}^{-}}{J_{\lambda}}(u).
\] 
Then $\tilde{J_{\lambda}}(u)\leq J_{\lambda}(u)$
for every $u$ and $\tilde{c}_1\leq{c}_1$. It is easy to see that
$u_\lambda$ is also a local minimum of $\tilde{J_\lambda}$. Next we define
\[
 U_1 :=\Big\{u=0\; \text{or}\; u\in W^{1,N}_0(\Omega):
\|u\|<t^{-}\big(\frac{u}{\|u\|}\big)\Big\}\]
and
\[
U_2 :=\Big\{u\in  W^{1,N}_0(\Omega): \|u\|
>t^{-}\big(\frac{u}{\|u\|}\big)\Big\}.
\]
Then we claim that 
$ \mathcal{N}^{-}_{\lambda}= \{u \in W^{1,N}_0\setminus \{0\}: \|u\|
=t^{-}(\frac{u}{\|u\|})\}$. Indeed, for $u\in \mathcal{N}^{-}_{\lambda}$ 
and $u\in H^{+}_0$ as $h\geq 0$. Let 
$v= \frac{u}{\|u\|}\in W^{1,N}_0(\Omega)$. Then by Lemma \ref{le5}, there
exists a unique $t^{-}(v)$ such that $t^{-}(v)v \in \mathcal{N}^{-}_{\lambda}$. 
Using this and the fact that $u\in\mathcal{N}^{-}_{\lambda}$ we have
 ${t^{-}(v)}=\|u\|$. For
other side, let $u\in W^{1,N}_0(\Omega)\setminus \{0\}$ such that
${t^{-}(v)}=\|u\|$. Then $t^{-}(v)v \in
\mathcal{N}^{-}_{\lambda}$ which implies $u\in \mathcal{N}^{-}_{\lambda}$.\\
So, combining above discussion, we have $
W^{1,N}_0(\Omega)\setminus\mathcal{N}_{\lambda}^{-}=U_1 \cup U_2$. Also, it is
easy to see that $\mathcal{N}_{\lambda}^{+}\subset U_1$. In particular, $u_\lambda
\in U_1$.  Fix $n_0\in \mathbb{N}$ and for any $M>0$, we define
\begin{equation}\label{eq:criclevel}
\rho_{M}=\min_{\gamma\in \mathcal{F}_M}\max_{t\in[0,1]}J_{\lambda}(\gamma(t))
\end{equation}
where $ \mathcal{F}_{M}=\{\gamma \in C([0,1]:  W^{1,N}_0(\Omega)) : 
\gamma(0)=u_\lambda,\; \gamma(1)= u_\lambda+M\phi_{n_0}\}$. Then we have the lemma:

\begin{lemma}
There exist $M>0$ such that $u_\lambda + M\phi_{n_0}\in U_2$ and
moreover, $\rho_{M}\geq c_1$.
\end{lemma}

