\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \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 $\mathbb{R}^N$, $N\geq 2$, $0< q0$. 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 $\Omega\subset\mathbb{R}^N$, $N\geq 2$: $$\label{PlaVh} \begin{gathered} -\Delta_N u+V(x)|u|^{N-2}u = \lambda h(x)|u|^{q-1}u+ u|u|^{p} e^{|u|^{\beta}} \quad \text{in }\Omega \\ u \geq 0 \quad \text{in } \Omega,\quad u\in W^{1,N}_0(\Omega),\\ u =0\quad \text{on } \partial \Omega \end{gathered}$$ where $\Delta_N u= \nabla\cdot(|\nabla u|^{N-2}\nabla u)$, $0< q0$. 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 $N\geq2$, $u\in W^{1,N}_0(\Omega)$ $$\label{eq002} \sup_{\| u\| \leq 1}\int_{\Omega} e^{\alpha |u|^{\frac{N}{N-1}}} dx <\infty$$ 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 $10$ 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 $$\label{eq1} J_{\lambda}(u)= \frac{1}{N} E_{V}(u)- \frac{\lambda}{q+1} H(u)- \int_{\Omega} G(u) dx,$$ 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 $$\label{def1} \int_{\Omega}\left(|\nabla u|^{N-2}\nabla u \nabla \phi + V(x)|u|^{N-2}u\phi\right)dx = \int_{\Omega}g(u)\phi dx +\lambda \int_{\Omega} h(x)|u|^{q-1}u\phi dx.$$ \end{definition} We remark that the similar existence results with some obvious modification can be proved for critical exponent problem for $p$-Laplacian with $p0$ 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 $00$ 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 \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} 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 $00$ 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 $t_1(u)0$, $\beta(V,h)>0$, 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 $\lambda\in(0,\lambda_0)$, $$\label{eq8} \Lambda_m:=\inf_{u\in \Lambda\setminus\{0\} \cap H^{+}_0}\Big\{\int_{\Omega} \left(p+2-N +\beta{|u|^{\beta}}\right)|u|^{p+2} e^{|u|^{\beta}}- (N-1-q) {\lambda} H(u) \Big\}>0.$$ \end{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 $t_1= t_1(u)0$ 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 $t_1=t_1(u)t_*$ 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 $$\label{eq3} J_{\lambda}(u) = \frac{1}{N} \int_{\Omega}{g(u)u}\, dx- \int_{\Omega} {G(u)}\, dx -{\lambda}\big(\frac{1}{q+1}-\frac{1}{N}\big) H(u).$$ 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 $$\frac{1}{N}g(u)u- G(u)\geq \big(\frac{1}{N}-\frac{1}{p+2}\big)|u|^{p+2+\beta} ,$$ 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 $$\label{eq5} \theta_{\lambda} \geq -C(p,q,N)\lambda^{\frac{k}{k-1}},$$ 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 $\mathcal{N}_{\lambda}$, \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} 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 $w\in W^{1,N}_0(\Omega)$, $$\label{eq:3.1} \langle\xi'(0),w\rangle =\frac{ N R(u,w)-\int_{\Omega}\big({g(u)+g'(u)u}\big)w\,dx - \lambda (q+1)\int_{\Omega} h(x)|u|^{q-1}uw\, dx} {(N-q-1)E_{V}(u)- \int_{\Omega}{g'(u) u^2}dx+ q\int_{\Omega} g(u) u \,dx},$$ 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 \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} 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 $$\label{eq30} \rho(s)\leq-\frac{(p+1-q)}{N(q+1)(p+2)}\left(p+2-N+\beta|s|^{\beta}\right) |s|^{p+2}e^{|s|^{\beta}}.$$ 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 $$\label{eq38} J_{\lambda}(u_n) =\big(\frac{1}{N}-\frac{1}{q+1}\big)E_{V}(u_n) +\int_{\Omega}\big(\frac{1}{q+1}g(u_n)u_n - G(u_n)\big) dx.$$ From \eqref{eq34} and \eqref{eq38}, we obtain $$\label{eq39} E_{V}(u_n)\geq \frac{(p+1-q)}{(N-q+1)(p+2)}\; C_3>0$$ 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 $$\label{eq35} H(u_n) \geq \frac{C_3}{\lambda N}>0\quad \forall n.