\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 154, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/154\hfil Positive ground state solutions for KGM systems] {Positive ground state solutions for quasicritical Klein-Gordon-Maxwell type systems with potential vanishing at infinity} \author[E. L. de Moura, O. H. Miyagaki, R. Ruviaro \hfil EJDE-2017/154\hfilneg] {Elson Leal de Moura, Olimpio Hiroshi Miyagaki, Ricardo Ruviaro} \address{Elson Leal de Moura\newline Universidade Federal dos Vales do Jequitinhonha e Mucuri, 39803-371 Te\'ofilo Otoni-MG, Brazil} \email{elson.moura@ufvjm.edu.br} \address{Olimpio Hiroshi Miyagaki \newline Universidade Federal de Juiz de Fora, Departamento de Matem\'atica, 36036-330 Juiz de Fora-MG, Brazil} \email{ohmiyagaki@gmail.com} \address{Ricardo Ruviaro \newline Universidade de Bras\'ilia, Departamento de Matem\'atica, 70910-900 Bras\'ilia-DF, Brazil} \email{ruviaro@mat.unb.br} \dedicatory{Communicated by Vicentiu Radulescu} \thanks{Submitted April 19, 2017. Published June 27, 2017.} \subjclass[2010]{35A15, 35Q61, 35B38, 35B09} \keywords{Klein-Gordon-Maxwell; positive solution; ground state; \hfill\break\indent vanishing potential} \begin{abstract} This article concerns the Klein-Gordon-Maxwell type system when the nonlinearity has a quasicritical growth at infinity, involving {\it zero mass} potential, that is, $V(x)\to 0$, as $|x|\to\infty$. The interaction of the behavior of the potential and nonlinearity recover the lack of the compactness of Sobolev embedding in whole space. The positive ground state solution is obtained by proving that the solution satisfies Mountain Pass level. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} This article concerns the existence of nontrivial solution to the Klein-Gordon-Maxwell system \begin{equation} \begin{gathered} -\Delta u +V(x)u-(2\omega + \phi)\phi u = K(x)f(u), \quad\text{in } \mathbb{R}^3,\\ \Delta \phi = (\omega +\phi)u^{2},\quad\text{in } \mathbb{R}^3, \end{gathered} \label{KGM} \end{equation} where $ u\in H^1(\mathbb{R}^3):=H$, $\omega >0$ is a parameter, and we assume that $V, K: \mathbb{R}^3\to \mathbb{R}$ and $f: \mathbb{R}\to\mathbb{R}$ are continuous functions, with $V, K$ nonnegative and $f$ having a quasicritical growth at infinity. We will treat problem \eqref{KGM} with {\it zero mass} potential, that is, $V(x)\to 0$, as $|x|\to\infty$. Problems involving zero mass potential, with $\phi=0$, have been studied by several researchers, and extended or improved in several ways; see for instance \cite{Montenegro,AS2012,AS,AW,BPR,BGM,BL,BV,GM,LW} and reference therein. In all these papers above, there are restrictions on $V$ and $K$ to get some compact embedding into a weighted $L^p$ space. In a remarkable work, Benci and Fortunato in \cite{BenciF1} considered problem \eqref{KGM}, with $V(x)=m_{0}^{2}-\omega^2$, as a model describing nonlinear Klein-Gordon