\begin{proof}
 Firstly, we can easily choose a suitable constant
$S>0$ such that $0<t^{-}(u)<S$ for all  $u:\|u\|=1$. Recall the
inequality: For any $p\geq 2$,  there exists a constant $C(p)>0$
such that 
\[
|\xi_2|^p > |\xi_1|^p+ p|\xi_1|^{p-2} \langle \xi_1,
\xi_2-\xi_1 \rangle + \frac{C(p)}{2^p-1}|\xi_2-\xi_1|^p, \quad
\text{for every } \xi_1, \xi_2\in \Omega.
\]
 Let $M>0$ be such that $\frac{C(N)}{2^{N}-1}M^N \geq S$. 
Then we show that $w_{n_0}:=u_\lambda+M\phi_{n_0}\in U_2$. 
By using $u_\lambda$ is a solution of \eqref{PlaVh}, $h\geq 0$ and 
support$({\phi_{n_0}})\subset
B_{\delta_n}(0)$, we have
\begin{align*}
\|w_{n_0}\|^N 
&= \int_{\Omega} |\nabla(u_\lambda+ M\phi_{n_0})|^N dx\\
&\geq \int_{\Omega} (|\nabla u_\lambda|^N + M N |\nabla u_\lambda|^{N-2}\nabla
u_\lambda \cdot \nabla\phi_{n_0} +\frac{C(N)}{2^{N}-1}|M\nabla\phi_{n_0}(x)|^N)dx\\
&\geq \int_{\Omega} \left(|\nabla u_\lambda|^N + MN\left(
g(u_\lambda)+\lambda h(x)|u_{\lambda}|^{q-1}u_{\lambda}-
V(x)|u_{\lambda}|^{N-2}u_{\lambda}\right)\phi_{n_0}\right) \\
&\quad + \frac{C(N)M^N}{2^{N}-1}\\
&\geq \|u_\lambda\|^{N} +\frac{C(N)M^N}{2^{N}-1}\\
& >  \frac{C(N)}{2^{N}-1}M^N\geq S
> t^{-}\big(\frac{w_{n_0}}{\|w_{n_0}\|}\big).
\end{align*}
Next, we show that $\rho_{M}\geq c_1$. For this, it is sufficient to
show that every path starting from $u_{\lambda}$ to 
$u_{\lambda}+ M\phi_{n_0}$ intersects $\mathcal{N}^{-}_{\lambda}$. 
As $u_\lambda\in U_1$,
$u_{\lambda}+M\phi_{n_0}\in U_2$ and $\gamma$ is a continuous, so there
exists some $t_0$ such that 
\[
\|u_\lambda+ t_0\phi_{n_0}\|=t^{-}\Big(\frac{u_\lambda+
t_0\phi_{n_0}}{\|u_\lambda+ t_0\phi_{n_0}\|}\Big).
\]
 Hence $u_\lambda+ t_0\phi_{n_0}\in \mathcal{N}_{\lambda}^{-} $.
\end{proof}

  Also, we note that $J_{\lambda}(u_\lambda + t v)\to -\infty$ as $t\to\infty$ 
for any $v\in  W^{1,N}_0(\Omega)\setminus \{0\}$.
We obtain an upper bound on $\rho_M$ in the following Lemma. Proof
here is adopted from \cite{JSK}.

\begin{lemma} 
Let $\rho_M$ be defined as in \eqref{eq:criclevel}. Then
$\rho_M< J_{\lambda}(u_\lambda)+\frac{1}{N} \alpha_{N}^{N-1}$.
\end{lemma}

\begin{proof} 
Let $\delta_n>0$ be such that $\delta_n \to 0$ as $n\to\infty$.
Then we define a sequence of Moser functions in $\Omega$ as
$$ 
\phi_{n}(x) = \frac{1}{w_{N-1}^{1/N}}
\begin{cases}
(\log n)^{\frac{N-1}{N}} & 0\leq \frac{|x|}{\delta_n}\leq {\frac{1}{n}};\\[3pt]
 \frac{\log{\frac{\delta_n}{|x|}}}{(\log n)^{1/N}}
 & \frac{1}{n}\leq \frac{|x|}{\delta_n}\leq  1;\\[3pt]
 0 & \frac{|x|}{\delta_n}\geq 1,
\end{cases}
$$
with $\operatorname{support}(\phi_n) \subset B_{\delta_n}(0)$.
 We choose this ball in
such a way that $V\leq 0$ on $B_{\delta_n}(0)$. It can be easily seen
that $\|\nabla \phi_n\|_{N} =1$ for all  $n$. We prove the Lemma by
contradiction argument. Suppose 
$\rho_M\geq J_{\lambda}(u_\lambda)+\frac{1}{N} \alpha_{N}^{N-1}$. Then for each $n$, there
exist $t_n$ such that
\begin{equation} \label{eqa1}
\sup_{t>0}J_{\lambda}(u_\lambda+t\phi_n)=J_{\lambda}(u_\lambda+t_n\phi_n)\geq
J_{\lambda}(u_\lambda)+\frac{1}{N}\alpha_{N}^{N-1}.
\end{equation}
Then from \eqref{eqa1}, we obtain $\{t_n\}$ is a bounded sequence,
otherwise $J_{\lambda}(u_\lambda+t_n\phi_n)\to -\infty$.

