\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 82, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/82\hfil Solitary waves] {Solitary waves for the coupled nonlinear Klein-Gordon and Born-Infeld type equations} \author[F. Wang\hfil EJDE-2012/82\hfilneg] {Feizhi Wang} \address{Feizhi Wang \newline School of Mathematics, Yantai University, Yantai, Shandong, China} \email{wangfz@ytu.edu.cn} \thanks{Submitted December 22, 2011. Published May 23, 2012.} \thanks{Supported by grant ZR2011AL009 from the Shandong Province Natural Science Foundation} \subjclass[2000]{35J15} \keywords{Nonlinear Klein-Gordon; Born-Infeld type equation; \hfill\break\indent electrostatic solitary wave; critical points} \begin{abstract} In this article we study the existence of solutions for a nonlinear Klein-Gordon-Maxwell equation coupled with a Born-Infeld equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} It is well known that the gauge potential $(\phi,\mathbf{A})$ can be coupled to a complex order parameter $\psi$ through the minimal coupling rule; that is the formal substitution \begin{align*} \frac{\partial}{\partial t} \mapsto\frac{\partial}{\partial t}+ie\phi,\\ \nabla \mapsto\nabla-ie\mathbf{A}, \end{align*} where $e$ is the electric charge, $\mathbf{A}:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}^3$ is a magnetic vector potential and $\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}$ is an electric potential. Therefore, in a flat Minkowskian space-time with metric $(g_{\mu\nu})=\operatorname{diag}[1,-1,-1,-1]$, we can define the Klein-Gordon-Maxwell Lagrangian density $$\mathcal{L}_{KGM}=\frac12\big[|\frac{\partial \psi}{\partial t}+ie\phi\psi|^2 -|\nabla\psi-ie\mathbf{A}|^2-m^2|\psi|^2\big]+\frac1q|\psi|^q,$$ where $m\geq0$ represents the mass of the charged field. The total action of the system is thus given by $$\label{BIKGM} \mathcal{S}=\iint(\mathcal{L}_{KGM}+\mathcal{L}_{\rm emf})\,dx\,dt,$$ where $\mathcal{L}_{\rm emf}$ is the Lagrangian density of the electro-magnetic field. In the Born-Infeld theory (see \cite{BI34}), with a suitable choice of constants, $\mathcal{L}_{\rm emf}$ can be written as $$\mathcal{L}_{\rm emf}=\mathcal{L}_{BI}:=\frac{b^2}{4\pi} \Big(1-\sqrt{1-\frac1{b^2}(|\mathbf{E}|^2-|\mathbf{B}|^2)}\Big),$$ where $b$ is the so-called Born-Infeld parameter, $b\gg 1$. By the Maxwell equations, $$\mathbf{E}=-\nabla \phi-\frac{\partial\mathbf{A}}{\partial t}$$ is the electric field, and $$\mathbf{B}=\nabla\times\mathbf{A}$$ is the magnetic induction field. If, as in \cite{BF02}, we consider the electrostatic solitary wave: $$\psi(x,t)=u(x)e^{-i\omega t},\quad \mathbf{A}=0,\quad \phi=\phi(x),$$ where $u:\mathbb{R}^3\to\mathbb{R}$ and $\omega\in\mathbb{R}$, then the total action in \eqref{BIKGM} takes the form \label{BIfunctional} \begin{aligned} {F}_\text{BI}(u,\phi) &=\frac12\int_{\mathbb{R}^3}|D