\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 98, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/98\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for a singular semilinear elliptic problem in $\mathbb{R}^2$} \author[Manass\'es de Souza \hfil EJDE-2011/98\hfilneg] {Manass\'es de Souza} \address{Manass\'es de Souza \newline Departamento de Matem\'atica, Universidade Federal da Para\'iba\\ 58.051-900 Jo\~ao Pessoa, PB, Brazil} \email{manasses@mat.ufpb.br} \thanks{Submitted May 2, 2011. Published August 3, 2011.} \thanks{Supported by the National Institute of Science and Technology of Mathematics INCT-Mat.} \subjclass[2000]{35J60, 35J20, 35B33} \keywords{Variational methods; Trudinger-Moser inequality; critical points; \hfill\break\indent critical exponents} \begin{abstract} Using minimax methods we study the existence and multiplicity of nontrivial solutions for a singular class of semilinear elliptic nonhomogeneous equation where the potentials can change sign and the nonlinearities may be unbounded in $x$ and behaves like $\exp(\alpha s^2)$ when $|s|\to+\infty$. We establish the existence of two distinct solutions when the perturbation is suitable small. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we consider the semilinear elliptic equation $$\label{P} -\Delta u + V(x) u = \frac{g(x)f(u)}{|x|^a}+h(x)\quad\text{in }\mathbb{R}^2,$$ where $a\in [0,2)$, the functions $V,g:\mathbb{R}^2\to \mathbb {R}$ and $f:\mathbb{R}\to\mathbb{R}$ are continuous with $f(0)=0$ and $h\in(H^1(\mathbb{R}^2))^*\equiv H^{-1}$ is a small perturbation, $h \not\equiv0$. We are interested in finding nontrivial solutions of \eqref{P} when the nonlinearity $f(s)$ has the maximal growth which allows to treat \eqref{P} variationally in the Sobolev space $H^1(\mathbb{R}^2)$. On the potentials we assume the hypothesis \begin{itemize} \item[(V1)] There exist $D>0$ such that $V(x)\geq -D$, for all $x\in\mathbb{R}^2$; \item[(V2)] $\lambda_{1}=\inf_{u\in E\backslash\{0\}} \|u\|^2_E/\|u\|_2^2>0$; \end{itemize} where $E$ is the following subspace of $H^1(\mathbb{R}^2)$ $E=\big\{u\in H^1(\mathbb{R}^2):\int_{\mathbb{R}^2}V(x)u^2\,\mathrm{d} x<\infty\big\},$ which is a Hilbert space endowed with the scalar product $\langle u,v\rangle_E=\int_{\mathbb{R}^2}[\nabla u\cdot\nabla v+V(x)uv]\mathrm{d}x$ to which corresponds the norm $\|u\|_E=\langle u,u\rangle^{1/2}_E$ (see \cite[Lemma 2.1 and Proposition 3.1]{SIRA00}). Here, as usual, $H^1(\mathbb{R}^2)$ denotes the Sobolev spaces modelled in $L^2(\mathbb{R}^2)$ with norm $\| u \|_{1,2}=\Big(\int_{\mathbb{R}^2}(|\nabla u|^2+|u|^2)\,\mathrm{d} x\Big)^{1/2}.$ To ensure the continuous imbedding of $E$ into $H^1(\mathbb{R}^2)$, we assume the condition (V2) on the first eigenvalue of the operator $A=-\Delta+V(x)$ (see \cite[Proposition 2.2]{SIRA00}). We use the following notation: if $\Omega\subset\mathbb {R}^2$ is open and $s\geq 2$, we set $\nu_{s}(\Omega)=\inf_{u\in H^1_0(\Omega)\backslash\{0\}}\frac{\int_{\Omega}[|\nabla u|^2 +V(x)u^2]\, \mathrm{d}x}{\big(\int_\Omega|u|^s\,\mathrm{d} x\big)^{2/s}},$ and we put $\nu_s(\emptyset)=\infty$. To obtain a compactness result, we shall consider the following assumptions: \begin{itemize} \item[(V3)] $\lim_{R\to\infty}{\nu_{s}(\mathbb{R}^2\backslash \overline{B}_R)}=\infty$. \item[(V4)] There exist a function $K(x)\in L^{\infty}_{\rm loc}(\mathbb{R}^2)$, with $K(x)\geq 1$, and constants $\alpha>1$, $c_0, R_0>0$ such that $K(x)\leq c_{0}[1+(V^+(x))^{1/\alpha}],$ for all $|x|\geq R_{0}$, where $V^+(x)=\max_{x\in\mathbb{R}^2}\{0,V(x)\}$. \end{itemize} It is also well known that assumptions (V3)--(V4) imply that the imbeddings of $E$ into $L^q(\mathbb{R}^2)$ are compact for all $2\leq q<\infty$ (see \cite[Proposition 3.1]{SIRA00}). Concerning the function $g$, we assume that it is strictly positive and does not have to be bounded in $x$ provided that the growth of $g$ is controlled by the growth of $V(x)$. More precisely: \begin{itemize} \item[(H1)] There exists $a_0,b_0>0$ such that $a_0\leq g(x)\leq b_0K(x)$ for all $x\in\mathbb{R}^2$. \end{itemize} Moreover, we suppose that $f(s)$ satisfies the following conditions: \begin{itemize} \item[(H2)] $\lim_{s\to 0}\frac{f(s)}{s}=0$. \item[(H3)] There is a number $\mu>2$ such that for all $s\in\mathbb{R}\backslash\{0\}$ $0<\mu F(s):= \mu \int_0^sf(t)\mathrm{d}t \leq sf(s).$ \end{itemize} Motivated by Trudinger-Moser inequality (see \cite{MOSE71,TRUD67}) and by pioneer works of Adimurthi \cite{AD90} and de Figueiredo et al. \cite{FMRU95} we treat the so-called subcritical case, which we define next. We say that a function $f(s)$ has \emph{subcritical growth} at $+\infty$ if for all $\beta>0$ $$\label{cres-sub} \lim_{|s|\to +\infty}\frac{|f(s)|}{e^{\beta s^2}}=0.$$ Throughout this paper, we denote by $H^{-1}$ the dual space of $H^1(\mathbb{R}^2)$ with the usual norm $\|\cdot\|_{H^{-1}}$. Next we state our existence result. \begin{theorem}\label{teoremaS1} If $f(s)$ has subcritical growth at $+\infty$ and {\rm (V1)--(V4), (H1)--(H3)} are satisfied then problem \eqref{P} has a weak solution with positive energy if $h\equiv0$. Moreover, if $h\not\equiv0$, there exists $\delta>0$ such that if $\|h\|_{H^{-1}}<\delta$, problem \eqref{P} has at least two weak solutions. One of them with positive energy, while the other one with negative energy. \end{theorem} The results in this paper were in part motivated by several recent papers on elliptic problems involving exponential growth. See for example de Souza \cite{DEDO2011} for the singular and homogeneous case, Giacomoni-Sreenadh \cite{GEA} for the singular and nonhomogeneous case, do \'O et al. \cite{JMMU08} and Tonkes \cite{TONK99} for the nonsingular and nonhomogeneous case, Cao \cite{CAO92}, de Figueiredo et al. \cite{FMRU95} and do \'O \cite{JMAR97} for the nonsingular and homogeneous case. Our paper is closely related to the recent works of do \'O et al. \cite{JMMU08} and Rabelo \cite{PRabelo}. Indeed, we improve and complement the results in do \'O et al. \cite{JMMU08} for the subcritical case in the sense that we use nonlinearities unbounded in $x$ and potentials which can change sign. Moreover in \cite{JMMU08} was studied the existence and multiplicity of weak solutions of \eqref{P} in terms of the Trudinger-Moser inequality for the nonsingular case. We point out that ours results are closely related with results in \cite{ADYANG2010,DEDO2011,MAN12011,MAN22011,MAN32011}. The proofs of our existence