Now using the one dimensional inequality: $(1+t^2+2t \cos \alpha)^{N/2}
\le 1+t^N+Nt \cos \alpha + O(t^2+t^{N-1})$ for $t\ge 0$ to estimate
$|\nabla(u_\lambda+t\phi_{n})|^N$ in $J_{\lambda}(u_\lambda+t_n\phi_n)$ as in
\cite{JSK}, we obtain
\[
J_{\lambda}(u_\lambda+t_n\phi_n) 
\le  \frac{t_{n}^{N}}{N}+ J_{\lambda}(u_\lambda)+ t_{n}^{2}O\Big(\delta_{n}^{N-2}(\log
n)^{\frac{-2}{N}}\Big)
+ t_{n}^{N-1}O\Big(\delta_n (\log n)^{{\frac{(1-N)}{N}}}\Big)
\]
Using \eqref{eqa1}, and choosing $\delta_n= (\log n)^{-1/N}$, we obtain
\begin{align}\label{eqa4}
{t_{n}^{N}}\geq \alpha_{N}^{N-1},
\end{align}
since $t_n$ is bounded. Now $t_n$ is a point of maximum for one
dimensional map $t\to J_{\lambda}(u_\lambda+t_n\phi_n)$ and hence,
$\frac{d}{dt}J_{\lambda}(u_\lambda+t\phi_n)|_{t=t_n}=0$. So,
\begin{equation} \label{eqa5}
\begin{aligned}
&\int_{\Omega}(|\nabla(u_\lambda+t_n\phi_n)|^{N-2} \nabla(u_\lambda+t_n\phi_n) \nabla
\phi_n + V(x)|u_\lambda+t_n\phi_n|^{N-2}(u_\lambda+t_n\phi_n) \phi_n) dx \\
& = \int_{\Omega}g(u_\lambda+t_n\phi_n)\phi_n \,dx
+{\lambda}\int_{\Omega}h(x)|u_\lambda+t_n\phi_n|^{q-1}(u_\lambda+t_n\phi_n)\phi_n\,dx
\end{aligned}
\end{equation}
Let $c_n=\min_{|x|\leq \frac{\delta_n}{n}}u_\lambda(x)$. Then
\begin{equation} \label{eqa6}
\begin{aligned}
\int_{\Omega} g(u_\lambda+t_n\phi_n)\phi_n\, dx
& \geq \int_{|x|\leq \frac{\delta_n}{n}}
 g(u_\lambda+t_n\phi_n)e^{|u_\lambda+t_n\phi_n|^{\frac{N}{N-1}}}\phi_n\, dx  \\
&\geq\frac{w_{N-1}}{N}
\big(\frac{\delta_n}{n}\big)^{N}\phi_{n}(0)|t_n\phi_{n}(0)|^{p+1}
e^{(c_n+t_n\phi_n(0))^{\frac{N}{N-1}}}.
\end{aligned}
\end{equation}
Now using Taylor's expansion and \eqref{eqa4}, for some
$K_0>0$, we obtain
\begin{equation} \label{eqa7}
\begin{aligned}
(c_n+t_n\phi_n(0))^{\frac{N}{N-1}}
&\geq(t_n\phi_n(0))^{\frac{N}{N-1}}+\frac{Nc_n}{N-1}(t_n\phi_n(0))^{\frac{1}{N-1}}  \\
&\geq N\log{n}+K_0 (\log{n})^{1/N}t_{n}^{\frac{1}{N-1}},
\end{aligned}
\end{equation}
since $t_n$ is bounded away from zero. Using \eqref{eqa6} and
\eqref{eqa7}, we obtain
\begin{equation} \label{eq09}
\begin{aligned}
\int_{\Omega}  g(u_\lambda+t_n\phi_n)\phi_n
& \geq \frac{w_{N-1}}{N}
\big(\frac{\delta_n}{n}\big)^{N}\phi_{n}(0)|t_n\phi_{n}(0)|^{p+1}
e^{(c_n+t_n\phi_n(0))^{\frac{N}{N-1}}} \\
&\geq\frac{w_{N-1}}{N}
\big(\frac{\delta_n}{n}\big)^{N}\phi_{n}(0)|t_n\phi_{n}(0)|^{p+1}
e^{(N\log{n} + K_0(\log{n})^{1/N})} \\
&\geq \frac{w_{N-1}}{N}|t_n\phi_{n}(0)|^{p+2}(\log
n)^{-1}e^{\frac{K_0}{2}(\log{n})^{1/N}}\\
 \to \infty\quad\text{as } n\to \infty.
\end{aligned}
\end{equation}
So, the right hand side of \eqref{eqa5} tends to $\infty$ as
$n\to \infty$ but the left hand side is bounded, which is a contradiction.
  Hence  $\rho_{M}< J_{\lambda}(u_\lambda)+\frac{1}{N}\alpha_{N}^{N-1}$.
\end{proof}