$$ 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 $$\label{liminf3.4} K:=\liminf_{n \to \infty}\Big\{(N-1-q) E_{V}(u_n) - \int_{\Omega} g'(u_n) {u^2_n} dx + q \int_{\Omega} g(u_n) u_n dx\Big\}>0.$$ 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, \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} 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 $$\label{eq36} \langle J_{\lambda}'(u_n),f \rangle\leq \frac{1}{n}\left(\|u_n\|\|\xi'_{n}(0)\|_{*}+1\right)\le \frac{1}{n}\frac{C_4\|f\|}{K},$$ 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 $t_1(u_{\lambda})=1 0$, 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 $t_1(u_\lambda-w)= t(w)<1+\epsilon1$, we obtain $J_{\lambda}(u_\lambda)0$ 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 $$\tilde{J_{\lambda}}(u)= \frac{1}{N} E_{V}(u)- \int_{\Omega}{\tilde{G}(x,u)}dx - {\lambda}\int_{\Omega} \tilde K(x,u)dx,$$ 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 $$\label{eq4.2} \tilde{J_{\lambda}}(u)= J_{\lambda}(u)- \int_{\Omega} ( g(u_\lambda)u_\lambda- G(u_\lambda))dx -\frac{\lambda q}{q+1} H(u_\lambda).$$ 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 $U_1 :=\Big\{u=0\; \text{or}\; u\in W^{1,N}_0(\Omega): \|u\|t^{-}\big(\frac{u}{\|u\|}\big)\Big\}.$ 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 $$\label{eq:criclevel} \rho_{M}=\min_{\gamma\in \mathcal{F}_M}\max_{t\in[0,1]}J_{\lambda}(\gamma(t))$$ 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 $00$ 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 $$\label{eqa1} \sup_{t>0}J_{\lambda}(u_\lambda+t\phi_n)=J_{\lambda}(u_\lambda+t_n\phi_n)\geq J_{\lambda}(u_\lambda)+\frac{1}{N}\alpha_{N}^{N-1}.$$ 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, \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} Let $c_n=\min_{|x|\leq \frac{\delta_n}{n}}u_\lambda(x)$. Then \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} Now using Taylor's expansion and \eqref{eqa4}, for some $K_0>0$, we obtain \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} since $t_n$ is bounded away from zero. Using \eqref{eqa6} and \eqref{eqa7}, we obtain \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} 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 \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} 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 $u_\lambda\equiv v_\lambda$, $c_00$ 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 $(1+\epsilon)\|v_n\|^{\frac{N}{N-1}} 1$ 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 $$\label{eq5.2} -\Delta_N u_{\lambda}+V(x)|u_{\lambda}|^{N-2}u_{\lambda} = {u_{\lambda}|u_{\lambda}|^{p}} e^{|u_{\lambda}|^{\frac{N}{N-1}}} +\lambda hu_{\lambda}^{q}, \quad \text{in } \Omega.$$ We can choose $\lambda>0$ large such that $$\label{eq5.1} \lambda_1(V) - \lambda h t^{q+1-N} - g(t)t^{1-N} < 0$$ 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} \begin{thebibliography}{00} \bibitem{A} Adimurthi; \emph{Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian}, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990) 393-413. \bibitem{GA} G. A. Afrouzi, S. Mahdavi, Z. Naghizadeh; \emph{The Nehari Manifold for p-Laplacian equation with Dirichlet boundry condition}, Nonlinear Analysis: Modeling and Control, 12 (2007) 143-155. \bibitem{COA} C. O. Alves, A. El Hamidi; \emph{Nehari manifold and existence of positive solutions to a class of quasilinear problem}, Nonlinear Anal., 60 no. 4 (2005) 611-624. \bibitem{WY} W. Allegretto, Y. Huang; \emph{A Picone's identity for $p$-Laplacian and applications}, Nonlinear Analysis, Theory, Methods