results rely on minimization methods in combination with the mountain-pass theorem. In the subcritical case we are able to prove that the associated functional satisfies the Palais-Smale compactness condition which allow us to obtain critical points for the functional. As a consequence we can distinguish the local minimum solution from the mountain-pass solution. \begin{remark} \label{rmk1.1} \rm The study of such a class of problem has been motivated in part by the search for standing waves for the nonlinear Schr\"odinger equation (see for instance \cite{BELI83-I} and \cite{Rabinowitz2}) $i\frac{\partial \psi }{\partial t}=-\Delta \psi +W(x)\psi -G(|\psi |)\psi-e^{i\lambda t}L(x),\quad x\in\mathbb{R}^2,$ where $\psi=\psi(t,x)$, $\psi:\mathbb{R}\times \mathbb{R}^2\to\mathbb{C}$, $\lambda$ is a positive constant, $W:\mathbb{R}^2\to \mathbb{R}$ is a given potential and for suitable functions $G:\mathbb{R}^+\to \mathbb{R},\; L:\mathbb{R}^2\to\mathbb{R}$. \end{remark} This article is organized as follows. Section \ref{prelim} contains some preliminary results including a singular Trudinger-Moser inequality. In Section \ref{geo}, contains the variational framework and we also check the geometric conditions of the associated functional. In Section \ref{palais-smale-seq}, we prove some properties of the Palais-Smale sequences. Finally, in section \ref{proof-main-results} we complete the proofs of our main results. \section{Preliminary results}\label{prelim} Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, we know by the Trudinger-Moser inequality that for all $\beta>0$ and $u\in H_0^1(\Omega)$, $e^{\beta u^2}\in L^1(\Omega )$ (see \cite{MOSE71,TRUD67}). Moreover, there exists a positive constant $C$ such that $\sup_{u\in H_0^1(\Omega) \; : \; \|\nabla u\|_2\leq 1}\int_{\Omega} e^{\beta u^2}\mathrm{d}x\leq C|\Omega| \quad\text{if }\beta \leq 4\pi,$ where $|\Omega|$ denotes Lebesgue measure of $\Omega$. This inequality is optimal, in the sense that for any growth $e^{\beta u^2}$ with $\beta > 4\pi$ the correspondent supremum is infinite. Adimurthi-Sandeep \cite{ADSA07} proved a singular Trudinger-Moser inequality, which in the case $N=2$ reads: $\int_{\Omega}\frac{e^{\beta u^2}}{|x|^a} \mathrm{d}x <\infty \quad \text{ for all } u \in H^1_0(\Omega),\;\beta >0,$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ containing the origin and $a \in [0,2)$. Moreover, there exists a positive constant $C(\beta,a)$ such that $$\label{adsa-singular} \sup_{u\in H_0^1(\Omega) : \|\nabla u\|_2\leq 1}\int_{\Omega}\frac{e^{\beta u^2}}{|x|^a} \mathrm{d}x \leq C(\beta,a) |\Omega| \quad\text{if and only if}\quad \beta/4\pi + a/2 \leq 1.$$ Here we shall use the following extension of these results for the whole space $\mathbb{R}^2$ obtained by Giacomoni and Sreenadh in \cite{GEA} (see also \cite{DEDO2011}): \begin{lemma}\label{Trudinger-Moser} If $\beta >0$, $a\in [0,2)$ and $u\in H^1( {\mathbb{R}}^2)$ then $$\label{TM1} \int_{\mathbb{R}^2}\frac{(e^{\beta u^2} -1)}{|x|^a}\mathrm{d}x <\infty.