u|^2\,dx+\frac12\int_{\mathbb{R}^3}\left(m^2-(e\phi-\omega)^2\right)u^2\,dx\\ &\quad - \frac1q\int_{\mathbb{R}^3} |u|^{q}\,dx-\frac{b^2}{4\pi}\int_{\mathbb{R}^3} \Big(1-\sqrt{1-\frac1{b^2}|\nabla\phi|^2}\Big)\,dx. \end{aligned} The critical point $(u,\phi)$ of ${F}_\text{BI}$ satisfies the Euler-Lagrange equations associted to \eqref{BIfunctional}. By standard calculations, we obtain: \label{P22} %\tag{P_\text{BI}} \begin{aligned} -\Delta u+[m^2-(\phi-\omega)^2]u=|u|^{q-2}u,\quad \text{in } \mathbb{R}^3,\\ \nabla\cdot\frac{\nabla\phi}{\sqrt{1-\frac1{b^2}|\nabla\phi|^2}} =4\pi (\phi-\omega)u^2,\quad \text{in } \mathbb{R}^3, \end{aligned} where we have taken $e=1$. We can see that the sign $\omega$ is not relevant for the existence of solutions for problem \eqref{P22}. In fact, if $(u,\phi)$ is a solution of \eqref{P22} with $\omega$, then $(u,-\phi)$ is also a solution corresponding to $-\omega$. So, without loss of generality, we can assume $\omega>0$. As we know, a large number of works have been devoted to the problem like \eqref{P22}. In the following we review some assumptions and the corresponding results. In \cite{APP09,AP10,BF02,BF09-1,BF09-2,BF10,DM04-1,DM04-2,Wang11-2}, the authors consider the first-order expansion of the second formula of \eqref{P22} for $b\to+\infty$. Therefore \eqref{P22} becomes $$\label{P1} %\tag{P_1} \begin{gathered} -\Delta u+[m^2-(\phi-\omega)^2]u=|u|^{q-2}u,\quad\text{in }\mathbb{R}^3,\\ \Delta\phi=4\pi(\phi-\omega)u^2,\quad\text{in }\mathbb{R}^3. \end{gathered}$$ About the problem \eqref{P1}, the pioneering work is given by Benci and Fortunato \cite{BF02}. They showed that \eqref{P1} has infinitely many solutions when $q\in(4,6)$ and $0<\omega\omega^2. $$d'Aprile and Mugnai \cite{DM04-1} also showed that \eqref{P1} has no nontrivial solutions when q\geq6 and 0<\omega\leq m or q\leq2. Recently, in \cite{APP09}, under the following conditions: \begin{gather*} (q-2)(4-q)m^2>\omega^2,\quad p\in(2,3),\\ m>\omega>0,\quad p\in[3,6), \end{gather*} Azzollini, Pisani and Pomponio showed that \eqref{P1} admits a nontrivial solution. It is easy to see that (p-2)(4-p)>(p-2)/2 for p\in(2,3]. In \cite{DP02,FOP02,Mugnai04}, the authors consider the second-order expansion of the second formula of \eqref{P22} for b\to+\infty. Therefore \eqref{P22} becomes $$\label{P2} %\tag{P_2} \begin{gathered} -\Delta u+[m^2-(\phi-\omega)^2]u=|u|^{q-2}u,\quad\text{in }\mathbb{R}^3,\\ \Delta\phi+\beta_2\Delta_4\phi=4\pi(\phi-\omega)u^2,\quad\text{in }\mathbb{R}^3, \end{gathered}$$ where \beta_2=1/(2b^2)\to0 and \Delta_4 \phi=D(|D\phi|^2D\phi). In \cite{FOP02}, Fortunato, Orsina and Pisani showed the existence of electrostatic solutions with finite energy, while in \cite{DP02} d'Avenia and Pisani proved that \eqref{P2} has infinitely many solutions, provided that 4\omega^2.