\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 268, pp. 1--18.\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-2017/268\hfil Nonlinear Choquard equations] {Multiple nodal solutions of nonlinear \\ Choquard equations} \author[Z. Huang, J. Yang, W. Yu \hfil EJDE-2017/268\hfilneg] {Zhihua Huang, Jianfu Yang, Weilin Yu} \address{Zhihua Huang \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{zhhuang2016@126.com} \address{Jianfu Yang \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{jfyang\_2000@yahoo.com} \address{Weilin Yu \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{williamyu2065@163.com} \dedicatory{Communicated by Claudianor O. Alves} \thanks{Submitted July 15, 2017. Published October 27, 2017.} \subjclass{35J61, 35B33, 35B38, 35B65} \keywords{Nonlinear Choquard equations; nodal solutions; nonlocal term} \begin{abstract} In this article, we consider the existence of multiple nodal solutions of the nonlinear Choquard equation \begin{gather*} -\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \quad \text{in }\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{gather*} where $p\in (5/2,5)$. We show that for any positive integer $k$, the above problem has at least one radially symmetrical solution changing sign exactly $k$-times. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we consider the existence of multiple nodal solutions for the nonlinear Choquard equation \begin{equation} \label{eP} \begin{gathered} -\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3) \end{gathered} \end{equation} where $p\in (5/2,5)$. In the case $p=2$, equation \eqref{eP} is the Choquard-Pekar equation introduced by Pekar in \cite{P}, see also Section 2.1 in \cite{DA}, to describe the quantum theory of a polaron at rest and proposed by Choquard \cite{L} in the study of a certain approximation to Hartree-Fock theory for one component plasma. Further physical consideration of \eqref{eP}, known as the Schr\"{o}dinger-Poisson equation, can be found in \cite{J, MPT} as a model of self-gravitating matter and in \cite{Len} as a non-relativistic model of boson stars. In the 1980's, the nonlinear Choquard equation \eqref{eP} was studied in \cite{L, Lions, Lions1, M} by the variational method, and recently, this problem and its generalization have attractive the attention of many researches. Existence and qualitative properties of solutions have been investigated in \cite{CCS, CS, CSS, GS, MS,MS1,MS2} and references therein. In particular, the existence of nodal solutions for the Choquard equation was investigated in \cite{CCS1,CSS,ClapS, GS}, by the variational method, that is, by seeking for critical points of an associated functional. The energy functional associated with the Choquard equation \eqref{eP} is defined for each $u$ in $H^1(\mathbb{R}^3)$ by \begin{equation}\label{eq:1.1} I(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+|u|^2)dx -\frac{1}{2p}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)|^p|u(y)|^p}{|x-y|} \,dx\,dy. \end{equation} By the Hardy-Littlewood-Sobolev inequality, the functional $I$ is well defined on $H^1(\mathbb{R}^3)$ if $p\in (\frac{5}{2},5)$. Hence, critical points of $I(u)$ are weak solutions of problem \eqref{eP}, and necessarily contained in the Nehari manifold $\mathcal{N}=\{u\in H^1(\mathbb{R}^3): u\neq0,\langle I'(u),u\rangle=0\}.$ A standard way to find critical points of $I$ is to seek for minimizers of the functional $I$ constraint to the Nehari manifold $\mathcal{N}$. This idea was used in \cite{GS} in constructing a sign-changing solution for the Choquard equation in an odd Nehari manifold. Another way to construct a nodal solution is to find a critical point of $I$ in the Nehari set $\mathcal{N}_0=\{u\in H^1(\mathbb{R}^3): u^{\pm}\neq0,\langle I'(u),u^{\pm}\rangle=0\}.