fields in $\mathbb{R}^3$ interacting with the electromagnetic field. Thus the solution represents a solitary wave of the type $\Phi(x,t)=u(x)e^{i \omega t}$ in equilibrium with a purely electrostatic field $\bf E= -\nabla \phi(x)$. There are a lot of works devoted to system \eqref{KGM}, and we would like to cite some of them. Benci and Fortunato \cite{BenciF2} proved the existence of infinitely many radially symmetric solutions when $m_0 > \omega$ and $K(x)f (u) = |u|^{p-2}u$, $ 4 6$. For the critical nonlinearity $K(x)f (u) = |u|^{p-2}u$, with $p=6$, Cassani \cite{Cassani} obtained a non-existence result for the above system, and he showed the existence of radially symmetric solution when $ 4 0$ such that \begin{equation}\label{V} 0< V_{0}\leq V(x)\leq a_1,\quad \forall x\in\mathbb{R}^3 \end{equation} and if $2<\theta <4$, then \begin{equation}\label{V0} 0< \frac{2(4-\theta)}{\theta -2}\leq V_{0},\quad \forall x\in\mathbb{R}^3\,; \end{equation} also \begin{equation}\label{K} 0< K(x)\leq \frac{a_2}{1+|x|^{\xi_0}},\quad \forall x\in\mathbb{R}^3. \end{equation} \item[(A2)] If $\{A_n\}\subset\mathbb{R}^3$ is a sequence of Borel sets such that the Lebesgue measure of $A_n$ is bounded uniformly, that is, $\mu(A_n)\leq R$, for all $n$ and some $R>0$, then \begin{equation}\label{eq1} \lim_{r\to+\infty}\int_{A_n\bigcap B^c_r(0)}K(x)\,dx=0,\quad \text{uniformly for } n\in\mathbb{N}. \end{equation} \item[(A3)] (behavior at zero) $\limsup_{s\to 0^+}f(s)/s=0$, \item[(A3')] (behavior at zero) there is a constant $p\in (2,6)$ such that $\limsup_{s\to 0^{+}}\frac{f(s)}{s^{p-1}}< +\infty$, \item[(A4)] (quasicritical growth) $\limsup_{s\to +\infty}f(s)/s^{5}=0$, \item[(A5)] (Ambrosetti-Rabinowitz) there exists $\theta>4$, such that $0< \theta F(u)\leq f(u)u$ for all $u>0$, where $F(u)=\int^u_0f(s)\,ds$. \end{itemize} \begin{remark} \rm From \eqref{V}, \eqref{K} and $p\in(2,6)$, we have \begin{equation}\label{K3} \frac{K(x)}{[V(x)]^{(6-p/4}}\to 0,\quad \text{as } |x|\to +\infty. \end{equation} \end{remark} Our main results are as follows. \begin{theorem}\label{teo1} Suppose that {\rm (A1)--(A5)} hold. Then problem \eqref{KGM} possess a positive ground state solution. \end{theorem} \begin{theorem}\label{teo2} Suppose that {\rm (A1), (A2), (A3'), (A4), (A5)} hold. Then problem \eqref{KGM} possess a positive ground state solution. \end{theorem} Let us briefly sketch the contents of this article. In the next section we present some preliminaries. In Section 3, we prove the boundedness of the Cerami sequence and in the Section 4, we prove of the main results. In the Section 5, we analyze the case when $V(x)\to 0$, as $|x|\to\infty$. \section{Preliminary results} By the reduction method described in \cite{BenciFC}, the Euler-Lagrange functional associated with the system \eqref{KGM}, $J:H\equiv H^{1}(\mathbb{R}^3)\to \mathbb{R}$, is $$ J(u) = \frac{1}{2}\| u\|^{2} -\frac{1}{2}\int_{\mathbb{R}^3}\omega \phi_{u}u^{2}\,dx - \int_{\mathbb{R}^3}K(x)F(u)\,dx, $$ where $F(u)=\int^u_0f(s)\,ds$. From the conditions on $f$ and by standard arguments, the functional $J\in C^{1}(H,\mathbb{R})$ has Frechet derivative $$ J'(u)v = \int_{\mathbb{R}^3}( \nabla u\nabla v+V(x)uv)\, dx -\int_{\mathbb{R}^3}( 2\omega+ \phi_{u})\phi_{u}uv\,dx - \int_{\mathbb{R}^3}K(x)f(u)v\,dx, $$ for all $v\in H$. The norm in $H$ given by $$ \| u \|^2 = \int_{\mathbb{R}^3 }(| \nabla u|^2+V(x)u^{2})dx $$ is equivalent to the usual norm in $H$. The induced inner product is $$ \langle u,v\rangle:=\int_{\mathbb{R}^3 }( \nabla u\nabla v+V(x)uv)\,dx, $$ We recall that the critical points of functional $J$ are precisely the weak solutions of \eqref{KGM}. We also assume that $f(s)=0$ for all $s\in (-\infty, 0]$. A fundamental tool in our analysis will be the following Lemma. \begin{lemma}\label{L2.1} For every $ u\in H$, there exists a unique $ \phi_{u}\in D^{1,2}(\mathbb{R}^3)$ which solves \begin{equation}\label{e22} \Delta \phi = (\omega +\phi)u^{2}. \end{equation} Furthermore, in the set $ \{x : u(x)\neq 0\}$ we have $-\omega \leq \phi_{u}\leq 0$ if $\omega>0$. \end{lemma} For a proof of the above lemma, see \cite[Proposition 2.1]{Mugai2}. From assumption (A3) and (A4) [or (A3') and (A4)] and combining with Lemma \ref{L2.1} follows that the functional $J$ satisfies the geometric conditions of the Mountain Pass Theorem of Ambrosetti and Rabinowitz in \cite{Ambrosetti}. So, there is a sequence $ (u_n)\subset H$ such that \begin{equation}\label{Cer} J(u_n)\to c \quad \text{and} \quad (1+\| u_n\|)\| J'(u_n)\|\to 0, \quad n\to \infty, \end{equation} where $$ c= \inf_{\gamma\in \Gamma}\max_{t\in [0,1 ]}J(\gamma (t)) $$ is the Mountain Pass level, with $$ \Gamma = \{\gamma \in C([0,1 ], H^{1}(\mathbb{R}^3)); \gamma (0)=0, J(\gamma (1))\leq 0 \} . $$ The second result in this section is the following Hardy-type inequality. \begin{lemma}\label{p21} Suppose that {\rm (A1)--(A4)} or {\rm (A1), (A3), (A4)} hold. Then, $H$ is compactly embedded into $$ \Gamma^{q}(\mathbb{R}^3):= \{\varphi: \mathbb{R}^3\to\mathbb{R}; \varphi \text{ is measurable and } \int_{\mathbb{R}^3}K(x)| \varphi|^{q}\,dx <\infty \}, $$ for all $q\in (2,6)$. \end{lemma} \begin{proof} Consider (A1), (A3) and (A4); thus fixed $q\in(2,6)$ and given $\varepsilon>0$, there are $00$ such that \begin{equation}\label{24} K(x)|s|^q\leq \varepsilon C(V(x)|s|^2+|s|^{6})+CK(x)\mathcal{X}_{[s_0,s_1]} (|s|)|s|^{6},\quad \forall s\in\mathbb{R}. \end{equation} Hence, \begin{equation}\label{25} \int_{B^c_r(0)} K(x)|u|^q\,dx\leq \varepsilon CQ(u) +C\int_{A\cap B^c_r(0)}K(x)\,dx,\quad \forall u\in H \end{equation} where \begin{gather*} Q(u)=\int_{\mathbb{R}^3}V(x)|u|^2\,dx+\int_{\mathbb{R}^3}|u|^{6}\,dx, \\ A=\{x\in\mathbb{R}^3: s_0\leq|u(x)|\leq s_1 \}. \end{gather*} If $(v_n)$ is a sequence such that $v_n\rightharpoonup v$ weakly in $H$, as $ n \to \infty$, there is some constant $M_1>0$ such that $$ \| v_n \|^2=\int_{\mathbb{R}^3}(|\nabla v_n|^2+V(x)|v_n|^2)\,dx \leq M_1, \quad \int_{\mathbb{R}^3}|v_n|^{6}\,dx \leq M_1,\quad \forall n\in \mathbb{N}, $$ implying that $(Q(v_n))$ is bounded. On the other hand, setting $$ A_n=\{x\in\mathbb{R}^3:s_0\leq |v_n(x)|\leq s_1 \}, $$ the above inequality implies $$ s_0^{6}\mu(A_n)\leq \int_{A_n}|v_n|^{6}\,dx\leq M_1,\quad \forall n\in\mathbb{N}, $$ showing that $\sup_{n\in\mathbb{N}}\mu(A_n)<+\infty$. Therefore, from (II), there is a $r>0$ such that \begin{equation}\label{26} \int_{A_n\cap B^c_r(0)}K(x)\,dx<\frac{\varepsilon}{s^{6}_1},\quad \forall n\in\mathbb{N}. \end{equation} Now, \eqref{25} and \eqref{26} lead to \begin{equation}\label{27} \int_{B^c_r(0)}K(x)|v_n|^q\,dx \leq \varepsilon CM_1+s^{6}_1\int_{A_n\cap B^c_r(0)}K(x)\,dx <(CM_1+1)\varepsilon,\quad \forall n\in\mathbb{N}. \end{equation} Since $q\in(2,6)$ and $K$ is a continuous function, from the Sobolev embeddings it follows that \begin{equation}\label{28} \lim_{n\to+\infty}\int_{B_r(0)}K(x)|v_n|^q\,dx=\int_{B_r(0)}K(x)|v|^q\,dx. \end{equation} In light of \eqref{27}$ and \eqref{28}$, we have \begin{equation}\label{29} \lim_{n\to+\infty}\int_{\mathbb{R}^3}K(x)|v_n|^q\,dx =\int_{\mathbb{R}^3}K(x)|v|^q\,dx. \end{equation} This means that $$ v_n\to v,\quad \text{in } \Gamma^{q}(\mathbb{R}^3), \quad n \to \infty, \;\forall q\in(2,6). $$ Now, we fix $x \in \mathbb{R}^3$ and $\forall s>0$ there is a constant $C=C(p)$ such that $$ CV(x)^{\frac{6-p}{4}}\leq V(x)s^{2-p}+s^{6-p}; $$ % it follows from the fact that the function $$ h(s)=V(x)s^{2-p}+s^{6-p}, \quad s>0, $$ has the minimum value $CV(x)^{\frac{6-p}{4}}$. Using (A1), (A3') and (A4),and choosing $\varepsilon \in (0,C)$ for some $C>0$ we infer that $$ K(x)| s|^{p}\leq \varepsilon (V(x)| s|^{2} +| s|^{6}), \quad \forall s \in \mathbb{R}, | x| \geq r . $$ Consequently, for all $u \in H$ we have $$ \int_{B^{c}_{r}(0)}K(x)| s|^{p}\,dx \leq \int_{B^{c}_{r}(0)}\varepsilon (V(x)| s|^{2} +| s|^{6})\,dx. $$ If $(v_n)$ is a sequence such that $v_n\rightharpoonup v$ weakly in $H$, as $n \to \infty$, there is $M_2>0$ such that \begin{equation}\label{32} \int_{B^c_r(0)}K(x)|v_n|^q\,dx\leq 2\varepsilon M_2. \end{equation} Since $q\in(2,6)$ and $K$ is a continuous function, it follows from the Sobolev embeddings \begin{equation}\label{33} \lim_{n\to+\infty}\int_{B_r(0)}K(x)|v_n|^q\,dx=\int_{B_r(0)}K(x)|v|^q\,dx. \end{equation} From \eqref{32} and \eqref{33}, we obtain $$ \lim_{n\to+\infty}\int_{\mathbb{R}^3}K(x)|v_n|^q\,dx =\int_{\mathbb{R}^3}K(x)|v|^q\,dx. $$ implying that $$ v_n\to v\quad \text{in } \Gamma^{q}(\mathbb{R}^3), \quad n \to \infty, \;\forall q\in(2,6). $$ \end{proof} \begin{lemma} \label{L22} Suppose that {\rm (A1)--(A4)} are satisfied, and consider a sequence $ (v_n)$ in $H$ such that $ v_n\rightharpoonup v$ weakly in $H$, as $n \to \infty$. Then $$ \lim_{n\to +\infty}\int_{\mathbb{R}^3}K(x)f(v_n)v_n\,dx =\int_{\mathbb{R}^3}K(x)f(v)v\,dx. $$ \end{lemma} \begin{proof} Assuming (A1), (A3) and (A4), for a fixed $q\in(2,6)$ and $\varepsilon>0$, there is $C>0$ such that \begin{equation}\label{212} |K(x)f(s)s|\leq\varepsilon C(V(x)|s|^2+|s|^{6})+K(x)|s|^q,\quad \forall s\in\mathbb{R}. \end{equation} From Lemma \ref{p21}, we have $$ \int_{\mathbb{R}^3}K(x)|v_n|^q\,dx\to\int_{\mathbb{R}^3}K(x)|v|^q\,dx, $$ then there exists $r>0$ such that \begin{equation}\label{213} \int_{B^c_r(0)}K(x)|v_n|^q\,dx<\varepsilon,\quad \forall n\in\mathbb{N}. \end{equation} Since $(v_n)$ is bounded in $H$, there exists $M_3>0$ such that $$ \int_{\mathbb{R}^3}V(x)|v_n|^2\,dx\leq M_3\quad \text{and}\quad \int_{\mathbb{R}^3}V(x)|v_n|^{6}\,dx\leq M_3. $$ Combining the last two inequalities with \eqref{212} and \eqref{213}, we obtain $$ \big|\int_{B^c_r(0)}K(x)f(v_n)v_n\,dx\big|<(2CM_3+1)\varepsilon,\quad \forall n\in\mathbb{N}. $$ To complete the proof we need to show that $$ \lim_{n\to+\infty}\int_{B_r(0)}K(x)f(v_n)v_n\,dx =\int_{B_r(0)}K(x)f(v)v\,dx. $$ However, this limit is obtained by using hypothesis (A4) and arguing as in \cite{Chabrowski}, setting $$ P(x,s)=K(x)f(s)s, \quad Q(x,u_n(x)) = | u_n(x)|^{6}. $$ \end{proof} \begin{lemma} \label{L23} Suppose that $f$ satisfies {\rm (A1), (A3'), (A4)}, and consider a sequence $ (v_n)$ in $H$ such that $ v_n\rightharpoonup v$ weakly in $H$, as $n \to \infty$. Then $$ \lim_{n\to +\infty}\int_{\mathbb{R}^3}K(x)f(v_n)v_n\,dx =\int_{\mathbb{R}^3}K(x)f(v)v\,dx. $$ \end{lemma} \begin{proof} Using the Lemma \ref{p21}, for $r>0$ sufficiently small, arguing as in \eqref{212} we infer that $$ K(x)\leq \varepsilon (V(x)| s|^{2-p} +| s|^{6-p}),\quad \forall | x|\geq r . $$ The rest of the proof follows similarly to the proof of Lemma \ref{L22}. \end{proof} \section{Boundedness of Cerami sequence} \begin{lemma}\label{ltda} The Cerami sequence $ (u_n)\subset H$ given in \eqref{Cer} is bounded.\\ \end{lemma} \begin{proof} We have a positive constant $ M $ such that \begin{equation}\label{LD} M+o_n{(1)}\| u_n\|\geq \theta J(u_n)-J'(u_n)u_n \end{equation} for $ 2< q < 6$. From (A1), (A5) and Lemma \ref{L2.1} the Cerami sequence $ (u_n)$ is such that \begin{align*} \theta J(u_n)-J'(u_n)u_n & = \big(\frac{\theta -2}{2}\big)\| u_n\|^{2} + \big(\frac{-\theta +4}{2}\big)\int_{\mathbb{R}^3} \omega \phi_{u_n}u_n^{2}\,dx + \int_{\mathbb{R}^3} \phi_{u_n}^{2}u_n^{2}\,dx \\ &\quad + \int_{\mathbb{R}^3}K(x)(f(u_n)u_n-\theta F(u_n))\,dx \\ & \geq \big(\frac{\theta -2}{2}\big)\| u_n\|^{2}, \quad \text{if } \theta >4. \end{align*} Similarly, if $2< \theta <4$ we use the hypothesis $$ 0< \frac{2(4-\theta)}{\theta -2}\leq V_{0}\leq V(x), $$ and Lemma \ref{L2.1} to obtain \begin{equation} \label{LD1} \begin{aligned} & \theta J(u_n)-J'(u_n)u_n \\ & \geq \big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}| \nabla