$$ Recently, Yu \cite{Yu10} studied the original Born-Infeld equations, i.e. \eqref{P22}. He proved the existence of the least-action solitary waves in both bounded smooth domain case and$\mathbb{R}^3$case whenever$q\in(2,6)$and $$\frac {q-2}q m^2>\omega^2.$$ In the present paper we consider the nonlinear Klein-Gordon equations coupled with the$N$-th order expansion of the second formula of \eqref{P22} for$b\to+\infty$: $$\label{Pn} %\tag{P_N} \begin{gathered} -\Delta u+[m^2-(\phi-\omega)^2]u=|u|^{q-2}u,\quad\text{in }\mathbb{R}^3,\\ \sum_{k=1}^N(\beta_k\Delta_{2k} \phi)=4\pi(\phi-\omega)u^2,\quad\text{in }\mathbb{R}^3, \end{gathered}$$ where$\beta_1=1$,$\beta_k=\frac{1\cdot3\cdot5\dots(2k-3)}{2^{k-1}(k-1)!}\frac1{b^{2(k-1)}}$and$\Delta_{2k}\phi=D(|D\phi|^{2k-2}D\phi)$, for$k=2,3,\dots,N$. It is well-known that$H^1(\mathbb{R}^3)$is the usual Sobolev space endowed with the norm $$\|u\|_{H^1(\mathbb{R}^3)}=\Big(\int_{\mathbb{R}^3} [|Du|^2+u^2]\,dx\Big)^{1/2}$$ (see \cite{Adams75}, \cite[Theorem~1.8]{Wi96}).$D^{N}(\mathbb{R}^3)$denotes the completion of$C_0^\infty(\mathbb{R}^3,\mathbb{R})$with respect to the norm $$\|\phi\|_{D^{N}(\mathbb{R}^3)} = \Big(\int_{\mathbb{R}^3} |D\phi|^{2}\,dx\Big)^{1/2} +\Big(\int_{\mathbb{R}^3} |D\phi|^{2N}\,dx\Big)^{1/(2N)}.$$ By a solution$(u,\phi)$of \eqref{Pn}, we understand$(u,\phi)\in H^1(\mathbb{R}^3)\times D^N(\mathbb{R}^3)$satisfying \eqref{Pn} in the weak sense. Obviously,$(u,\phi)=(0,0)$is a trivial solution of \eqref{Pn}. We define a functional$F_N:H^1(\mathbb{R}^3)\times D^N(\mathbb{R}^3)\to\mathbb{R}$by $F_N(u,\phi) =\int_{\mathbb{R}^3} \big[\frac12|D u|^2-\frac1{4\pi}\sum_{k=1}^N \big(\frac1{2k}\beta_k|D\phi|^{2k}\big) +\frac12(m^2-(\phi-\omega)^2)u^2-\frac1q|u|^{q}\big]\,dx\,.$ It is easy to see that$F_N\in C^1(H^1(\mathbb{R}^3)\times D^{N}(\mathbb{R}^3),\mathbb{R})$. Therefore solutions of \eqref{Pn} correspond to critical points of the functional$F_N$. Next we give our main result. \begin{theorem}\label{main2} Problem \eqref{Pn} has at least a nontrivial solution$(u,\phi)\in H^1(\mathbb{R}^3)\times D^N(\mathbb{R}^3)$, provided one of the following conditions is satisfied \begin{itemize} \item[(i)]$ q\in(3,6)$and$m>\omega>0$. \item[(ii)]$ q\in(2,3]$and$(q-2)(4-q)m^2>\omega^2>0$. \end{itemize} \end{theorem} Set$|u|_q:=\{\int_{\mathbb{R}^3} |u|^q dx\}^{1/q}$for$10\}, the following equalities hold: \begin{gather}\label{ident1} \begin{aligned} &\int_{B_R}-\Delta u\langle x, Du\rangle\,dx\\ &=\frac{2-n}2\int_{B_R}|D u|^2\,dx -\frac1R\int_{\partial B_R}\langle x, Du\rangle^2\,d\sigma +\frac R2\int_{\partial B_R}|Du|^2\,d\sigma; \end{aligned}\\ \label{ident3} \begin{aligned} &\int_{B_R}(a+b\phi)\phi u\langle x,Du\rangle\,dx\\ &=-\int_{B_R}\big(\frac a2+b\phi\big)u^2\langle x,D\phi\rangle\,dx\\ &\quad -\frac n2\int_{B_R}(a+b\phi)\phi u^2\,dx +\frac R2\int_{\partial B_R}(a+b\phi)\phi u^2\,d\sigma; \end{aligned}\\ \label{ident4} \int_{B_R}g(u)\langle x,Du\rangle\,dx =-n\int_{B_R}G(u)\,dx+R\int_{\partial