$$ Moreover, if $\beta/4\pi + a/2 < 1$ and $\|u\|_2 \leq M$, then there exists a positive constant $C=C(\beta,M)$ such that $$\sup_{\|\nabla u\|_{2}\leq 1}\int_{\mathbb{R}^2}\frac{(e^{\beta u^2} -1)}{|x|^a}\mathrm{d}x \leq C(\beta,M). \label{TM2}$$ \end{lemma} Our choice of the variational setting $E$ ensures that the imbedding is continuous in $H^1(\mathbb{R}^2)$ and compact in $L^s(\mathbb{R}^2)$, for $s\geq 2$ (see \cite[Lemma 2.1 and Proposition 3.1]{SIRA00}). This lemma in \cite{SIRA00} provides a inequality which will be needed throughout the paper: $$\label{eq.emb} \|u\|_E^2\geq\zeta\int_{\mathbb{R}^2}|\nabla u|^{2}\,\mathrm{d}x,$$ for some $\zeta>0$ and for all $u\in E$. \begin{lemma}\label{estim-import} Let $\beta>0$ and $r\geq1$. Then for each $\theta>r$ there exists a positive constant $C=C(\theta)$ such that for all $s\in \mathbb{R}$ $(e^{\beta s^2}-1)^r\leq C(e^{\theta\beta s^2}-1).$ In particular, for $r\in[1,\alpha)$, we have that $K(x)^r\dfrac{(e^{\beta u^2}-1)^r}{|x|^a}$ belongs to $L^1(\mathbb{R}^2)$ for all $u\in H^1(\mathbb{R}^2)$. \end{lemma} \begin{proof} The proof of the inequality above is a consequence of L'Hospital Rule (see \cite[Lemma 2.2]{JMMU08} for a proof). Now, as $K(x)\in L^\infty_{\rm loc}(\mathbb{R}^2)$, for $R>1$ we have that % \begin{align*} &\int_{\mathbb{R}^2}K(x)^r\frac{(e^{\beta u^2}-1)^r}{|x|^a}\, \mathrm{d}x\\ &\leq C_1\int_{|x|\leq R}\frac{(e^{\beta u^2}-1)^r}{|x|^a}\,\mathrm{d}x +\int_{|x|>R}K(x)^r(e^{\beta u^2}-1)^r\,\mathrm{d}x\\ &\leq C_2\int_{|x|\leq R}\frac{(e^{\theta\beta u^2}-1)}{|x|^a}\,\mathrm{d}x+C_3\int_{|x|>R}K(x)^r(e^{\theta\beta u^2}-1)\,\mathrm{d}x. \end{align*} From Lemma \ref{Trudinger-Moser} it follows that the first term is integrable. To estimate the other term, we note that \begin{align*} \int_{|x|>R}K(x)^r(e^{\theta\beta u^2}-1)\,\mathrm{d}x =\sum_{m=1}^{\infty}\frac{(\theta\beta)^m}{m!} \int_{|x|>R} K(x)^r|u|^{2m}\,\mathrm{d}x. \end{align*} By (V4) and H\"older inequality, we have \begin{align*} &\int_{\mathbb{R}^2}K(x)^r|u|^{2m}\,\mathrm{d}x\\ &\leq C_4\|u\|^{2m}_{2m}+C_5\int_{|x|>R_0}\big(V^+(x)\big) ^{r/\alpha}|u|^{2m}\,\mathrm{d}x\\ &\leq C_4\|u\|^{2m}_{2m}+C_5\Big[\int_{|x|>R_0}V^+(x)|u|^2\,\mathrm{d} x\Big]^{r/\alpha} \Big[\int_{|x|>R_0}|u| ^{2(m\alpha-r)/(\alpha-r)} \Big]^{(\alpha-r)/\alpha}. \end{align*} By (V2) and the continuous imbedding $E\hookrightarrow L^s(\mathbb{R}^2)$, for all $s\geq 2$, we can conclude that $\int_{\mathbb{R}^2}K(x)^r|u|^{2m}\,\mathrm{d}x \leq C\|u\|^{2m}_E.$ Thus, we obtain \label{6001} \begin{aligned} &\int_{\mathbb{R}^2}K(x)^r\frac{(e^{\beta u^2}-1)^r}{|x|^a} \,\mathrm{d}x \\ & \leq C_1\int_{|x|\leq R}\frac{(e^{\theta\beta u^2}-1)}{|x|^a}\,\mathrm{d}x+ C\sum_{m=1}^{\infty}\frac{1}{m!}\left(\theta\beta\|u\|_E^2\right)^m\\ & \leq C_1\int_{|x|\leq R}\frac{(e^{\theta\beta u^2}-1)}{|x|^a}\,\mathrm{d}x+ C\left[\exp\left(\theta\beta\|u\|_E^2\right)-1\right]<\infty, \end{aligned} which completes the proof. \end{proof} \begin{corollary}\label{desigualdade importante} If $v\in E$, $\beta>0$, $q>0$ and $\|v\|_E\leq M$ with $\frac{\beta M^2}{4\pi\zeta} + \frac{a}{2} <1$, then there exists $C=C(\beta,M,q,\zeta)>0$ such that $\int_{\mathbb{R}^2}K(x)|v|^q\frac{(e^{\beta v^2}-1)}{|x|^a}\,\mathrm{d} x\leq C\|v\|_E^q.