\begin{lemma}\label{le32}
Let  $\alpha(V,h)>0$, $\beta(V,h)>0$, then given
 $u\in \mathcal{N}_{\lambda}^{-}\cap H^{+}$, there exist $\epsilon>0$ and a 
differentiable function $\xi^{-} :  B(0,\epsilon)\subset W^{1,N}_0(\Omega) \to
\mathbb{R}$ such that $\xi^{-}(0)=1$ and the function
$\xi^{-}(w)(u-w)\in \mathcal{N} _{\lambda}^{-}$ and for all 
$w\in W^{1,N}_0(\Omega)$,
\begin{align*}
&\langle(\xi^{-})'(0),w\rangle  \\
&=\frac{ N R(u,w) -\int_{\Omega}\left(g(u)w+g'(u)uw\right)dx - \lambda(q+1)
\int_{\Omega} h(x)|u|^{q-1}uw dx}{(N-q-1) E_{V}(u)-
\int_{\Omega}{g'(u) u^2}dx+ q\int_{\Omega} g(u) u dx}.
\end{align*}
\end{lemma}

\begin{proof} 
First, we note that if $u\in\mathcal{N}_{\lambda}^{-}$, then
$u\in\Lambda\setminus\{0\}$, satisfies \eqref{eq8}. 
Then  Lemma \ref{le31}, there exist $\epsilon>0$ and a differentiable 
function $\xi^{-} :  B(0,\epsilon)\subset W^{1,N}_0(\Omega) \to \mathbb{R}$
such that $\xi^{-}(0)=1$ and the function 
$\xi^{-}(w)(u-w)\in \mathcal{N} _{\lambda}$
for all $w\in  B(0,\epsilon)$. Since
 $u\in \mathcal{N}_{\lambda}^{-}$, we have
\[
(N-1-q) E_{V}(u)+q \int_{\Omega} g(u)u\, dx - \int_{\Omega}{g'(u)u^2}\,dx <0.
\]
Thus by continuity of  $J_{\lambda}'$ and $\xi^{-}$, we have
\begin{align*}
&\phi''_{(\xi^{-}(w)(u-w))}(1) \\
&= (N-1-q) E_{V}(\xi^{-}(w)(u-w))+q \int_{\Omega}
g(\xi^{-}(w)(u-w))\xi^{-}(w)(u-w)  \\  
&\quad - \int_{\Omega} g'(\xi^{-}(w)(u-w))(\xi^{-}(w)(u-w))^2 <0 ,
\end{align*}
if $\epsilon$ is sufficiently small. This concludes the proof.
\end{proof}

 We recall the following results which will be used later.

\begin{proposition}[\cite{li}] \label{pl}
Let $\{v_n : \|v_n\|=1\}$ be a sequence in $W^{1,N}_0(\Omega)$
converging weakly to a non-zero $v$. Then for every
$p<(1-\|v\|^N)^{\frac{-1}{N-1}}$,\[\sup_{n}\int_{\Omega} exp(p \alpha_{N}
|v_n|^{\frac{N}{N-1}})< \infty.\]
\end{proposition}

\begin{lemma}[\cite{do6, JSK}] \label{lions} 
Let  $\{v_n\} \subset W^{1,N}_0(\Omega)$ be
Palais-Smale sequence; that is, $J(v_n) \to c$, 
$J'(v_n) \to 0$ as $n\to \infty$. Then there exists a
subsequence $\{v_n\}$ of $\{v_n\}$ and $v\in W^{1,N}_0(\Omega)$ such
that $G(v_n) \to G(v)$, $g(v_n)\to g(v)$ strongly in
$L^{1}(\Omega)$.
\end{lemma}

 Now we show the existence of a second non-trivial solution which
is different from $u_\lambda$.