B_R}G(u)d\sigma; \\ \label{ident2} \begin{aligned} \int_{B_R}\Delta_{2k}\phi\, \langle x, D\phi\rangle\,dx &=\int_{B_R}D(|D \phi|^{2k-2}D \phi)\langle x,D \phi\rangle\,dx\\ &=\frac {n-2k}{2k}\int_{B_R}|D\phi|^{2k}dx -\frac R{2k}\int_{\partial B_R}|D\phi|^{2k}d\sigma\\ &\quad +\frac1R\int_{\partial B_R}|D\phi|^{2k-2}\langle x,D\phi\rangle^2\, d\sigma, \end{aligned} \end{gather} where\Delta_{2k}\phi=D(|D\phi|^{2k-2}|D\phi|)$and$g:\mathbb{R}\to\mathbb{R}$is a continuous function such that$g(0)=0$and$G(s)=\int_0^sg(t)\,dt$. \end{lemma} \begin{proof} The proofs of \eqref{ident1}, \eqref{ident3} and \eqref{ident4} can be found in \cite[Lemma 3.1]{DM04-1}. In the following we show \eqref{ident2}. For fix$i_1,\dots,i_{k-1}$,$j,l=1,2,\dots,n, we see from the integration by parts formula that \begin{align*} &\int_{B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2\phi_{x_jx_j}x_l\phi_{x_l}dx\\ &=-\int_{B_R}(\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l\phi_{x_l})_{x_j}\phi_{x_j}dx +\int_{\partial B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l\phi_{x_l}\phi_{x_j} \frac{x_j}{|x|}d\sigma\\ &=-\int_{B_R}(\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2)_{x_j}x_l\phi_{x_l}\phi_{x_j}dx -\int_{B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2\phi_{x_l}\phi_{x_j}\delta_{lj}dx\\ &\quad -\int_{B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l\phi_{x_l{x_j}}\phi_{x_j}dx +\int_{\partial B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l\phi_{x_l}\phi_{x_j} \frac{x_j}{|x|}d\sigma, \end{align*} whered\sigma$indicates the$(n-1)$-dimensional area element in$\partial B_R$and$\delta_{lj}$are the Kroneker symbols. Summing up for$i_1,\dots,i_{k-1}$,$j,l=1,2,\dots,n, we have \label{KGM1} \begin{aligned} &\int_{B_R}|D \phi|^{2k-2}\Delta\phi\langle x,D \phi\rangle\,dx\\ &=-\int_{B_R}\langle D|D \phi|^{2k-2},D \phi\rangle\langle x,D \phi\rangle\,dx -\int_{B_R}|D\phi|^{2k}\,dx\\ &\quad -\int_{B_R}|D\phi|^{2k-2}\langle x,D^2\phi D\phi\rangle\, dx +\frac1R\int_{\partial B_R}|D\phi|^{2k-2}\langle x,D\phi\rangle^2\, d\sigma. \end{aligned} Similarly, for fixi_1,\dots,i_{k-1}$,$j,l=1,2,\dots,n, we see from the integration by parts formula that \begin{align*} &2\int_{B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l\phi_{x_l{x_j}}\phi_{x_j}dx =\int_{B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l(\phi_{x_j}^2)_{x_l}dx\\ &=-\int_{B_R}(\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2x_l)_{x_l}\phi_{x_j}^2dx +\int_{\partial B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2\phi_{x_j}^2 \frac{x_l^2}{|x|}d\sigma\\ &=-\int_{B_R}(\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2)_{x_l}x_l\phi_{x_j}^2dx -\int_{B_R}(\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2)_{x_l}\phi_{x_j}^2dx\\ &\quad +\int_{\partial B_R}\phi_{x_{i_1}}^2\dots\phi_{x_{i_{k-1}}}^2\phi_{x_j}^2 \frac{x_l^2}{|x|}d\sigma. \end{align*} Summing