u_n|^{2}\,dx + \big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}V(x)u_n^{2}\,dx +\omega \big(\frac{-\theta +4}{2}\big) \int_{\mathbb{R}^3} \phi_{u_n}u_n^{2}\,dx \\ &\geq \big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}| \nabla u_n|^{2}\,dx +\big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}V_{0}u_n^{2}\,dx +\omega \big(\frac{-\theta +4}{2}\big) \int_{\mathbb{R}^3} \phi_{u_n}u_n^{2}\,dx \\ &\geq \big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}| \nabla u_n|^{2}\,dx +\big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}V_{0}u_n^{2}\,dx +\omega^{2} \big(\frac{\theta -4}{2}\big) \int_{\mathbb{R}^3} u_n^{2}\,dx\\ &= \big(\frac{\theta -2}{2}\big)\int_{\mathbb{R}^3}| \nabla u_n|^{2}\,dx + \big[\frac{(\theta -2)V_{0}+(\theta -4)\omega^{2}}{2}\big] \int_{\mathbb{R}^3}u_n^{2}\,dx\\ & \geq C \| u_n\|^{2}. \end{aligned} \end{equation} In light of \eqref{LD} and \eqref{LD1} we conclude that $(u_n)$ is bounded. \end{proof} \begin{lemma} \label{L26} If $ u_n\rightharpoonup u$ weakly in $H$, as $n \to \infty$, then passing to a subsequence if necessary, $\phi_{u_n}\rightharpoonup \phi_{u} $ weakly in $ D^{1,2}(\mathbb{R}^3)$, as $n \to \infty$. \end{lemma} \begin{proof} Consider $ (u_n), u \in H$ such that $ u_n\rightharpoonup u$ weakly in $H$, as $n \to \infty$. It follows that \begin{gather*} u_n\rightharpoonup u \quad \text{ weakly in $L^{p}(\mathbb{R}^3)$, as $n \to \infty$, }2\leq p\leq 6, \\ u_n\to u \quad \text{in $L^{p}_{\rm loc}(\mathbb{R}^3)$, as $n \to \infty$, }2\leq p< 6. \end{gather*} From Lemma \ref{L2.1}, note that for all $n\geq 1$ we have \begin{align*} \| \phi_{u_n}\|^{2}_{D^{1,2}(\mathbb{R}^3)} & = -\int_{\mathbb{R}^3}\omega \phi_{u_n}u_n^{2}\,dx - \int_{\mathbb{R}^3}\phi^{2}_{u_n}u_n^{2}\,dx\\ & \leq -\int_{\mathbb{R}^3}\omega \phi_{u_n}u_n^{2}\,dx \leq C \| \phi_{u_n}\|_{D^{1,2}(\mathbb{R}^3)} \| u_n\|^{2}_{\frac{12}{5}}. \end{align*} It means that $ (\phi_{u_n})$ is bounded in $ D^{1,2}(\mathbb{R}^3)$. Since $ D^{1,2}(\mathbb{R}^3)$ is a Hilbert space, there is a $\xi \in D^{1,2}(\mathbb{R}^3)$ such that \begin{gather*} \phi_{u_n}\rightharpoonup \xi \quad \text{ weakly in } L^{p}(\mathbb{R}^3), \text{ as } n \to \infty,\; 2\leq p\leq 6, \\ \phi_{u_n}\to \xi \quad \text{in } L^{p}_{\rm loc}(\mathbb{R}^3),\text{ as } n \to \infty,\quad 2\leq p< 6. \end{gather*} We desire to prove the following equality $\phi_{u}=\xi$. For this, it is necessary to show, in the sense of distributions, $$ \Delta \xi = (\omega +\xi)u^{2} $$ and use the uniqueness of the solution given in Lemma \ref{L2.1}. Consider a test function $\psi \in C_{0}^{\infty}(\mathbb{R}^3)$. We know by Lemma \ref{L2.1} we have $$ \Delta \phi_{u_n} = (\omega +\phi_{u_n})u^{2}_n. $$ Then we just need to see how each term of the equality above converges. To verify that \begin{gather*} \int_{\mathbb{R}^3}\nabla \phi_{u_n}\nabla \psi\,dx \to \int_{\mathbb{R}^3}\nabla \xi\nabla \psi\,dx, \quad \text{as } n \to \infty, \\ \int_{\mathbb{R}^3} \phi_{u_n}u^{2}_n\psi\, dx \to \int_{\mathbb{R}^3}\xi u^{2}\psi\, dx, \quad \text{as } n \to \infty, \end{gather*} it is sufficient to note that it is a consequence of the definition the weak convergence. By the strong convergence in $L^{p}_{\rm loc}(\mathbb{R}^3),2\leq p< 6$, we obtain $$ \int_{\mathbb{R}^3}u^{2}_n\psi\,dx \to \int_{\mathbb{R}^3} u^{2}\psi\,dx, \quad \text{as } n \to \infty. $$ We consider a test function $\varphi \in C_{0}^{\infty}(\mathbb{R}^3)$. Using boundedness of $(\phi_{u_n})$, the strong convergences in $ L^{p}_{\rm loc}(\mathbb{R}^3),2\leq p< 6$ and the Sobolev embeddings follows that as $n\to+\infty$, we have \begin{align*} \int_{\mathbb{R}^3}( \phi_{u_n}u_n-\xi u)\varphi\, dx & = \int_{\mathbb{R}^3} \phi_{u_n}(u_n-u)\varphi\, dx + \int_{\mathbb{R}^3}u(\phi_{u_n}-\xi)\varphi\, dx\\ & \leq C\| \phi_{u_n}\|_{D^{1,2}(\mathbb{R}^3)} \Big( \int_{\mathbb{R}^3}| u_n-u|^{6/5}|\varphi|^{6/5} \,dx \Big)^{5/6} \\ &\quad +\int_{\mathbb{R}^3}(\phi_{u_n}-\xi)u\varphi\, dx \to 0,\quad \text{as }n\to \infty\to+\infty. \end{align*} For the same reasons, it follows that \begin{align*} \int_{\mathbb{R}^3}( \phi^{2}_{u_n}u_n-\xi^{2} u)\varphi \,dx & = \int_{\mathbb{R}^3} \phi^{2}_{u_n}(u_n-u)\varphi\, dx + \int_{\mathbb{R}^3}u(\phi^{2}_{u_n}-\xi^{2})\varphi \,dx\\ & \leq C \| \phi_{u_n}\|_{D^{1,2}(\mathbb{R}^3)} \Big( \int_{\mathbb{R}^3}| u_n-u|^{3/2} |\varphi|^{3/2} \,dx \Big)^{2/3}\\ &\quad +\int_{\mathbb{R}^3}(\phi^{2}_{u_n}-\xi^{2})u\varphi \,dx \to 0,\quad \text{as } n\to+\infty. \end{align*} From density, for all $\varphi \in H$ we infer that \begin{gather*} \int_{\mathbb{R}^3}( \nabla u_n\nabla\varphi +V(x)u_n\varphi)\, dx \to \int_{\mathbb{R}^{N}}( \nabla u\nabla \varphi+V(x)u\varphi)\, dx, \\ \int_{\mathbb{R}^3}( 2\omega+ \phi_{u_n})\phi_{u_n}u_n\varphi \,dx \to \int_{\mathbb{R}^3}( 2\omega+ \xi)\xi u \varphi\, dx, \end{gather*} as $n\to +\infty$, thus we prove the lemma. \end{proof} \section{Proof of the main results} \begin{proof}[Proof of Theorem \ref{teo1}] Let $(u_n)$ be a Cerami sequence as given in \eqref{Cer}. From Lemma \ref{ltda} follows that $(u_n)$ is bounded and, up to subsequence, we can assume that there is $u\in H $, such that $$ u_n\rightharpoonup u,\quad \text{weakly in $H$, as } n \to \infty. $$ We will show that $u_n\to u$, as $n \to + \infty$. From Lemma \ref{L22}, we have $$ \lim_{n\to +\infty}\int_{\mathbb{R}^3}K(x)f(u_n)u_n\,dx =\int_{\mathbb{R}^3}K(x)f(u)u\,dx. $$ On the other hand, we know that $$ J'(u)v = \int_{\mathbb{R}^3}( \nabla u.