\begin{theorem}\label{th41}
Let $\alpha(V,h)>0$, $\beta(V,h)>0$ and $\lambda$ satisfies \eqref{eq8}.
Then there exist a minimizing sequence $\{v_n\}$ in $\mathcal{N}^{-}_{\lambda}$ 
and $v_\lambda$ such that $v_n \rightharpoonup v_\lambda$
weakly in $W^{1,N}_0(\Omega)$, $v_\lambda$ is a non-negative solution for
\eqref{PlaVh} and moreover it is different from $u_\lambda$.
\end{theorem}

\begin{proof} 
We note that $\mathcal{N}_{\lambda}^{-}$ is a closed set, as $t^{-}(u)$
is a continuous function of $u$ and  $\tilde{J}_{\lambda}$ is bounded
below on $\mathcal{N}_{\lambda}^{-}$. Therefore, by Ekeland's Variational
principle, we can find a sequence $\{v_n\}\in \mathcal{N}_{\lambda}^{-}$ such
that
\begin{equation} \label{eq46}
\begin{aligned}
\tilde{J}_{\lambda}(v_n)
&\leq \inf_{u\in \mathcal{N}_{\lambda}^{-}}\tilde{J}_{\lambda}(u) +\frac{1}{n} \\
\tilde{J}_{\lambda}(v)&\geq\tilde{J}_{\lambda}(v_n)-\frac{1}{n}\|v-v_n\|
\quad \forall v\in \mathcal{N}_{\lambda}^{-}.
\end{aligned}
\end{equation}

Now $v_n\in \mathcal{N}_{\lambda}\cap H^{+}_0$ then by Lemma \ref{le01}, we
have $\{v_n\}$ is a bounded sequence in $W^{1,N}_0(\Omega)$. From
 \eqref{eq46}, we have $J_{\lambda}(v)\geq
J_{\lambda}(v_n)-\frac{1}{n}\|v-v_n\|$ for all  $v\in \mathcal{N}_{\lambda}^{-}$. It
is easy to see that $v_n\in\mathcal{N}_{\lambda}^{-}$ implies 
$v_n\in \Lambda\setminus \{0\}$. Then $\liminf_{n\to\infty}E_{V}(v_n)>0$,
follows from the Step 1 of Lemma \ref{le3}. Thus by Lemma \ref{le32}
and following the proof of Proposition \ref{pro1}, we obtain
$\|\tilde{J}_{\lambda}'(v_n)\|_{*}\to 0$ as $n\to \infty$. Thus
following the proof as in Lemma \ref{le34}, we have $v_\lambda$,  weak
limit of sequence $\{v_n\}$, is a solution of \eqref{PlaVh}.
Taking $\phi=v_{\lambda}^{-}$ as a test function in \eqref{def1}, we
have
\begin{align*}
&\int_{\Omega}\left(|\nabla v_{\lambda}|^{N-2}\nabla v_{\lambda} \nabla v_{\lambda}^{-} +
V(x)|v_{\lambda}|^{N-2}v_{\lambda} v_{\lambda}^{-} \right)dx \\
&= \int_{\Omega}g(v_{\lambda}) v_{\lambda}^{-} dx +\lambda \int_{\Omega}
h(x)|v_{\lambda}|^{q-1}v_{\lambda} v_{\lambda}^{-} dx.
\end{align*}
Using this we obtain
\begin{align*}
 E_{V}(v_{\lambda}^{-})
&=\int_{\Omega}(|\nabla v_{\lambda}^{-}|^{N} + V(x)|v_{\lambda}^{-}|^{N})dx \\
&= -\int_{\Omega}g(v_{\lambda}) v_{\lambda}^{-} dx - \lambda \int_{\Omega}
h(x)|v_{\lambda}|^{q-1}v_{\lambda} v_{\lambda}^{-} dx \leq 0.
\end{align*}
since $\tilde{g}(x,s)\geq 0$ and $\tilde{k}(x,s)\geq 0$ for all
$s\in \mathbb{R}$. As $H(v_{\lambda}^{-})\geq 0$, $\alpha(V,h)>0$ and
$\beta(V,h)>0$ so by using Lemma \ref{le2} we obtain that
$\|v_{\lambda}^{-}\|=0$.
Hence $v_\lambda\geq 0$ in $\Omega$.

 Finally, we show $u_\lambda\not\equiv v_\lambda$. By \eqref{eq4.2}, we
see that $\tilde{J_\lambda} (v_n) \to \tilde{c_1}$ which is equivalent
to $J_\lambda (v_n) \to c_1$ as $n\to \infty$.