$ However, $\mathcal{N}_0$ is not a manifold. The argument then among other things, lies in showing that there is a minimizer of $I$ constraint on $\mathcal{N}_0$, and verifying that the minimizer is a critical point of $I$. Using this approach, a sign-changing solution is constructed in \cite{GS} for the Choquard equation, and in \cite{AS,WZ} for the nonlinear Schr\"{o}dinger-Poisson system and in \cite{BS, FN} for the Kirchhoff equation, further results can found in references therein. In this paper, we intend to show that for every fixed integer $k$, there exists a radial solution of problem \eqref{eP} which changes sign exactly $k$ times. Particularly, for $k=2$, there is a radially sign changing solution of problem \eqref{eP}. For every integer $k\geq0$, it was proved in \cite{BW} and \cite{CZ} independently that, there is a pair of solutions $u^\pm_k$ having exact $k$ nodes of \begin{equation}\label{eq:1.2} \begin{gathered} -\Delta u+V(|x|)u=f(|x|,u)\quad\text{in }\mathbb{R}^N,\\ u\in H^1(\mathbb{R}^N). \end{gathered} \end{equation} Such solutions of \eqref{eq:1.2} are obtained by gluing solutions of the equation in each annulus, including every ball and the complement of it. However, this approach cannot be applied directly to problems with nonlocal terms, because nonlocal terms need the global information of $u$. This difficulty was overcome by regarding the problem as a system of $k+1$ equations with $k+1$ unknown functions $u_i$, each $u_i$ is supported on only one annulus and vanishes at the complement of it. This argument relies on, among other things, constructing a functional $E_k$ and a Nehari type manifold $\mathcal{N}_k$, then finding a minimizer of $E_k$ constraint on $\mathcal{N}_k$. In this way, Kim and Seok \cite{KS} found infinitely many nodal solutions for Schr\"{o}dinger-Poisson system, and then Deng et at \cite{DPS} treated Kirchhoff problems in $\mathbb{R}^3$ in a similar way. However, this argument can not be simply carried out to deal with the Choquard equation \eqref{eP}, because in the proof of $\mathcal{N}_k$ being a manifold for problems considered in \cite{DPS} and \cite{KS}, a key ingredient used is that the related matrix is diagonally dominant at each point of $\mathcal{N}_k$, but this is not the case for the Choquard equation \eqref{eP}. In this paper, we find a way to show that the matrix associated to our Nehari type set $\mathcal{N}_k$ is nonsingular, the fact eventually allows us to verify that $\mathcal{N}_k$ is a manifold. This method might be possible to apply to analogous problems. Our main result in this paper is stated as follows. \begin{theorem}\label{thm:1.1} Suppose $5/2 0})^{k+1}$ if and only if \begin{equation}\label{eq:2.3} \begin{aligned} &t_i^2\|u_{i}\|_{i}^{2} -t_i^{2p}\int_{B_i}\int_{B_i} \frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy\\ &-\sum_{j\neq i}^{k+1}\int_{B_i}\int_{B_j}\frac{t_i^pt_j^p|u_i(x)|^p|u_j(y)|^p}{|x-y|} \,dx\,dy=0 \end{aligned} \end{equation} for $i=1,\dots,k+1$. Hence, the problem is reduced to verify that there is only one solution $(t_{1},\dots,t_{k+1})$ of system \eqref{eq:2.3} with $t_i>0$, for each $i=1,\dots,k+1$. To this end, we introduce a parameter $0\leq\mu\leq1$, and consider the solvability of the following system of $(k+1)$ equations \begin{equation}\label{eq:2.4} \begin{aligned} G_i(t_1,\dots,t_{k+1}) :=&t_i^2\|u_{i}\|_{i}^{2}-t_i^{2p}\int_{B_i}\int_{B_i} \frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy\\ &-\mu\sum_{j\neq i}^{k+1}\int_{B_i}\int_{B_j} \frac{t_i^pt_j^p|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy=0, \end{aligned} \end{equation} for $i=1,\dots,k+1$. Let \begin{equation}\label{eq:2.5} \mathcal{Z}=\big\{\mu: 0\leq\mu\leq1 \text{ and \eqref{eq:2.4} is uniquely solvable in } (\mathbb{R}_{>0})^{k+1}\big\}. \end{equation} Apparently, $0\in \mathcal{Z}$, so the set $\mathcal{Z}$ is nonempty in $[0,1]$. We claim that $\mathcal{Z}= [0,1]$, which implies the result. To prove the claim, it is sufficient to show that $\mathcal{Z}$ is both open and closed in $[0,1]$. We first prove that the set $\mathcal{Z}$ is open in $[0,1]$. Suppose that $\mu_0\in\mathcal{Z}$ and $(\bar{t}_{1},\dots,\bar{t}_{k+1})\in(\mathbb{R}_{>0})^{k+1}$ is the unique solution of \eqref{eq:2.4} with $\mu=\mu_0$. To apply the implicit function theorem at $\mu_0$, we calculate the matrix \begin{equation}\label{eq:2.5a} M=(M_{ij})=(\partial_{t_j}G_i)_{i,j=1,\dots,k+1}. \end{equation} Each component of the matrix $M$ is then given by \begin{align*} M_{ii}&=2\bar{t}_i\|u_{i}\|_{i}^{2}-2p\bar{t}^{2p-1}_i \int_{B_i}\int_{B_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy\\ &\quad-\mu_0p\bar{t}^{p-1}_i\sum_{j\neq i}^{k+1}\int_{B_i} \int_{B_j}\frac{\bar{t}_j^p|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy\\ &=(2-p)\bar{t}_i\|u_{i}\|_{i}^{2}-p\bar{t}^{2p-1}_i\int_{B_i} \int_{B_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy \end{align*} for $i=1,\dots,k+1$, where we have used \eqref{eq:2.4}, and $M_{ij}=-\mu_0p\bar{t}^{p}_i\bar{t}^{p-1}_j\int_{B_i} \int_{B_j}\frac{|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy$ for $i\neq j$, $i,j=1,\dots,k+1$. Therefore, \begin{equation}\label{eq:2.7} \det M=\frac{(-1)^{k+1}}{\bar{t}_1\dots \bar{t}_{k+1}} \det \widetilde{M}, \end{equation} where components of the matrix $\widetilde{M}=(\widetilde{M}_{ij})$ are given by $\widetilde M_{ii}=(p-2)\bar{t}_i^2\|u_{i}\|_{i}^{2}+p\bar{t}^{2p}_i \int_{B_i}\int_{B_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy$ for $i=1,\dots,k+1$, and $\widetilde M_{ij}=\mu_0p\bar{t}^{p}_i\bar{t}^{p}_j \int_{B_i}\int_{B_j}\frac{|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy,\quad \text{for } i\neq j,\ i,j=1,\dots,k+1.$ By Lemma \ref{lem:A} in the appendix, we obtain $\det M\neq0$. Hence, the implicit function theorem implies that there are an open neighborhood $U_0$ of $\mu_0$ and a neighborhood $A_0 \subset (\mathbb{R}_{>0})^{k+1}$ of $(\bar{t}_{1},\dots,\bar{t}_{k+1})$ such that system \eqref{eq:2.4} is uniquely solvable in $U_0\times A_0$. Now we show \eqref{eq:2.4} is uniquely solvable in $U_0\times (\mathbb{R}_{>0})^{k+1}$, this means $U_0\subset \mathcal{Z}$, and $\mathcal{Z}$ is open. Suppose, on the contrary, that there is $\mu_1 \in U_0$ such that there exists the second solution $(\tilde{t}_{1},\dots,\tilde{t}_{k+1}) \in (\mathbb{R}_{>0})^{k+1}\setminus A_0$ of \eqref{eq:2.4}. By the implicit function theorem, we can find a solution curve $(\mu,(\tilde{t}_{1}(\mu),\dots,\tilde{t}_{k+1}(\mu)))$ in $(\mu_1-\varepsilon,\mu_1+\varepsilon)\times \big((\mathbb{R}_{>0})^{k+1}\setminus A_0\big)$. If $\mu_0<\mu_1$, we extend this curve as much as possible. Since it cannot be defined at $\mu_0$ and enter into $U_0\times A_0$, there should have a point $\mu_2\in [\mu_0,\mu_1)$ such that $(t_1(\mu),\dots,t_{k+1}(\mu))$ being defined in $(\mu_2,\mu_1]$ and blowing up as $\mu\to \mu_2^+$. However, this is impossible, since if $(t_1,\dots,t_{k+1})$ has sufficiently large norm, the left-hand side of \eqref{eq:2.4} is strictly negative for at least one $i$. This gives a contradiction. Thus, $U_0\subset \mathcal{Z}$. The case $\mu_0 >\mu_1$ can be proved in the same way. Next, we show that the set $\mathcal{Z}$ is closed in $[0,1]$. Let $\{\mu_n\}$ be a sequence in $\mathcal{Z}$ converging to $\mu_0\in[0,1]$ and $(t_{1}^n,\dots,t_{k+1}^n)\in(\mathbb{R}_{>0})^{k+1}$ be the solution of \eqref{eq:2.4} for $\mu_n$. By the preceding argument, we see that the sequence $(t_{1}^n,\dots,t_{k+1}^n)$ is bounded above. Thus we may assume that $(t_{1}^n,\dots,t_{k+1}^n)$ converges to a solution $(t_{1}^0,\dots,t_{k+1}^0)\in(\mathbb{R}_{\geq0})^{k+1}$ of \eqref{eq:2.4} for $\mu_0$. Let $v^n=t_1^n u_1+\dots+t^n_{k+1}u_{k+1}$. Since $\{v_n\}$ is uniformly bounded in $\mathcal{H}_k$, by \eqref{eq:2.4} and the Hardy-Littlewood-Sobolev inequality, we derive \begin{equation}\label{eq:2.8} \begin{aligned} (t_i^n)^2\|u_{i}\|_{i}^{2} &=(t_i^n)^{2p}\int_{B_i}\int_{B_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy\\ &\quad+\mu_n\sum_{j\neq i}^{k+1}\int_{B_i}\int_{B_j}\frac{(t_i^n)^p(t_j^n)^p|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy\\ &\leq(t_i^n)^{2p}\int_{B_i}\int_{B_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|}\,dx\,dy\\ &\quad+\sum_{j\neq i}^{k+1}\int_{B_i}\int_{B_j}\frac{(t_i^n)^p(t_j^n)^p|u_i(x)|^p|u_j(y)|^p}{|x-y|}\,dx\,dy\\ &=\int_{B_i}\int_{\mathbb{R}^3}\frac{|t^n_iu_i(x)|^p|v^n(y)|^p}{|x-y|}\,dx\,dy\\ &\leq C_1(t_i^n)^p\|u_{i}\|_{\frac{6p}{5}}^{p}\|v^n\|_{\frac{6p}{5}}^{p}\leq C_2(t_i^n)^p\|u_{i}\|_{i}^{p}. \end{aligned} \end{equation} This implies that $00$ for $i=1,\dots,k+1$, that is, $(t_{1}^0,\dots,t_{k+1}^0)\in(\mathbb{R}_{>0})^{k+1}$. By the implicit function theorem again, $(t_{1}^0,\dots,t_{k+1}^0)$ is the unique solution of \eqref{eq:2.4} in $(\mathbb{R}_{>0})^{k+1}$. Hence, $\mathcal{Z}$ is closed. The conclusion of Lemma \ref{lem:2.1} then follows. \end{proof} \begin{lemma}\label{lem:2.2} For any $5/20})^{k+1}\to \mathbb{R}$ defined as $$\phi(c_1,\dots,c_{k+1})=E(c_{1}u_{1},\dots,c_{k+1}u_{k+1}).$$ \end{lemma} \begin{proof} By Lemma \ref{lem:2.1}, we know that $(t_{1},\dots,t_{k+1})$ is the unique critical point of $\phi$ in $(\mathbb{R}_{>0})^{k+1}$. Since $p\in(\frac{5}{2},5)$, it is observed that $\phi(c_{1},\dots,c_{k+1})\to-\infty$ uniformly as $|(c_{1},\dots,c_{k+1})|\to+\infty$, so it is sufficient to check that a maximum point cannot be achieved on the boundary of $(\mathbb{R}_{>0})^{k+1}$. Choose $(c^0_{1},\dots,c^0_{k+1})\in\partial(\mathbb{R}_{>0})^{k+1}$, without loss of generality, we may assume that $c_1^0=0$. Since \begin{align*} \phi(t,c^0_{2},\dots,c^0_{k+1})&=E(tu_1,c^0_{2}u_2,\dots,c^0_{k+1}u_{k+1})\\ &=\frac{t^2}{2}\|u_1\|_1^2-\frac{t^{2p}}{2p}\int_{B_1}\int_{B_1} \frac{|u_1(x)|^p|u_1(y)|^p}{|x-y|}\,dx\,dy\\ &\quad-\frac{t^{p}}{p}\sum_{i=2}^{k+1}\int_{B_1} \int_{B_i}\frac{|u_1(x)|^p|c_i^0u_i(y)|^p}{|x-y|}\,dx\,dy +\frac{1}{2}\sum_{i=2}^{k+1}\|c_i^0u_i\|_i^2\\ &\quad-\frac{1}{2p}\sum_{i,j=2}^{k+1}\int_{B_i} \int_{B_j}\frac{|c_i^0u_i(x)|^p|c_i^0u_j(y)|^p}{|x-y|}\,dx\,dy \end{align*} is increasing with respect to $t$ if $t$ is small enough, $(0,c^0_{2},\dots,c^0_{k+1})$ is not a maximum point of $\phi$ in $(\mathbb{R}_{>0})^{k+1}$. The assertion follows. \end{proof} Finally, we have the following existence result for problem \eqref{ePi}. \begin{lemma}\label{lem:2.4} For any $5/20$ such that $\|u_i\|_i\geq\alpha_i>0$, $i=1,\dots,k+1$. If $(u_1,\dots,u_{k+1})\in \mathcal{N}_k$, there holds \begin{equation}\label{eq:2.12} E(u_1,\dots,u_{k+1}) =(\frac{1}{2}-\frac{1}{2p})\sum_{i=1}^{k+1}\|u_i\|_i^2\geq\alpha>0 \end{equation} for some $\alpha>0$. This implies that any minimizing sequence $\{(u_{1}^{n},\dots,u_{k+1}^{n})\}$ of $E\big|_{\mathcal{N}_k}$ is bounded in $\mathcal{H}_k$. We assume that the minimizing sequence $(u_{1}^{n},\dots,u_{k+1}^{n})$ weakly converges to an element $(u_{1}^{0},\dots,u_{k+1}^{0})$ in $\mathcal{H}_k$. We claim that $u_{i}^{0}\neq0$ for each $i=1,\dots,k+1$. Indeed, if $(u_{1}^{n},\dots,u_{k+1}^{n})$ strongly converges to $(u_{1}^{0},\dots,u_{k+1}^{0})$ in $\mathcal{H}_k$, we may show in the same way as the proof of \eqref{eq:2.8} that $\|u_i^n\|_i^2\leq C\|u_i^n\|_i^p$ for each $i$, In other words, $\|u_i^n\|_i\geq\mu_i>0$, thereby $\|u_i^0\|_i\geq\mu_i^0>0$ for $i=1,\dots,k+1$. Suppose now that $(u_{1}^{n},\dots,u_{k+1}^{n})\not\to (u_{1}^{0},\dots,u_{k+1}^{0})$ strongly in $\mathcal{H}_k$ as $n\to\infty$. That