up fori_1,\dots,i_{k-1}$,$j,l=1,2,\dots,n, we have \begin{align*} &2\int_{B_R}|D\phi|^{2k-2}\langle x,D^2\phi D\phi\rangle dx\\ &=-\int_{B_R}\langle x,D(|D\phi|^{2k-2})\rangle|D\phi|^2 dx -n\int_{B_R}|D\phi|^{2k}dx+R\int_{\partial B_R}|D\phi|^{2k}d\sigma\\ &=-2(k-1)\int_{B_R}|D\phi|^{2k-2}\langle x,D^2\phi D\phi\rangle dx -n\int_{B_R}|D\phi|^{2k}dx+R\int_{\partial B_R}|D\phi|^{2k}d\sigma. \end{align*} Then $$\label{KGM2} \int_{B_R}|D\phi|^{2k-2}\langle x,D^2\phi D\phi\rangle dx =-\frac n{2k}\int_{B_R}|D\phi|^{2k}dx +\frac R{2k}\int_{\partial B_R}|D\phi|^{2k}d\sigma.$$ Using \eqref{KGM1} and \eqref{KGM2}, we obtain \begin{align*} &\int_{B_R}\Delta_{2k}\phi\, \langle x, D\phi\rangle\,dx\\ &=\int_{B_R}D(|D \phi|^{2k-2}D \phi)\langle x,D \phi\rangle\,dx\\ &=\int_{B_R}|D \phi|^{2k-2}\Delta\phi\,\langle x,D \phi\rangle\,dx +\int_{B_R}\langle D|D \phi|^{2k-2},D \phi\rangle\langle x,D \phi\rangle\,dx\\ &=-\int_{B_R}|D\phi|^{2k}\,dx -\int_{B_R}|D\phi|^{2k-2}\langle x,D^2\phi D\phi\rangle\, dx +\frac1R\int_{\partial B_R}|D\phi|^{2k-2}\langle x,D\phi\rangle^2\, d\sigma\\ &=\frac {n-2k}{2k}\int_{B_R}|D\phi|^{2k}dx -\frac R{2k}\int_{\partial B_R}|D\phi|^{2k}d\sigma +\frac1R\int_{\partial B_R}|D\phi|^{2k-2}\langle x,D\phi\rangle^2\, d\sigma. \end{align*} \end{proof} Set \Omega=m^2-w^2 $. From the above Lemma we have the following result. \begin{lemma} If$(u,\phi)$is a solution of the system \eqref{Pn}, then$(u,\phi)satisfies the Poho\v{z}aev type identity: \label{Pohozaev} \begin{aligned} &\int_{\mathbb{R}^3}|Du|^2\,dx+3\int_{\mathbb{R}^3}u^2\,dx +\frac1{4\pi}\sum_{k=2}^N\Big(\beta_k\frac {3(k-1)}{k}\int_{\mathbb{R}^3} |D\phi|^{2k}dx\Big)\\ &\quad -2\int_{\mathbb{R}^3}\phi^2 u^2\,dx+5\int_{\mathbb{R}^3}\omega\phi u^2\,dx -\frac6q\int_{\mathbb{R}^3}|u|^q\,dx=0. \end{aligned} \end{lemma} \begin{proof} Multiplying the first formula of \eqref{Pn} by\langle x, Du\rangle$, integrating on$B_Rand using the above Lemma, we conclude that \label{Pohoproof1} \begin{aligned} &-\frac12\int_{B_R}|Du|^2\,dx-\frac32\Omega\int_{B_R}u^2\,dx\\ &+\int_{B_R}(\phi-\omega)u^2\langle x,D\phi\rangle\,dx +\frac32\int_{B_R}(\phi-2\omega)\phi u^2\,dx+\frac3q\int_{B_R}|u|^q\,dx\\ &=\frac1R \int_{\partial B_R}\langle x,Du\rangle^2d\sigma -\frac R2\int_{\partial B_R}|Du|^2d\sigma\\ &\quad-\frac {\Omega R}2\int_{\partial B_R}u^2d\sigma +\frac{R}2\int_{B_R}(\phi-2\omega)\phi u^2\,d\sigma +\frac R{q}\int_{\partial B_R}|u|^q\,dx. \end{aligned} Multiplying the second formula of \eqref{Pn} by\langle x,D\phi\rangle$, integrating on$B_Rand using the above Lemma, we obtain \label{Pohoproof2} \begin{aligned} &4\pi\int_{B_R}(\phi-\omega)u^2\langle x,D\phi\rangle\,dx\\ &=\int_{B_R}\sum_{k=1}^N(\beta_k\Delta_{2k} \phi)\langle x,D\phi\rangle\,dx\\ &=\sum_{k=1}^N\beta_k\int_{B_R}\Delta_{2k} \phi\langle x,D\phi\rangle\,dx\\ &=\sum_{k=1}^N\beta_k\Big(\frac {3-2k}{2k}\int_{B_R}|D\phi|^{2k}dx -\frac R{2k}\int_{\partial