 Case 1:  Suppose $u_\lambda\equiv v_\lambda$, $c_0=c_1$, then
\begin{align*}
J_{\lambda}(u_{\lambda})
&=c_0=c_1= \lim_{n\to\infty}{J}_{\lambda}(v_n)\\
&= \lim_{n\to\infty}\frac{1}{N}\|v_n\|^{N}
 -\int_{\Omega} G(v_n)dx -\frac{\lambda}{q+1} H(v_n)\\
&= \lim_{n\to\infty}\frac{1}{N}\|v_n\|^{N}-\int_{\Omega}
G(u_\lambda)dx -\frac{\lambda}{q+1} H(u_\lambda)\\
&=\lim_{n\to\infty}\frac{1}{N}\|v_n\|^{N}+
J_{\lambda}(u_{\lambda})-\frac{1}{N}\|u_{\lambda}\|^{N}.
\end{align*}
Thus $ \lim_{n\to\infty}\|v_n\|=\|u_\lambda\|$. Since 
$\nabla v_{n}(x)\to \nabla u_{\lambda}(x)$ a.e. pointwise then we have 
$v_n\to u_{\lambda}$ strongly in $W^{1,N}_0(\Omega)$ and
 $u_\lambda \in \mathcal{N}_{\lambda}^{-}$, as $\mathcal{N}_{\lambda}^{-}$ is a closed set. 
This is a contradiction as $u_\lambda \in \mathcal{N}_{\lambda}^{+}$.

 Case 2: Suppose $u_\lambda\equiv v_\lambda$, $c_0<c_1$.
For proving this case, we use the same idea as in \cite{JSK}. Using
$J_{\lambda}(v_n)\to c_1$ and Lemma \ref{lions}, we have
\begin{equation}\label{eq41}
\lim_{n\to \infty}\|v_n\|^{N}= c_1 N-\int_{\Omega} V(x)|u_{\lambda}|^N + N
\int_{\Omega} {G(u_\lambda)} + \frac{N\lambda}{q+1} H(u_\lambda).
\end{equation}
Setting 
\[
w_n=\frac{v_n}{\|v_n\|_{N}}\quad\text{and}\quad 
w_{\lambda}=\frac{u_\lambda}{\lim_{n\to\infty}\|v_n\|},
\]
we have $\|w_{\lambda}\|\leq 1$ and $w_{n}\rightharpoonup w_{\lambda}$ weakly in
$W^{1,N}_0(\Omega)$.
Now, the following two possibilities occurs:

(i) $\|w_{\lambda}\| =1$, in this case we have $\lim_{n\to \infty}\|
v_n\|=\|u_\lambda\|$. Then we have $v_n \to u_\lambda$ in $W^{1,N}_0(\Omega)$
and $u_\lambda \in \mathcal{N}_{\lambda}^{-}$, which gives a contradiction as 
$u_\lambda \in \mathcal{N}_{\lambda}^{+}$.