is, $\|u_i^0\|_i<\liminf_{n\to\infty}\|u_i^n\|_i$ for at least one $i\in\{1,\dots,k+1\}$. Again, we have $u_{i}^{0}\neq0$ for each $i=1,\dots,k+1$. Indeed, since $(u_{1}^{n},\dots,u_{k+1}^{n})\in\mathcal{N}_k$, $\|u_i^n\|_i^2=\int_{\mathbb{R}^3} \int_{B_i}\frac{|u^n(x)|^p|u_i^n(y)|^p}{|x-y|}\,dx\,dy$ and the inclusion $H^1_r(\mathbb{R}^3)\hookrightarrow L^q(\mathbb{R}^3)$ is compact for $20$ such that $\|u_i^0\|_i\geq\mu_0>0$. Since each component of $(u_{1}^{0},\dots,u_{k+1}^{0})$ is nonzero, by Lemma \ref{lem:2.1}, one can find $(t_{1}^{0},\dots,t_{k+1}^{0})\in(\mathbb{R}_{>0})^{k+1}$ and $(t_{1}^{0},\dots,t_{k+1}^{0})\neq(1,\dots,1)$ such that \\ $(t_{1}^{0}u_{1}^{0},\dots,t_{k+1}^{0}u_{k+1}^{0})\in\mathcal{N}_k$. But, in this case, by \eqref{eq:2.12a} and Lemma \ref{lem:2.3} we derive that \begin{align*} &\inf_{(u_1,\dots,u_{k+1})\in \mathcal{N}_k}E(u_1,\dots,u_{k+1})\\ &\leq E(t_1^0u_1^0,\dots,t_{k+1}^{0}u_{k+1}^{0})\\ &<\liminf_{n\to\infty}\{\frac{1}{2}\sum_{i=1}^{k+1}(t_i^0)^2\|u_i^n\|_i^2 -\frac{1}{2p}\sum_{i=1}^{k+1}(t_i^0)^{2p}\int_{B_i}\int_{B_i} \frac{|u_i^n(x)|^p|u_i^n(y)|^p}{|x-y|}\,dx\,dy\\ &\quad-\frac{1}{2p}\sum_{j\neq i}^{k+1}\int_{B_i} \int_{B_j}\frac{(t_i^0)^p(t_j^0)^p|u_i^n(x)|^p|u_j^n(y)|^p}{|x-y|}\,dx\,dy\}\\ &\leq\liminf_{n\to\infty}E(u_1^n,\dots,u_{k+1}^n)\\ &=\inf_{(u_1,\dots,u_{k+1})\in \mathcal{N}_k}E(u_1,\dots,u_{k+1}), \end{align*} which is a contradiction. Therefore, $(u_1^n,\dots,u_{k+1}^n)$ converges strongly to \\ $(u_1^0,\dots,u_{k+1}^0)$ in $\mathcal{H}_k$ and $(u_1^0,\dots,u_{k+1}^0)\in\mathcal{N}_k$ is a minimizer of $E\big|_{\mathcal{N}_k}$. Furthermore, we can check that $(w_1,\dots,w_{k+1}):=(|u_1^0|,-|u_2^0|,\dots,(-1)^k|u_{k+1}^0|)$ is also in $\mathcal{N}_k$ and is a minimizer of $E\big|_{\mathcal{N}_k}$. Hence, it is a critical point of $E\big|_{\mathcal{N}_k}$. By Lemma \ref{lem:2.2}, it is also a critical point of $E$ and satisfies \eqref{ePi}. The strong maximum principle yields that each $(-1)^{i+1}w_i$ is positive in $B_i$. The assertion follows. \end{proof} \section{Existence of sign-changing radial solutions} It is known that for any $\mathbf{r}_k=(r_1,\dots ,r_k) \in \boldsymbol{\Gamma}_k$, there is a solution $w^{\mathbf{r}_k}=(w_1^{\mathbf{r}_k},\dots, w_{k+1}^{\mathbf{r}_k})$ of \eqref{ePi} which consists of sign changing components. We will find a ${\bf \bar r}_k=(\bar r_1,\dots ,\bar r_k) \in \boldsymbol{\Gamma}_k$ such that $w^{{\bf \bar r}_k}=(w^{{\bf \bar r}_k}_1,\dots,w^{{\bf \bar r}_k}_{k+1})$ is a solution of \eqref{ePi} which is characterized as a least energy solution among all elements in $\boldsymbol{\Gamma}_k$ with nonzero components. Using this solution as a building block, we will construct a radial solution of \eqref{eP} that changes sign exactly $k$ times. Denote by $B_i^{\mathbf{r}_k}$ the nodal domain and by $E^{\mathbf{r}_k}$ the functional related to $\mathbf{r}_k$. Note that $w^{\mathbf{r}_k}_i$ is $\mathcal{C}^2(B_i^{\mathbf{r}_k})$ for each $i$ by standard elliptic regularity results. Hence, it is enough to match the first derivative with respect to the radial variable, of adjacent components $w^{\mathbf{r}_k}_i$ and $w^{\mathbf{r}_k}_{i+1}$ at the point $r_i$ to ensure the existence of a solution of equation \eqref{eP} with $k$ times sign changing. T find a least energy radial solution of \eqref{ePi} among elements in $\boldsymbol{\Gamma}_k$ with nonzero components, we need to estimate the energy of the solution $(w^{\mathbf{r}_k}_1,\dots ,w^{\mathbf{r}_k}_{k+1})$ of \eqref{ePi}. To this end, we first define the function $\psi:\mathbf{\Gamma}_k\to \mathbb{R}$ by \begin{equation}\label{eq:3.1} \begin{split} \psi(\mathbf{r}_k) &=\psi(r_1,\dots,r_k)=E^{\mathbf{r}_k}(w_1^{\mathbf{r}_k},\dots, w_{k+1}^{\mathbf{r}_k})\\ &=\inf_{(u_1^{\mathbf{r}_k},\dots,u_{k+1}^{\mathbf{r}_k})\in \mathcal{N}_k^{\mathbf{r}_k}}E^{\mathbf{r}_k}(u_1^{\mathbf{r}_k},\dots, u_{k+1}^{\mathbf{r}_k}). \end{split} \end{equation} \begin{lemma}\label{lem:3.1} Suppose $5/20$, such that $|u(x)|\leq C \frac{\|u\|}{|x|},\quad\text{a.e. in }\mathbb{R}^3,$ we deduce, as in \eqref{eq:3.2}, that \begin{align*} \|w_{k+1}^{\mathbf{r}_k}\|_{k+1}^2 &=\int_{\mathbb{R}^3}\int_{B_{k+1}^{\mathbf{r}_k}} \frac{|w^{\mathbf{r}_k}(x)|^p|w_{k+1}^{\mathbf{r}_k}(y)|^p}{|x-y|}\,dx\,dy\\ &\leq C\Big(\int_{B_{k+1}^{\mathbf{r}_k}}|w_{k+1}^{\mathbf{r}_k}(x) |^{\frac{6p}{5}}dx\Big)^{5/6}\\ &\leq C\|w_{k+1}^{\mathbf{r}_k}\|_{k+1}^p\frac{5}{6p-15} r_k^{\frac{15-6p}{5}}; \end{align*} that is, $r_k^{\frac{6p-15}{5}}\leq C\frac{5}{6p-15}\|w_{k+1}^{\mathbf{r}_k}\|_{k+1}^{p-2}.$ Since $5/20})^{k+1}$ such that $(z_1^{\bar{\mathbf{s}}},\dots,z_{k+1}^{\bar{\mathbf{s}}}) :=(\hat{t}_1\bar{z}_1,\dots,\hat{t}_{k+1}\bar{z}_{k+1}) \in \mathcal{N}_k^{\bar{\mathbf{s}}}$ with $\bar{\mathbf{s}}=(\bar{r}_1,\dots,\bar{r}_{l-1},\bar{s},\bar{r}_{l+1}, \dots,\bar{r}_k)$. On the other hand, we can verify that \begin{equation}\label{eq:3.10} (\hat{t}_1,\dots,\hat{t}_{k+1})\to(1,\dots,1) \end{equation} as $\delta\to0$. Let $W(t):=\sum_{i=1}^{k+1}w_i^{\bar{\mathbf{r}}_k}(t)\in H^1_r(\mathbb{R}^3)$ and $Z(t):=\sum_{i=1}^{k+1}z_i^{\bar{\mathbf{s}}}(t)\in H^1_r(\mathbb{R}^3)$. Then \begin{equation}\label{eq:3.11} E(W)=E^{\bar{\mathbf{r}}_k}(w_1^{\bar{\mathbf{r}}_k},\dots, w_{k+1}^{\bar{\mathbf{r}}_k}) \leq E^{\bar{\mathbf{s}}}(z_1^{\bar{\mathbf{s}}},\dots, z_{k+1}^{\bar{\mathbf{s}}})=E(Z). \end{equation} On the other hand, for any $f\in H^1_r(\mathbb{R}^3)$, the solution $\varphi$ of $-\Delta\varphi=f$ is radial and it can be expressed as $\varphi(t)=\frac{1}{t}\int_{0}^{\infty}f(s)s\min\{s,t\}\,ds$ for $t>0$. Therefore, $W$ satisfies \begin{equation}\label{eq:3.12} \int_0^\infty t^2(W'^{2}+W^2)dt =\int_0^\infty\int_0^\infty|W(s)|^p|W(t)|^p st\min\{s,t\}\,ds\,dt \end{equation} and \begin{equation}\label{eq:3.13} \begin{split} E(W)&=\frac{1}{2}\int_0^\infty (W'^{2}+W^2)t^2dt\\ &\quad-\frac{1}{2p}\int_0^\infty\int_0^\infty|W(s)|^p|W(t)|^pst \min\{s,t\}\,ds\,dt\\ &=\big(\frac{1}{2}-\frac{1}{2p}\big)\int_0^\infty \int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt. \end{split} \end{equation} We deduce from $w_-= \lim_{\delta\to 0}\frac{W(\bar r_l-\delta)-W(\bar r_l)}{-\delta}$ that \begin{equation}\label{eq:3.14} W(\bar r_l-\delta)=-\delta w_-+o(\delta). \end{equation} Since $W$ satisfies $-\big(t^2W'\big)'+t^2W=\int_0^\infty |W(s)|^pst\min\{s,t\}\,ds|W|^{p-2}W(t)$ for $\bar{r}_l-\delta\leq t\leq \bar{r}_l$, and $W(\bar{r}_l)= 0$, thereby $\big(t^2W'\big)'(\bar{r}_l)=0$, we obtain \begin{equation}\label{eq:3.15} (\bar{r}_l-\delta)^2W'(\bar{r}_l-\delta)=\bar{r}_l^2w_-+o(\delta). \end{equation} We write \begin{align*} E(Z)&=\frac{1}{2}\int_0^\infty (Z'^{2}+Z^2)t^2dt -\frac{1}{2p}\int_0^\infty\int_0^\infty|Z(s)|^p|Z(t)|^pst \min\{s,t\}\,ds\,dt\\ &=\frac{1}{2}\Big(\int_0^{\bar{r}_l-\delta} +\int_{\bar{r}_l+\delta}^\infty\Big)(Z'^{2}+Z^2)t^2dt +\frac{1}{2}\int_{\bar{r}_l-\delta}^{\bar{r}_l+\delta}(Z'^{2}+Z^2)t^2dt\\ &\quad-\frac{1}{2p}\int_0^\infty\int_0^\infty|Z(s)|^p|Z(t)|^pst\min\{s,t\}\,ds\,dt. \end{align*} By \eqref{eq:3.10}, we see that $\int_0^{\bar{r}_l-\delta}(Z'^{2}+Z^2)t^2\,dt =\int_0^{\bar{r}_l-\delta}(W'^{2}+W^2)t^2\,dt+o(\delta).