\nabla v+V(x)uv)\,dx -\int_{\mathbb{R}^3}( 2\omega+ \phi_{u})\phi_{u}uv\,dx - \int_{\mathbb{R}^3}K(x)f(u)v\,dx. $$ Since $J'(u_n)u_n=o_n(1)$, we get \begin{equation}\label{316} \lim_{n\to+\infty}\| u_n \|^2=\lim_{n\to+\infty} \Big[ \int_{\mathbb{R}^3}( 2\omega+ \phi_{u_n})\phi_{u_n}u^2_n \,dx + \int_{\mathbb{R}^3}K(x)f(u_n)u_n\,dx\Big]. \end{equation} By Lemma \ref{L22}, we have $$ \lim_{n\to+\infty}\int_{\mathbb{R}^3}K(x)f(u_n)u\,dx =\int_{\mathbb{R}^3}K(x)f(u)u\,dx $$ and from Lemma \ref{L26}, we obtain that $$ \lim_{n\to \infty} \int_{\mathbb{R}^3}( 2\omega+ \phi_{u_n})\phi_{u_n}u^2_n \,dx = \int_{\mathbb{R}^3}( 2\omega+ \xi)\xi u^2\,dx. $$ Then \begin{equation}\label{3170} \lim_{n\to + \infty}\| u_n \|^2 =\int_{\mathbb{R}^3}( 2\omega+ \xi)\xi u^2\,dx + \int_{\mathbb{R}^3}K(x)f(u)u\,dx. \end{equation} Moreover, since $J'(u_n)u=o_n(1)$, we have \begin{equation}\label{317} \| u \|^2=\int_{\mathbb{R}^3}( 2\omega+ \xi)\xi u^2\,dx +\int_{\mathbb{R}^3}K(x)f(u)u\,dx. \end{equation} Therefore, from \eqref{3170} and \eqref{317}, we obtain $\lim_{n\to + \infty}\| u_n \|^2=\| u \|^2$, showing that $$ u_n\to u,\quad \text{in } H, \text{ as } n \to \infty. $$ Consequently, $$ J(u)=c\quad \text{and}\quad J'(u)=0, $$ implying that $u$ is a ground state solution for $J$. Since $u_n\geq 0$, we have that $u\geq 0$. The positivity of $u$ follows by using the maximum principle. \end{proof} \begin{proof}[Proof of Theorem \ref{teo2}] It is similar to that of Theorem \ref{teo1}. However using the Lemma \ref{L23} instead of Lemma \ref{L22}. We omit the proof here. \end{proof} \section{Case $V(x)\to 0$, as $|x|\to\infty$} In this section, we study the problem \eqref{KGM}, inspired by \cite{Ambro}, replacing the hypothesis (A1) by \begin{itemize} \item[(A1')] $V,K:\mathbb{R}^3\to\mathbb{R}$ are smooth functions, $K\in L^{\infty}(\mathbb{R}^3)$ and there are constant $\tau, \xi_1, a_1, a_2, a_3>0$, such that \begin{equation}\label{Vk1} \frac{a_1}{1+|x|^{\tau}}\leq V(x)\leq a_2\quad\text{and}\quad 0< K(x)\leq \frac{a_3}{1+|x|^{\xi_1}},\quad \forall x\in\mathbb{R}^3. \end{equation} with $\tau, \xi_1$ satisfying $$ 5-\frac{4\xi_1}{\tau} 0$ and $n_0 \in \mathbb{N}$ such that $$ J(u_n)-\frac{1}{\theta}J'(u_n)u_n\leq M+\|u_n\|_V, \ \ \forall n\geq n_0. $$ On the other hand, it is certain that $u_n>0$ for each $x\in\mathbb{R}^3$ and using the assumption $(f_{2})$ for $\theta>4$ combined with Lemma \ref{L2.1}, we have \begin{align*} J(u_n)-\frac{1}{\theta}J'(u_n)u_n &\geq \big(\frac{1}{2}-\frac{1}{\theta}\big)\|u_n\|^2_V -\frac{1}{2}\int_{\mathbb{R}^3}\omega \phi_{u_n}u_n^{2}\,dx +\frac{2\omega}{\theta}\int_{\mathbb{R}^3}\phi_{u_n}u^2_n\,dx\\ &\quad +\frac{1}{\theta}\int_{\mathbb{R}^3}\phi^2_{u_n}u^2_n\,dx\\ &\geq \big(\frac{1}{2}-\frac{1}{\theta}\big)\|u_n\|^2_V +\omega\big(\frac{4-\theta}{2\theta}\big)\int_{\mathbb{R}^3}\phi_{u_n}u^2_n\,dx\\ &\geq \big(\frac{1}{2}-\frac{1}{\theta}\big)\|u_n\|^2_V, \end{align*} which shows that $(u_n)$ is bounded. \end{proof} In this way, we obtained the same results as those presented of Theorems \ref{teo1} and \ref{teo2}, using (A1') instead of (A1). \subsection*{Acknowledgments} O. 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