$ \end{corollary} \begin{proof} By H\"older inequality, % $$\label{1001} \int_{\mathbb{R}^2}K(x)|v|^{q}\frac{(e^{\beta v^2}-1)}{|x|^a}\,\mathrm{d} x \leq\|v\|^q_{qs}\Big[\int_{\mathbb{R}^2}K(x)^r \frac{(e^{\beta v^2}-1)^r}{|x|^{ar}}\mathrm{d}x\Big]^{1/r},$$ where $r>1$ is close to $1$ and $s=r/(r-1)$. Now, we consider $\theta>r$ close to $r$ such that $\frac{\theta\beta M^2}{4\pi\zeta}+\frac{ar}{2} <1$. By $(\ref{6001})$ and Lemma \ref{Trudinger-Moser}, we have that % \label{1002} \begin{aligned} &\int_{\mathbb{R}^2}K(x) |v|^q\frac{(e^{\beta v^2}-1)}{|x|^a}\,\mathrm{d}x\\ &\leq \Big\{C_1\int_{|x|\leq R}\frac{[e^{\frac{\theta\beta M^2}{\zeta}\left(\frac{v}{\|\nabla v\|_2}\right)^2}-1]}{|x|^{ar}} \,\mathrm{d}x+C_2\left[\exp\left(\theta\beta M^2\right)-1\right]\Big\}^{1/r}\|v\|^q_{qs}\\ &\leq C_3\|v\|^q_E. \end{aligned} \end{proof} To show that the weak limit of a Palais-Smale sequence in $E$ is a weak solution of \eqref{P} we will use the following convergence result, which is a version of Lemma~2.1 in \cite{FMRU95}. \begin{lemma}\label{Djairo} Let $\Omega\subset\mathbb{R}^2$ be a bounded domain and $f: \mathbb{R}\to\mathbb{R}$ a continuous function. Then for any sequence $(u_n)$ in $L^1(\Omega)$ such that $u_n\to u$ in $L^1(\Omega)$, $\frac{g(x)f(u_n)}{|x|^a}\in L^1(\Omega)\quad \text{and}\quad \int_\Omega\frac{g(x)|f(u_n)u_n|}{|x|^a}\,\mathrm{d} x\leq C_1,$ up to a subsequence we have $\frac{g(x)f(u_n)}{|x|^a}\to \frac{g(x)f(u)}{|x|^a}\quad \text{in } L^1(\Omega).$ \end{lemma} \begin{proof} It suffices to prove $\int_{\Omega}\frac{|g(x)f(u_n)|}{|x|^a}\,\mathrm{d}x \to \int_{\Omega}\frac{|g(x)f(u)|}{|x|^a} \,\mathrm{d}x.$ Since $u, {g(x)f(u)}/{|x|^a}\in L^1(\Omega)$, for each $\epsilon >0$ there is a $\delta >0$ such that for any measurable subset $A\subset \Omega$, $$\label{dj214} \int_{A}|u| \,\mathrm{d}x <\epsilon \quad \text{and}\quad \int_{A}\frac{|g(x)f(u)|}{|x|^a} \, \mathrm{d}x < \epsilon \quad\text{if } |A|\leq \delta.$$ Next using the fact that $u \in L^1(\Omega)$ we find $M_1 > 0$ such that $$\label{dj215} |\{x \in \Omega:|u(x)| \geq M_1\}|\leq \delta.$$ Let $M=\max \{M_1, C_1/\epsilon\}$. We write $\Big|\int_{\Omega}\frac{|g(x)f(u_n)|}{|x|^a}\,\mathrm{d}x - \int_{\Omega}\frac{|g(x)f(u)|}{|x|^a} \,\mathrm{d}x\Big| \leq I_{1,n} + I_{2,n} + I_{3,n},$ where \begin{gather*} I_{1,n} = \int_{[|u_n| \geq M]}\frac{|g(x)f(u_n)|}{|x|^a}\,\mathrm{d}x,\\ I_{2,n} = \Big|\int_{[|u_n| < M]}\frac{|g(x)f(u_n)|}{|x|^a}\,\mathrm{d} x - \int_{[|u| < M]}\frac{|g(x)f(u)|}{|x|^a}\,\mathrm{d}x\Big| ,\\ I_{3,n} = \int_{[|u| \geq M]}\frac{|g(x)f(u)|}{|x|^a}\,\mathrm{d}x. \end{gather*} Now we estimate each integral separately. $I_{1,n} = \int_{[|u_n| \geq M]}\frac{|g(x)f(u_n)|}{|x|^a}\,\mathrm{d}x = \int_{[|u_n| \geq M]}\frac{|g(x)f(u_n)u_n|}{|u_n||x|^a}\,\mathrm{d}x \leq \frac{C_1}{M}\leq \epsilon.$ From \eqref{dj214} and \eqref{dj215}, we have $I_{3,n} \leq \epsilon$. Next we claim $I_{2,n} \to 0$ as $n\to +\infty$. Indeed, \begin{align*} I_{2,n} &\leq \Big|\int_{\Omega}\frac{\mathcal{X}_{[|u_n| < M]}(|g(x)f(u_n)| - |g(x)f(u)|)}{|x|^a}\,\mathrm{d}x \Big| \\ &\quad + \Big|\int_{\Omega}\frac{(\mathcal{X}_{[|u_n| 0$there exists$\delta>0$such that$|f(s)|\leq\varepsilon |s|$always that$|s|<\delta$. On the other hand, for$\beta>0$we have that there exists$C>0$such that$|f(s)|\leq C(e^{\beta s^2}-1)$for all$s\geq\delta$. Thus $$\label{eq.bd1} |f(s)|\leq\varepsilon|s|+C_1(e^{\beta s^2}-1),$$ for