(ii) $\|w_{\lambda}\|<1$ and we have $c_1<c_0 +\frac{1}{N}
\alpha_{N}^{N-1}$. Then there exists $\epsilon>0$ small enough such that
\[
(1+\epsilon)<\frac{\alpha_N}{[N(c_1-J_{\lambda}(u_{\lambda}))]^{\frac{1}{N-1}}}.
\]
Set 
\[
\beta_0 := -\frac{1}{N}\int_{\Omega} V(x)|u_\lambda|^N dx
+\int_{\Omega} G(u_{\lambda})dx + \frac{\lambda}{q+1}H(u_\lambda).
\]
Then from \eqref{eq41} we have
$\lim_{n\to \infty}\|v_n\|^{N}=  N(c_1+\beta_0)$.
Also for sufficiently large $n$,
\[
(1+\epsilon)\|v_n\|^{\frac{N}{N-1}}
<\frac{\alpha_N\|v_n\|^{\frac{N}{N-1}}}{[N(c_1-J_{\lambda}
(u_{\lambda}))]^{\frac{1}{N-1}}}=\frac{\alpha_N}{(1-\|w_\lambda\|^N)^{1/N}},
\]
 since $w_{\lambda}= \frac{u_{\lambda}}{(N(c_1+\beta_0))^{1/N}}$.
 Choosing $p$ such that $(1+\epsilon)\|v_n\|^{\frac{N}{N-1}} <p<
\frac{\alpha_N}{(1-\|w_\lambda\|^N)^{1/N}}$. Now,
\begin{align*}
\int_{\Omega} f(v_n)(v_n-u_{\lambda})dx
&=\int_{\Omega}|v_n|^{p+1}e^{|v_n|^{\frac{N}{N-1}}}(v_n-u_{\lambda} )dx\\
&\leq\Big(\int_{\Omega}|v_n-u_{\lambda}|^t\Big)^{1/t}
\Big(\int_{\Omega}e^{(1+\epsilon)t'\|v_n\|^{\frac{N}{N-1}}|w_n|^{\frac{N}{N-1}}}
\Big)^{1/t'}
\end{align*}
choose $t>1$ sufficiently close to $1$ such that
$(1+\epsilon)t'\|v_n\|^{\frac{N}{N-1}} \leq p<
\frac{\alpha_N}{(1-\|w_\lambda\|^N)^{1/N}}$. Then by Proposition
\ref{pl}, we have 
\[
\int_{\Omega} f(v_n)(v_n-u_{\lambda})dx\to 0\quad\text{as } n\to\infty.
\]  
From this and
$J_{\lambda}'(u_n)(v_n-u_{\lambda})\to 0$ as $n\to \infty$ we obtain
\[
\int_{\Omega}|\nabla v_n|^{N-2}\nabla v_n (\nabla v_n -\nabla u_\lambda) dx \to 0\quad
\text{as } n\to\infty
\]
Moreover, since $v_n\rightharpoonup u_{\lambda}$ we have
\[
\int_{\Omega}|\nabla u_\lambda|^{N-2}\nabla u_\lambda (\nabla v_n -\nabla u_\lambda) dx \to 0\quad
\text{as } n\to\infty
\]
Hence, by using $|a-b|^N\leq 2^{N-2}(|a|^{N-2}a-|b|^{N-2}b)(a-b)$
for all $a,b\in \mathbb{R}^N$ we have
\[
\int_{\Omega}|\nabla v_n-\nabla u_\lambda|^{N}\to 0\quad \text{as } n\to\infty.
\]
Thus $v_n\to u_{\lambda}$ strongly in $W^{1,N}_0(\Omega)$, which is again
a contradiction as $u_\lambda \in \mathcal{N}_{\lambda}^{+}$. Thus $u_\lambda$ and
$v_\lambda$ are distinct.
\end{proof}


The proof of Theorem \ref{th13} follows from Theorems \ref{th3.6} and
\ref{th41} .

\section{Non-existence of solutions}

We derive non-existence results for \eqref{PlaVh}. Let
$\Omega_{h}^{+}$ be the largest domain where $h>0$ and $\phi({h}^{+})$
be the eigenfunction corresponding to the first eigenvalue
$\lambda_1(\Omega_{h}^{+})$ of $\Delta_{N}+V$ in $W^{1,N}_0(\Omega_{h}^{+})$.
We also assume that $\Omega_{h}^{+}\ne \emptyset$. 
First, we recall the Picone's identity (see \cite{WY}),

\begin{theorem}\label{th01}
 Let $v>0$, $u\geq 0$ in $W^{1,N}_0(\Omega)$. Then
\[
|\nabla u|^N -\nabla \big(\frac{u^N}{v^{N-1}}\big)|\nabla v|^{N-2}
\nabla v \geq 0\quad \text{a.e. in }\Omega;
\]
moreover, equality holds if and only if $u$ is a multiple of
$v$.
\end{theorem}

In the following Lemma, we only show the non-existence of solutions
that are positive in $\Omega_{h}^{+}$.