B_R}|D\phi|^{2k}d\sigma\\ &\quad +\frac1R\int_{\partial B_R}|D\phi|^{2k-2}\langle x,D\phi\rangle^2\, d\sigma\Big). \end{aligned} By \eqref{Pohoproof1}, \eqref{Pohoproof2} and the proof of \cite[Theorem 1.1, pp. 316-317]{DM04-1}, we deduce the equality \begin{align*} &-\frac12\int_{\mathbb{R}^3}|Du|^2\,dx -\frac{3}2\Omega\int_{\mathbb{R}^3}u^2\,dx +\frac1{4\pi}\sum_{k=1}^N\Big(\beta_k\frac {3-2k}{2k} \int_{\mathbb{R}^3}|D\phi|^{2k}dx\Big)\\ &+\frac32\int_{\mathbb{R}^3}(\phi-2\omega)\phi u^2\,dx +\frac3q\int_{\mathbb{R}^3}|u|^q\,dx=0. \end{align*} Then, noting \eqref{Pn}, we have \begin{align*} &\int_{\mathbb{R}^3}|Du|^2\,dx+3\Omega\int_{\mathbb{R}^3}u^2\,dx +\frac1{2\pi}\sum_{k=2}^N\Big(\beta_k\frac {3(k-1)}{2k} \int_{\mathbb{R}^3}|D\phi|^{2k}dx\Big)\\ &- 2\int_{\mathbb{R}^3}\phi^2 u^2\,dx+5\omega\int_{\mathbb{R}^3}\phi u^2\,dx -\frac6q\int_{\mathbb{R}^3}|u|^q\,dx=0. \end{align*} \end{proof} \section{Proof of the main theorem} First, we give a abstract result which is due to Jeanjean \cite{Jeanjean99}. \begin{proposition}\label{Jean-Prop} Let(X,\|\cdot\|)$be a Banach space and let$I\subset \mathbb{R}^+$be an interval. Consider the family of$C^1$functionals on$X$$$\Psi_\lambda(u)=A(u)-\lambda B(u),\quad \forall\lambda\in I,$$ with$B(u)$nonnegative and either$A(u)\to+\infty$or$B(u)\to+\infty$, as$\|u\|\to\infty$and such that$\Psi_\lambda(0)=0$. For any$\lambda\in I$we set $$\Gamma_\lambda=\{\gamma\in C([0,1],X):\gamma(0)=0,\Psi_\lambda(\gamma(1))\leq0\}.$$ If for every$\lambda\in I$the set$\Gamma_\lambda$is nonempty and $$c_\lambda:=\inf_{\gamma\in\Gamma_\lambda}\max_{t\in[0,1]}\Psi_\lambda(\gamma(t))>0,$$ then for almost every$\lambda\in I$there is a sequence$\{(u_\lambda)_n\}\subset X$such that \begin{itemize} \item[(i)]$\{(u_\lambda)_n\}$is bounded in$X$; \item[(ii)]$\Psi_\lambda((u_\lambda)_n)\to c_\lambda$; \item[(iii)]$\Psi_\lambda'((u_\lambda)_n)\to 0$in the dual$X^*$of$X$. \end{itemize} \end{proposition} \subsection*{Proof Theorem \ref{main2}} Denote $$M(\phi):=\frac1{4\pi}\sum_{k=2}^N\Big(\beta_k\frac {k-1}{k}\int_{\mathbb{R}^3}|D\phi|^{2k}\Big)dx.$$ Then, noting the definition of$\Phi(u)$we can write \eqref{Pohozaev} and$J(u)by: \begin{align*} &\int_{\mathbb{R}^3}|Du|^2\,dx+3\Omega\int_{\mathbb{R}^3}u^2\,dx+3M(\Phi(u))- 2\int_{\mathbb{R}^3}\Phi^2(u) u^2\,dx\\ &+5\omega\int_{\mathbb{R}^3}\Phi(u) u^2\,dx-\frac6q\int_{\mathbb{R}^3}|u|^q\,dx=0 \end{align*} and \begin{align*} J_N(u)&=\frac12\int_{\mathbb{R}^3}|D u|^2\,dx+\frac12\Omega\int_{\mathbb{R}^3} u^2\,dx+\frac \omega2\int_{\mathbb{R}^3}\Phi(u)u^2\,dx\\ &\quad +\frac12M(\Phi(u))- \frac1q\int_{\mathbb{R}^3} |u|^{q}\,dx, \end{align*} respectively. For\lambda\in [\frac12,1]$, we define the family of functionals$J_{N,\lambda}:H_r^1(\mathbb{R}^3)\to \mathbb{R}by \begin{align*} J_{N,\lambda}(u) &=\frac12\int_{\mathbb{R}^3}|D u|^2\,dx+\frac12\Omega\int_{\mathbb{R}^3} u^2\,dx +\frac \omega2\int_{\mathbb{R}^3}\Phi(u)u^2\,dx\\ &\quad +\frac12M(\Phi(u))- \frac\lambda q\int_{\mathbb{R}^3} |u|^{q}\,dx \end{align*} Using a slightly modified version of \cite[Lemmas 2.3 and 2.4]{APP09}, it can be proved that: for every\lambda\in[\frac12,1]$, there exist$\alpha_\lambda, \rho_\lambda>0$and$\nu_\lambda\in H_r^1(\mathbb{R}^3)$such that \begin{itemize} \item[(i)]$\inf_{\|u\|=\rho_\lambda}J_{N,\lambda}(u)>\alpha_\lambda$. \item[(ii)]$\|\nu_\lambda\|>\rho_\lambda$and$J_{N,\lambda}(\nu_\lambda)<0$. \end{itemize} Thus$J_{N,\lambda}$has the mountain pass geometry. So we can define the Mountain Pass level$c_\lambda$by $$c_\lambda:=\inf_{\gamma\in\Gamma_\lambda} \max_{0\leq t\leq1}J_{N,\lambda}(\gamma(t)),$$ where $$\Gamma_\lambda=\{\gamma\in C([0,1],H_r^1(\mathbb{R}^3)):\gamma(0)=0,\gamma(1)=\nu_\lambda\}.$$ Set$X=H_r^1(\mathbb{R}^3), I=[\frac12,1], \Psi_\lambda=J_{N,\lambda}$, $$A(u)=\frac12\int_{\mathbb{R}^3}|D u|^2\,dx+\frac12\Omega\int_{\mathbb{R}^3} u^2\,dx +\frac \omega2\int_{\mathbb{R}^3}\Phi(u)u^2\,dx+\frac12M(\Phi(u))$$ and $$B(u)=\frac1 q\int_{\mathbb{R}^3} |u|^{q} \,dx.$$ It is easy to see that$B(u)\geq0$for all$u\in H_r^1(\mathbb{R}^3)$and$A(u)\to +\infty$as$\|u\|\to\infty$. Thus, by Proposition~\ref{Jean-Prop}, for almost every$\lambda\in I$there is a sequence$\{(u_\lambda)_n\}\subset X$such that \begin{itemize} \item[(i)]$\{(u_\lambda)_n\}$is bounded in$H_r^1(\mathbb{R}^3)$; \item[(ii)]$J_{N,\lambda}((u_\lambda)_n)\to c_\lambda$; \item[(iii)]$J_{N,\lambda}'((u_\lambda)_n)\to 0$in the dual$(H_r^1(\mathbb{R}^3))^*$of$H_r^1(\mathbb{R}^3)$. \end{itemize} There exists$u_\lambda\in H_r^1(\mathbb{R}^3)$such that $$J_\lambda'(u_\lambda)=0,\ \ \ J_\lambda(u_\lambda)=c_\lambda,$$ for almost every$\lambda\in I$. Now we can choose a suitable$\lambda_n\to1$and$u_{\lambda_n}$such that $$J_{\lambda_n}'(u_{\lambda_n})=0,\quad J_{\lambda_n}(u_{\lambda_n})=c_{\lambda_n}\to c_1,$$ For simplicity we denoted$u_{\lambda_n}$by$u_n$. Since$J_{\lambda_n}'(u_n)=0$,$u_nsatisfies the Poho\v{z}aev equality \label{Pohozaev2} \begin{aligned} &\int_{\mathbb{R}^3}|Du_n|^2\,dx+3\Omega\int_{\mathbb{R}^3}u_n^2\,dx+3M(\Phi(u_n))- 2\int_{\mathbb{R}^3}\Phi^2(u_n) u_n^2\,dx\\ &+5\omega\int_{\mathbb{R}^3}\Phi(u_n) u_n^2\,dx-\frac{6\lambda_n}q\int_{\mathbb{R}^3}|u_n|^q\,dx=0. \end{aligned} ByJ_{\lambda_n}'(u_n)=0$and$J_{\lambda_n}(u_n)=c_{\lambda_n}\to c_1, we have \label{Nehari2} \begin{aligned} &\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx+\Omega\int_{\mathbb{R}^3} u_n^2\,dx +2\omega\int_{\mathbb{R}^3}\Phi(u_n)u^2_n\,dx\\ &-\int_{\mathbb{R}^3} \Phi^2(u_n)u_n^2\,dx- \lambda_n\int_{\mathbb{R}^3} |u_n|^{q}\,dx=0 \end{aligned} and, fornlarge enough, \begin{align*} &\frac12\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx+\frac12\Omega\int_{\mathbb{R}^3} u_n^2\,dx+\frac12M(\Phi(u_n))\\ &+\frac\omega2\int_{\mathbb{R}^3} \Phi(u_n)u_n^2\,dx- \frac{\lambda_n} q\int_{\mathbb{R}^3}|u_n|^{q}\,dx\leq