all$s\in\mathbb{R}. By (H1), (H3), (V4) and H\"older inequality, we obtain \begin{align*} \int_{\mathbb{R}^2}\frac{g(x)F(u)}{|x|^a}\,\mathrm{d}x&\leq \varepsilon\int_{\mathbb{R}^2}\frac{K(x)u^2}{|x|^a}\,\mathrm{d}x +C_2\int_{\mathbb{R}^2}\frac{K(x)|u| (e^{\beta u^2}-1)}{|x|^a}\,\mathrm{d}x\\ &\leq C_1 \int_{|x|\leq 1}\frac{u^2}{|x|^a}\,\mathrm{d}x + \varepsilon\int_{|x|>1}K(x)u^2\,\mathrm{d}x\\ &\quad +C_2\|u\|_s \Big[\int_{\mathbb{R}^2}K(x)^r\frac{(e^{\beta u^2}-1)^r}{|x|^{ar}}\,\mathrm{d}x\Big]^{1/r}, \end{align*} % wherer\in[1,\alpha)$and$s=r/(r-1)$, with$ar<2$. Considering the continuous imbedding$E\hookrightarrow L^s_{K(x)}(\mathbb{R}^2)$for$s\geq 2$,$a\in [0,2)$and Lemma \ref{estim-import}, it follows that${g(x)F(u)}/{|x|^a}\in L^1(\mathbb{R}^2)$which implies that$I$is well defined. Next, we show that$I$is in$C^1$on$E$. Indeed, letting$ N(u)=\int_{\mathbb{R}^2}{g(x)F(u)}/{|x|^a}\,\mathrm{d}x, we have by dominated convergence theorem that \begin{align*} \langle I'(u),\phi\rangle &= \langle u,\phi\rangle_E-\lim_{t\to 0}\frac{1}{t} [N(u+t\phi)-N(u)]-\int_{\mathbb{R}^2}h(x)\phi\,\mathrm{d}x\\ &= \langle u,\phi\rangle_E-\int_{\mathbb{R}^2} \frac{g(x)f(u)\phi}{|x|^a}\,\mathrm{d}x -\int_{\mathbb{R}^2}h(x)\phi\,\mathrm{d}x, \end{align*} % for all\phi\in E$. As$I'(u)$is linear and bounded, it suffices to show that the Gateaux derivative of$I$is continuous. It is clear that the first and last term are$C^1$. Hence, it remains to prove that$N$is$C^1$. Let$u_n\to u$in$E$. By Proposition~2.7 in \cite{JMMU08}, there exists a subsequence$(u_{n_k})$in$E$and$\ell(x)\in H^1(\mathbb{R}^2)$such that$u_{n_k}(x)\to u(x)$and$|u_{n_k}(x)|\leq \ell(x)$almost everywhere in$\mathbb{R}^2$. Given$\xi\in E$, we define $H_{n_k}(x)=\frac{g(x)f(u_{n_k}(x))\xi(x)}{|x|^a}.$ Then $H_{n_k}(x)\to H(x)=\frac{g(x)f(u(x))\xi(x)}{|x|^a}\quad \text{almost everywhere in } \mathbb{R}^2.$ Using$(\ref{eq.bd1})$and Lemma \ref{Trudinger-Moser}, we obtain that$H_{n_k}(x)$is integrable, it follows by dominated convergence theorem that $\lim_{k\to\infty}\int_{\mathbb{R}^2}H_{n_k}(x)\,\mathrm{d} x=\int_{\mathbb{R}^2}H(x)\,\mathrm{d}x.$ Thus, for each$\xi\in E$with$\|\xi\|_E=1, we obtain \begin{align*} \lim_{k\to\infty}\|N'(u_{n_k})-N'(u)\|_{E^*} &= \lim_{k\to\infty}\sup_{\|\xi\|_E=1}| \langle N'(u_{n_k})-N'(u),\xi\rangle|\\ &= \sup_{\|\xi\|_E=1}\lim_{k\to\infty}\int_{\mathbb{R}^2}\frac{g(x) [f(u_{n_k})-f(u)]\xi}{|x|^a}\,\mathrm{d} x = 0 \end{align*} and the proof is complete. The geometric conditions of the mountain-pass theorem for the functionalI$are established by our next two lemmas. \begin{lemma}\label{lema31} Suppose that (V1)-(V2), (V4), (H1)-(H3) and \eqref{cres-sub} are satisfied. Then there exists$\delta>0$such that for each$h\in H^1(\mathbb{R}^2)$with$\|h\|_{H^{-1}}<\delta$, there exists$\rho_h>0$such that $I(u)>0\quad whenever\quad\|u\|_E=\rho_h.