\begin{lemma}\label{lem5.2}
If $\lambda_1(V,\Omega_{h}^{+})<0$, then for every $\lambda>0$, \eqref{PlaVh}
 has no solution such that $u>0$ in $\Omega_{h}^{+}$.
\end{lemma}

\begin{proof} 
We extend $\phi({h}^{+})$ by zero outside $\Omega_{h}^{+}$ so
that $\phi({h}^{+})\in W^{1,N}_0(\Omega)$. Also we have
\[
\int_{\Omega_{h}^{+}} (|\nabla \phi({h}^{+})|^N +V |\phi(h^+)|^N ) dx
= \lambda_1(\Omega_{h}^{+}) \int_{\Omega_{h}^{+}} \phi({h}^{+})^{N}dx.
\]
If $u$ is a solution of \eqref{PlaVh}. Then for $\epsilon>0$, 
consider $\frac{\phi({h}^{+})^N}{(u+\epsilon)^{N-1}}$ as
a test function and we obtain
\begin{align*}
&\int_{\Omega_{h}^{+}} \Big(|\nabla u|^{N-2}\nabla u  \cdot \nabla
\big(\frac{\phi({h}^{+})^{N}}{(u+\epsilon)^{N-1}}\big) + V|u|^{N-2}u
\frac{\phi({h}^{+})^N}{(u+\epsilon)^{N-1}}\Big)dx\\
&= \lambda\int_{\Omega_{h}^{+}}h(x) u^{q}
\frac{\phi({h}^{+})^N}{(u+\epsilon)^{N-1}} dx+
\int_{\Omega_{h}^{+}}{g(u)}\frac{\phi({h}^{+})^N}{(u+\epsilon)^{N-1}} dx.
\end{align*}
After subtracting the above two equations and taking limit as $\epsilon\to 0$,
 we see that the left-hand side is non-negative by Theorem \ref{th01} 
and the right hand side is negative which is a contradiction.
\end{proof}

\begin{lemma}\label{lem5.3}
Let $h>0$ and  $\lambda_1(V)>0$. Then there exists $\lambda^0>0$ such that
\eqref{PlaVh} has no solution for $\lambda>\lambda^0$.
\end{lemma}

\begin{proof} 
Assume by contradiction that for all
$\lambda>0$, $(P_\lambda)$ has a solution $u_{\lambda}$. Then
\begin{equation} \label{eq5.2}
-\Delta_N u_{\lambda}+V(x)|u_{\lambda}|^{N-2}u_{\lambda} = {u_{\lambda}|u_{\lambda}|^{p}}
e^{|u_{\lambda}|^{\frac{N}{N-1}}} +\lambda hu_{\lambda}^{q}, \quad \text{in }
\Omega.
\end{equation}
We can choose $\lambda>0$ large such that
\begin{equation}\label{eq5.1}
\lambda_1(V) - \lambda h t^{q+1-N} - g(t)t^{1-N} < 0
\end{equation}
 for all $t>0$, and for almost every $x\in \mathbb{R}^N$. Also we have
\[
\int_{\Omega} (|\nabla \phi_1|^N +V |\phi_1|^N) dx
= \lambda_1(V) \int_{\Omega} \phi_1^{N} dx,
\]
where $\lambda_1(V)$ is an eigenvalue of $-\Delta_N+V$ corresponding to
$\phi_1$. Now multiply \eqref{eq5.2}
by $\frac{\phi_1^{N}}{(u_{\lambda}+\epsilon)^{N-1}}$ and integrating by parts we obtain
\begin{align*}
&\int_{\Omega}\Big(|\nabla u_{\lambda}|^{N-2} \nabla u_{\lambda} \cdot \nabla
\Big(\frac{\phi_1^{N}}{(u_{\lambda}+\epsilon)^{N-1}}\Big) + V
|u_{\lambda}|^{N-2}u_\lambda
\Big(\frac{\phi_1^{N}}{(u_{\lambda}+\epsilon)^{N-1}}\Big)\Big)dx\\
&=\lambda\int_{\Omega}h(x)
u_{\lambda}^{q}\Big(\frac{\phi_1^{N}}{(u_{\lambda}+\epsilon)^{N-1}}\Big)dx +
\int_{\Omega}{g(u_{\lambda})}\Big(\frac{\phi_1^{N}}{(u_{\lambda}+\epsilon)^{N-1}}\Big)
dx.
\end{align*}
After subtracting the above two equations and letting $\epsilon\to 0$ we
obtain that the left hand side is non-negative by Theorem \ref{th01},
and the right hand side
\[
\int_{\Omega} \left(\lambda_1(V) - \lambda h(x) u_{\lambda}^{q+1-N} -
{g(u_{\lambda})u_{\lambda}^{1-N}}\right) \phi_1^N dx <0,
\]
by \eqref{eq5.1}, a contradiction and hence the result follows.
\end{proof}

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\end{document}