$ Integrating by parts and using \eqref{eq:3.14} and \eqref{eq:3.15}, we obtain that \begin{align*} &\int_0^{\bar{r}_l-\delta}(W'^{2}+W^2)t^2\,dt+o(\delta)\\ &=W'(\bar{r}_l-\delta)W(\bar{r}_l-\delta)(\bar{r}_l-\delta)^2\\ &\quad +\int_0^{\bar{r}_l-\delta}\int_0^\infty|W(s)|^p|W(t)|^pst \min\{s,t\}\, \,ds\,dt\\ &=-\delta (w_-)^{2}\bar{r}_l^2+\int_0^{\bar{r}_l-\delta}\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt. \end{align*} Thus, \begin{equation}\label{eq:3.16} \begin{split} &\int_0^{\bar{r}_l-\delta}(Z'^{2}+Z^2)t^2\,dt\\ &=-\delta (w_-)^{2}\bar{r}_l^2+\int_0^{\bar{r}_l -\delta}\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt+o(\delta). \end{split} \end{equation} In the same way, \begin{equation}\label{eq:3.17} \begin{split} &\int_{\bar{r}_l+\delta}^\infty(Z'^{2}+Z^2)t^2\,dt\\ &=-\delta (w_+)^{2}\bar{r}_l^2+\int_{\bar{r}_l +\delta}^{\infty}\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt+o(\delta). \end{split} \end{equation} It is readily to verify that \begin{gather}\label{eq:3.18} \int_{\bar{r}_l-\delta}^{\bar{r}_l+\delta}Z'^{2}t^2\,dt =\frac{1}{2}\bar{r}_l^2(w_++w_-)^2\delta+o(\delta), \\ \label{eq:3.19} \int_{\bar{r}_l-\delta}^{\bar{r}_l+\delta}Z^2t^2\,dt=o(\delta). \end{gather} From \eqref{eq:3.16}-\eqref{eq:3.19}, we obtain \begin{equation}\label{eq:3.20} \begin{aligned} E(Z) =&-\frac{\delta}{2}(w_-)^{2}\bar{r}_l^2 +\frac{1}{2}\int_0^{\bar{r}_l-\delta}\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt\\ &-\frac{\delta}{2}(w_+)^{2}\bar{r}_l^2 +\frac{1}{2}\int_{\bar{r}_l+\delta}^{\infty}\int_0^\infty|W(s)|^p|W(t)|^p st\min\{s,t\}\,ds\,dt\\ &+\frac{\delta}{4}\bar{r}_l^2(w_++w_-)^2 -\frac{1}{2p}\int_0^\infty\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt\\ &+o(\delta). \end{aligned} \end{equation} Consequently, \begin{equation}\label{eq:3.21} \begin{aligned} &E(Z)-E(W)\\ &=-\frac{\delta}{4}\bar{r}_l^2(w_+-w_-)^2\\ &\quad+\frac{1}{2}\Big(\int_0^{\bar{r}_l-\delta}+\int_{\bar{r}_l+\delta}^\infty\Big)\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt\\ &\quad-\frac{1}{2p}\int_0^\infty\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt\\ &\quad-\big(\frac{1}{2}-\frac{1}{2p}\big)\int_0^\infty\int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt +o(\delta)\\ &=-\frac{\delta}{4}\bar{r}_l^2(w_+-w_-)^2 -\frac{1}{2}\int_{\bar{r}_l-\delta}^{\bar{r}_l+\delta} \int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt\\ &\quad +o(\delta). \end{aligned} \end{equation} This and the fact $\int_{\bar{r}_l-\delta}^{\bar{r}_l+\delta} \int_0^\infty|W(s)|^p|W(t)|^pst\min\{s,t\}\,ds\,dt=o(\delta)$ yields $E(Z)-E(W)=-\frac{\delta}{4}\bar{r}_l^2(w_+-w_-)^2+o(\delta)<0$ if $\delta>0$ sufficiently small, which contradicts \eqref{eq:3.11}. The proof is complete. \end{proof} \section{Appendix: Non-singularity of matrices} We show in this section that the matrices $M$ and $N$ defined in \eqref{eq:2.5a} and \eqref{eq:2.10} respectively are nonsingular. For $f, g\in L^1_{\rm loc}(\mathbb{R}^3)$, we recall that the Coulomb energy is defined in \cite{LL} by $D_N(f,g) = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}f(x)g(y)|x-y|^{2-N}\,dx\,dy.$ It is proved in \cite[Theorem 9.8]{LL} the following result. \begin{lemma}[{\cite[Theorem 9.8]{LL}}]\label{lem:A.1} Let $N\geq1$ and $f,g\in L^{\frac{2N}{N+2}}$, then $|D_N(f,g)|^2\leq D_N(f,f)D_N(g,g),$ with equality for $g\neq0$ if and only if $f=Cg$ for some constant $C$. \end{lemma} Denote $D(f,g)= D_3(f,g)$. Let $$\mathcal{A}(\mathbb{R}^3):=\big\{f\in L_{\rm loc}^1(\mathbb{R}^3): D(f,f)<\infty\big\}.$$ \begin{lemma}\label{lem:A.2} $\mathcal{A}(\mathbb{R}^3)$ is a linear subspace of $L_{\rm loc}^1(\mathbb{R}^3)$ with the inner product $D(f,f)$. \end{lemma} \begin{proof} By Lemma \ref{lem:A.1}, for any $f, g \in \mathcal{A}(\mathbb{R}^3)$, we have $D(f+g,f+g)\leq D(f,f)+D(g,g)+2\sqrt{D(f,f)D(g,g)}.$ It is then readily to verify that $\mathcal{A}(\mathbb{R}^3)$ is a linear subspace of $L_{\rm loc}^1(\mathbb{R}^3)$. It is also standard to see that $D(f,g)$ is an inner product in $\mathcal{A}(\mathbb{R}^3)$. \end{proof} Now, we show that the matrices $M$ and $N$ defined in \eqref{eq:2.5a} and \eqref{eq:2.10} respectively are nonsingular. We only prove the matrix $N$ is nonsingular, since for the matrix $M$, the proof is similar. \begin{lemma}\label{lem:A} The matrix $N$ defined in \eqref{eq:2.10} is nonsingular. \end{lemma} \begin{proof} Denote $v_i:=|u_i(x)|^p$. Then $v_i\in \mathcal{A}(\mathbb{R}^3)$, for $i=1,\dots,k+1$. Apparently, $v_1,\dots,v_{k+1}$ are linear independent. Let $L=\operatorname{span}\{v_1,\dots,v_{k+1}\}.