c_1+1. \end{align*} Set\alpha$and$\beta$two real number (which we will estimate later). Then from$\alpha\times\eqref{Pohozaev2}+\beta\times\eqref{Nehari2}, we obtain \begin{align*} &\frac{\lambda_n}q\int_{\mathbb{R}^3} |u_n|^{q}\,dx\\ &=\frac 1{6\alpha+q\beta}\Big\{(\alpha+\beta)\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx+(3\alpha+\beta)\Omega\int_{\mathbb{R}^3} u_n^2\,dx+3\alpha M(\Phi(u_n))\\ &\quad +(5\alpha+2\beta)\int_{\mathbb{R}^3} \omega\Phi_{u_n}u_n^2\,dx -(2\alpha+\beta) \int_{\mathbb{R}^3} \Phi_{u_n}^2u_n^2\,dx\Big\}. \end{align*} Thus \begin{align*} &c_1+1\geq J_{\lambda_n}(u_n)\\ &=\frac12\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx+\frac12\Omega\int_{\mathbb{R}^3} u_n^2\,dx\\ &\quad +\frac12M(\Phi(u_n))+\frac12\int_{\mathbb{R}^3} \omega\phi_{u_n}u_n^2\,dx- \frac{\lambda_n}q\int_{\mathbb{R}^3} |u_n|^{q}\,dx\\ &=\Big(\frac12-\frac {\alpha+\beta}{6\alpha+q\beta}\Big)\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx +\Big(\frac12-\frac {3\alpha+\beta}{6\alpha+q\beta}\Big)\Omega\int_{\mathbb{R}^3} u_n^2\,dx\\ &\quad+\Big(\frac12-\frac {5\alpha+2\beta}{6\alpha+q\beta}\Big) \int_{\mathbb{R}^3} \omega\phi_{u_n}u_n^2\,dx +\Big(\frac12-\frac {3\alpha}{6\alpha+q\beta}\Big)M(\Phi(u_n))\\ &\quad +\frac{2\alpha+\beta}{6\alpha+q\beta}\int_{\mathbb{R}^3} \Phi^2(u_n)u_n^2\,dx\\ &=\Big(\frac12-\frac {\tau+1}{6\tau+q}\Big)\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx\\ &\quad +\Big(\frac12-\frac {3\tau}{6\tau+q}\Big)M(\Phi(u_n)) +\frac{2\tau+1}{6\tau+q}\int_{\mathbb{R}^3}\Phi^2(u_n)u_n^2\,dx\\ &\quad +\Big(\frac12-\frac {3\tau+1}{6\tau+q}\Big)\Omega\int_{\mathbb{R}^3} u_n^2\,dx +\Big(\frac12-\frac {5\tau+2}{6\tau+q}\Big)\int_{\mathbb{R}^3}\omega\Phi(u_n)u_n^2\,dx, \end{align*} where\tau=\frac{\displaystyle\alpha}{\displaystyle\beta}$. Under one of the following conditions: \begin{itemize} \item[(i)]$ q\in(4,6)$,$\tau\in((2-q)/4,-1/2)$and$m>\omega>0$; \item[(ii)]$q\in(3,4]$,$\tau\in((2-q)/4,(q-4)/4)$and$ m>\omega>0$; \item[(iii)]$q\in(2,3]$,$\tau\in((2-q)/4,+\infty)$and$m\sqrt{(q-2)(4-q)}>\omega>0$, \end{itemize} we conclude that $$\frac12-\frac {\tau+1}{6\tau+q}>0, \quad \frac12-\frac {3\tau}{6\tau+q}>0$$ and $$\frac{2\tau+1}{6\tau+q}t^2+\Big(\frac12-\frac {5\tau+2}{6\tau+q}\Big)\omega t +\Big(\frac12-\frac {3\tau+1}{6\tau+q}\Big)\Omega\geq0,\quad \text{for } t\in[0,\omega].$$ So we obtain that$\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx$is bounded for all$n$. Then, as in \cite[Proof of Teorem 1.1, pp. 9]{APP09} we have$\{u_n\}$is bounded in$H^1_r(\mathbb{R}^3)$. Thus$\{u_{_n}\}$is a bounded$(PS)_{c_1}$-sequence for$J_N$. So$J_N$has a nontrivial critical point$u_N$. \begin{thebibliography}{00} \bibitem{Adams75} R. A. Adams; \emph{Sobolev spaces}, Academic Press, New York, 1975. \bibitem{APP09} A. Azzollini, L. Pisani, A. 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