$ \end{lemma} \begin{proof} In the same manner that \eqref{eq.bd1} was obtained, we can see that $$\label{555} |f(s)|\leq\varepsilon|s|+C_1|s|^q(e^{\beta s^2}-1),$$ with$q>2$. Thus, considering the continuous imbedding$E\hookrightarrow L^s_{K(x)}(\mathbb{R}^2)$for$s\geq 2$(see \cite[Proposition 3.1]{SIRA00}), we obtain for$\varepsilon>0sufficiently small \begin{align*} I(u)& \geq\frac{1}{2}\|u\|^2_E - \varepsilon\int_{\mathbb{R}^2}\frac{K(x)u^2}{|x|^a}\,\mathrm{d}x- C_1\int_{\mathbb{R}^2}\frac{K(x)|u|^{q+1}(e^{\beta u^2}-1)}{|x|^a} \,\mathrm{d}x-\int_{\mathbb{R}^2}h(x)u\,\mathrm{d}x\\ & \geq \big(\frac{1}{2}-\varepsilon\big)\|u\|^2 _E-C_1\int_{\mathbb{R}^2}\frac{K(x)|u|^{q+1}(e^{\beta u^2}-1)}{|x|^a}\,\mathrm{d}x -\int_{\mathbb{R}^2}h(x)u\,\mathrm{d}x \end{align*} and since\frac{\beta\sigma^2}{4\pi\zeta}+\frac{a}{2}<1$if$\|u\|_E<\sigma$is sufficiently small, we can apply Corollary \ref{desigualdade importante} to conclude that $I(u)\geq \big(\frac{1}{2}-\varepsilon\big)\|u\|^2_E -C\|u\|_E^{q+1}-\|h\|_{H^{-1}}\|u\|_E.$ Thus there exists$\rho_h>0$such that$I(u)>0$whenever$\|u\|_E=\rho_h$and$\|h\|_{H^{-1}}$is sufficiently small. Indeed, for$\varepsilon >0$sufficiently small and$q>2$, we may choose$\rho_h >0$such that $\big(\frac{1}{2}-\varepsilon\big)\rho_h-C_1\rho_h^{q}>0.$ Thus, for$\|h\|_{H^{-1}}$sufficiently small there exists$\rho_h>0$such that$I(u)>0$if$\|u\|_E=\rho_h$. \end{proof} \begin{lemma} Assume that {\rm (H1), (H3)} and \eqref{cres-sub} are satisfied. Then there exists$e\in E$with$\|e\|_E>\rho_h$such that $I(e)<\inf_{\|u\|=\rho_h}I(u).$ \end{lemma} \begin{proof} Let$u\in E\backslash\{0\}$with compact support and$u\geq 0$. Integrating (H3) we obtain that there exist$c,d>0$such that $F(s)\geq cs^\mu-d$ for all$s\in\mathbb{R}$. Thus, denoting$K=supp(u)and using (H1), we have that \begin{align*} I(tu) &\leq \frac{t^2}{2}\|u\|^2_E-ct^\mu\int_{K}\frac{g(x)u^\mu}{|x|^a} \,\mathrm{d}x +d\int_{K}\frac{g(x)}{|x|^a}\,\mathrm{d}x-t\int_{\mathbb{R}^2}h(x)u\, \mathrm{d}x\\ &\leq \frac{t^2}{2}\|u\|^2_E-C_1t^\mu\int_{K}\frac{u^\mu}{|x|^a}\,\mathrm{d}x +C_2(|K|)-t\int_{\mathbb{R}^2}h(x)u\,\mathrm{d}x, \end{align*} for allt>0$, which implies that$I(tu)\to-\infty$as$t\to\infty$. Setting$e=tu$with$t$large enough, the proof is complete. \end{proof} To find an appropriate ball to use a minimization argument we need the following result. \begin{lemma} If$f(s)$satisfies$(\ref{cres-sub})$and$h\neq0$, there exist$\eta>0$and$v\in E$with$\|v\|_E=1$such that$I(tv)<0$for all$00\quad\text{for each } h\neq 0. \] Since $f(0)=0$, by continuity, it follows that there exists $\eta>0$ such that $\frac{\mathrm{d}}{\mathrm{d}t}I(tv)=t\|v\|^2_E-\int_{\mathbb{R}^2} \frac{g(x)f(tv)v}{|x|^a}\,\mathrm{d}x-\int_{\mathbb{R}^2} h(x)v\,\mathrm{d} x<0,$ for all $01$ close to 1 such that $ar<2$ and $s=r/(r-1)$. Since $f(s)$ has subcritical growth and $E\hookrightarrow L^s(\mathbb{R}^2)$ is compact for $s\geq 2$, the second term converges to zero. Now, to estimate the other term we will use H\"older inequality, Young inequality and that $\|u_n\|_E \leq C$, thus we obtain \label{14001} \begin{aligned} &\int_{\mathbb{R}^2}K(x)|x|^{-a}(|u_n|+|u|)|u_n-u|\,\mathrm{d}x\\ &\leq \sqrt{2}\Big(\int_{\mathbb{R}^2}\frac{K(x)|u_n|^2}{|x|^a}\,\mathrm{d} x+\int_{\mathbb{R}^2}\frac{K(x)|u|^2}{|x|^a}\,\mathrm{d} x\Big)^{1/2}\Big(\int_{\mathbb{R}^2}\frac{K(x)|u_n-u|^2}{|x|^a}\,\mathrm{d} x\Big)^{1/2}\\ &\leq C_1\left\{C_2 \|u_n-u\|_s^2 + \int_{\mathbb {R}^2}K(x)|u_n-u|^{2}\,\mathrm{d}x \right\}^{1/2}. \end{aligned} Using (V4), we have \label{1400} \begin{aligned} &\int_{\mathbb {R}^2}K(x)|u_n-u|^{2}\,\mathrm{d}x\\ &=\int_{|x|\leq R_0}K(x)|u_n-u|^{2}\,\mathrm{d}x +\int_{|x|>R_{0}}K(x)|u_n-u|^{2}\,\mathrm{d}x\\ &\leq \max_{|x|\leq R_{0}}\{K(x)\}\int_{|x| \leq R_{0}}|u_n-u|^{2}\,\mathrm{d}x\\ &\quad +\int_{|x|>R_0}c_{0}[1+(V^+(x))^{1/\alpha}] |u_n-u|^{2}\,\mathrm{d}x\\ &\leq C\big\{\|u_n-u\|^2_{2}+ \int_{|x|>R_0}V^+(x)^{1/\alpha}|u_n-u|^2\,\mathrm{d}x\big\}. \end{aligned} By H\"older inequality, we obtain \label{1500} \begin{aligned} & \int_{|x|>R_0}V^+(x)^{1/\alpha} |u_n-u|^2\,\mathrm{d}x\\ & \leq \Big[\int_{|x|>R_0}V^+(x)|u_n-u|^{2}\,\mathrm{d} x\Big]^{1/\alpha}\Big[\int_{|x|>R_0}|u_n-u|^{(2\alpha-2)/(\alpha-1)} \,\mathrm{d}x\Big]^{(\alpha-1)/\alpha} \end{aligned} and by (V1), we have \label{1600} \begin{aligned} &\int_{|x|>R_0}V^+(x)|u_n-u|^{2}\,\mathrm{d}x\\ &= \int_{\mathbb{R}^2}V(x)|u_n-u|^{2}\,\mathrm{d}x-\int_{|x|\leq R_0} V(x)|u_n-u|^{2}\,\mathrm{d}x-\int_{|x|>R_0}V^{-}(x)|u_n-u|^2\,\mathrm{d}x\\ &\leq\int_{\mathbb {R}^2}\left[|\nabla (u_n-u)|^2+ V(x)|u_n-u|^2\right]\mathrm{d}x. \end{aligned} From \eqref{1500}, \eqref{1600} in \eqref{1400} and using (V3), we obtain \label{1700} \begin{aligned} &\int_{\mathbb {R}^2}K(x)|u_n-u|^{2}\,\mathrm{d}x\\ &\leq C\big\{\|u_n-u\|^2_{2}+\left(\|u_n-u\|^2_E +D\|u_n-u\|^2_{2}\right)^{1/\alpha}\|u_n-u\|^{(2\alpha-2) /\alpha}_{(2\alpha-2)/(\alpha-1)}\big\}\\ &\leq C\big\{\|u_n-u\|^2_{2} +\big(1+\frac{D}{\lambda_1}\big)^{1/\alpha}\|u_n-u\|^{2/\alpha}_E \|u_n-u\|^{(2\alpha-2)/\alpha}_{(2\alpha-2)/(\alpha-1)}\big\}. \end{aligned} Thus, by \eqref{14001}, \begin{align*} &\int_{\mathbb{R}^2}K(x)|x|^{-a}(|u_n|+|u|)|u_n-u|\,\mathrm{d}x\\ &\leq C_1\big\{C_2\|u_n-u\|^2_s + \|u_n-u\|^2_2+C_3\|u_n-u\|^{2/\alpha}_E\|u_n-u\|_2^{2(\alpha-1) /\alpha}\big\}^{1/2}. \end{align*} By compact embedding of $E$ in $L^s(\mathbb{R}^2)$ for any $s\geq2$, we obtain $\int_{\mathbb{R}^2}K(x)|x|^{-a}(|u_n|+|u|)|u_n-u|\,\mathrm{d} x\to 0\quad\text{as } n\to+\infty.$ Hence the Palais-Smale condition is satisfied. Therefore, the functional $I$ has a critical point $u_M$ at minimax level \begin{gather*} c_M=\inf_{\gamma\in \Gamma}\max_{t\in[0,1]}I(\gamma(t))>0, \\ \Gamma=\{\gamma\in C(E,\mathbb{R}) : \gamma(0)=0,\; \gamma(1)=e\}. \end{gather*} On the other hand, if $h\not\equiv 0$, then we obtain a second solution of \eqref{P} with negative energy. Indeed, let $\rho_h$ be as in Lemma~\ref{lema31}. Since $\overline{B}_{\rho_h}$ is a complete metric space with the metric given by norm of $E$, convex and the functional $I$ is of class $C^1$ and bounded below on $\overline{B}_{\rho_h}$, it follows by Ekeland variational principle that there exists a sequence $(u_n)$ in $\overline{B}_{\rho_h}$ such that $$\label{4141} I(u_n)\to c_0=\inf_{\|u\|_E\leq\rho_h}I(u)\quad \text{and}\quad \|I'(u_n)\|_{E'}\to 0.$$ We now apply the argument above again to conclude that \eqref{P} possesses a solution $u_0$ such that $I(u_0)=c_0<0$. \subsection*{Acknowledgments} The author would like to thank the anonymous referees for their valuable comments and suggestions which improved this article. \begin{thebibliography}{99} \bibitem{AD90} Adimurthi; \emph{Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $N$-Laplacian}, Ann. Scuola Norm. Sup. Pisa Cl. 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