and Applications, 32 7 (1998) 819-830. \bibitem {ABC} A. Ambrosetti, H. Brezis, G. Cerami; \emph{Combined effects of concave and convex nonlinearities in some elliptic problems}, J. Funct. Anal. 122 no. 2 (1994) 519-543. \bibitem{bd} I. Birindelli, F. Demengel; \emph{Existence of solutions for semilinear equations involving the $p$-Laplacian: the non coercive case}, Cal. var, 20 (2004) 343-366. \bibitem{KY} K. J. Brown, Y. Zhang; \emph{The Nehari manifold for a semilinear elliptic problem with a sign changing weight function}, J. Differential Equations, 193 (2003) 481-499. \bibitem{WUFI} K. J. Brown, T. F. Wu; \emph{A fibering map approach to a semilinear elliptic boundry value problem}, Electronic Journal of Differential Equations, 69 (2007) 1-9. \bibitem{cqu} M. Cuesta, H. Ramos Quoirin; \emph{A weighted eigenvalue problem for the $p$-Laplacian plus a potential}, Non-linear Differ. Equ. Appl- NODEA 16 (2009) 469-491. \bibitem{DP} P. Drabek, S. I. Pohozaev; \emph{Positive solutions for the p-Laplacian: application of the fibering method}, Proc. Royal Soc. Edinburgh Sect A, 127 (1997) 703-726. \bibitem{AE} A. El Hamidi; \emph{Multiple solutions with changing sign energy to a nonlinear elliptic equation}, Commun. Pure Appl. Anal., 3 (2004) 253-265. \bibitem{AEJ} A. El Hamidi, J. M. Rakotoson; \emph{Compactness and quasilinear problems with critical exponents}, Differential and Integral Equations, 18 (2005) 1201-1220. \bibitem{fgu} de Figueiredo, Gossez, Ubilla; \emph{Local superlinearity and sublinearity for $p-$Laplacian}, Journal of Functional Analysis, 257 (2009) 721-752. \bibitem{JSK} J. Giacomoni, S. Prashanth, K. Sreenadh; \emph{A global multiplicity result for $N$-Laplacian with critical nonlinearity of concave-convex type}, J. Differential Equations, 232 (2007) 544-572. \bibitem{li} P. L. Lions; \emph{The concentration compactness principle in the calculus of variations part-I} Rev. Mat. Iberoamericana, 1 (1985) 185-201. \bibitem{moser} J. Moser; \emph{A sharp form of an inequality by N. Trudinger}, Indiana Univ. Math. J., 20 (1971) 1077-1092. \bibitem{do6} J. Marcus do \'{O}; \emph{Semilinear Dirichlet problems for the $N$-Laplacian in $\Omega$ with nonlinearities in critical growth range}, Differential Integral Equations, 9 (1996) 967-979. \bibitem{PS} S. Prashanth, K. Sreenadh; \emph{Multiplicity of Solutions to a Nonhomogeneous elliptic equation in $\mathbb{R}^2$}, Differential and Integral Equations, 18 (2005) 681-698. \bibitem{qu} Humberto Ramos Quoirin; \emph{Lack of coercivity in a concave-convex type equation}, Calculus of variations, 37 (2010) 523-546. \bibitem{hqu} Humberto Ramos Quoirin, Pedro Ubilla; \emph{On some indefinite and non-powerlike elliptic equations,} Nonlinear Analysis, 79 (2013) 190-203. \bibitem{TA} G. Tarantello; \emph{On nonhomogeneous elliptic equations involving critical Sobolev exponent}, Ann. Inst. H. Poincare- Anal. non lineaire, 9 (1992) 281-304. \bibitem{WU} T. F. Wu; \emph{On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function}, J. Math. Anal. Appl., 318 (2006) 253-270. \bibitem{WU6} T. F. Wu; \emph{A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential}, Electron. J. Differential Equations, 2006 No. 131 (2006) 1-15. \bibitem{WU9} T. F. Wu; \emph{Multiplicity results for a semilinear elliptic equation involving sign-changing weight function}, Rocky Mountain J. Math., 39 (3) (2009) 995-1011. \bibitem{WU10} T. F. Wu; \emph{Multiple positive solutions for a class of concave-convex elliptic problems in $\Omega$ involving sign-changing weight}, J. Funct. Anal., 258 (1) (2010) 99-131. \end{thebibliography} \end{document}