$ So $L$ is a subspace of $\mathcal{A}(\mathbb{R}^3)$. Denote by $\{e_1,\dots , e_{k+1}\}$ the orthogonal basis obtained from $\{v_1,\dots,v_{k+1}\}$ by the Gram-Schmidt Orthogonalization procedure. We may assume $v_i=\Sigma^{k+1}_{j=1} a_{ij} e_j$ for $i=1,\dots ,k+1$. Then, the matrix $$A_{k+1}=\begin{pmatrix} a_{11} & a_{12} & \dots & a_{1(k+1)}\\ \vdots & \vdots & \ddots& \vdots\\ a_{(k+1)1} & a_{(k+1)2} & \dots & a_{(k+1)(k+1)} \end{pmatrix}$$ is invertible. Denote $D_{ij}= v_iv_j = D(v_i,v_j)$ for $i,j=1,\dots,k+1$. The matrix $(D_{ij})_{(k+1)\times (k+1)}$ can be written as $( D_{ij})_{(k+1)\times (k+1)} =\begin{pmatrix} v_1\\ \vdots\\ v_{k+1} \end{pmatrix} \begin{pmatrix} v_1&v_2 & \dots & v_{k+1} \end{pmatrix}.$ Using the fact that $v_i=\Sigma^{k+1}_{j=1} a_{ij} e_j$ for $i=1,\dots ,k+1$ and $(e_1,\dots, e_{k+1})$ is a orthogonal basis, we deduce \begin{align*} &\begin{pmatrix} v_1\\ \vdots\\ v_{k+1} \end{pmatrix} \begin{pmatrix} v_1&v_2 & \dots & v_{k+1} \end{pmatrix}\\ &=\begin{pmatrix} a_{11} & a_{12} & \dots & a_{1(k+1)}\\ \vdots & \vdots & \ddots & \vdots\\ a_{(k+1)1} & a_{(k+1)2} & \dots & a_{(k+1)(k+1)} \end{pmatrix}\begin{pmatrix} a_{11} & a_{21} & \dots & a_{(k+1)1}\\ \vdots & \vdots & \ddots & \vdots\\ a_{1(k+1)} & a_{2(k+1)} & \dots & a_{(k+1)(k+1)} \end{pmatrix}. \end{align*} Therefore, $(D_{ij})_{(k+1)\times (k+1)}=A_{k+1} A^T_{k+1}$ Since $A_{k+1}$ is invertible, the matrix $(D_{ij})_{(k+1)\times (k+1)}$ is positive definite. Let $d_i=\|u_i\|^2_i$, $i=1,\dots,k+1$. It is obvious that \begin{equation}\label{eq:2.17} \det N=(-1)^{k+1}\det \widetilde{N}, \end{equation} where \begin{align*} \widetilde{N}&=\begin{pmatrix} pD_{11}+(p-2)d_1 & pD_{12} & \dots & pD_{1(k+1)}\\ pD_{21} & pD_{22}+(p-2)d_2& \dots & pD_{2(k+1)}\\ \vdots & \vdots & \ddots & \vdots\\ pD_{(k+1)1} & pD_{(k+1)2}& \dots & pD_{(k+1)(k+1)}+(p-2)d_{k+1} \end{pmatrix}\\ &=p(D_{ij})_{(k+1)\times (k+1)}+ (p-2)\begin{pmatrix} d_1 & & & \\ & d_2 & & \\ & & \ddots & \\ & & & d_{k+1} \end{pmatrix}. \end{align*} So $\widetilde{N}$ is positive definite if $5/20$ for all $i$ and $(D_{ij})_{(k+1)\times (k+1)}$ is positive definite. The conclusion then follows. \end{proof} \subsection*{Acknowledgments} This work was supported by NNSF of China, Nos. 11671179 and 11371254. \begin{thebibliography}{00} \bibitem{AS} C. O. Alves, M. A. S. Souto; Existence of least energy nodal solution for a Schr\"{o}dinger-Poisson system in bounded domains, {\it Z. Angew. Math. Phys.} 65 (2014), 1153-1166. \bibitem{BS} C. J. Batkam, J. R. Santos Junior; Schr\"{o}dinger-Kirchhoff-Poisson type systems, {\it Commun. Pure Appl. Anal.}, 15 (2016), 429-444. \bibitem{BW} T. Bartsh, M. Willem; Infinitely many radial solutions of a semilinear elliptic problem on $\mathbb{R}^N$, {\it Arch. Ration. Mech. Anal}. 124 (1993), 261-276. \bibitem {BGS} C. Bonanno, P. d'Avenia, M. Ghimenti, M. Squassina; Soliton dynamics for the generalized Choquard equation, {\it J. Math. Anal. Appl}. 417 (2014), 180-199. \bibitem{CZ} D. Cao, X. Zhu; On the existence and nodal character of semilinear elliptic equations, {\it Acta Math. Sci.}, 8 (1988), 345-359. \bibitem{CCS} S. Cingolani, M. Clapp, S. Secchi; Multiple solutions to a magnetic nonlinear Choquard equation, {\it Z. Angew. Math. Phys}. 63 (2012), 233-248. \bibitem{CCS1} S. Cingolani, M. Clapp, S. Secchi; Intertwining semiclassical solutions to a Schr\"{o}dinger-Newton system, {\it Discrete Contin. Dyn. Syst.}, Ser. S 6(2013), 891-908. \bibitem{CS} S. Cingolani, S. Secchi; Multiple $\mathbb{S}^1$-orbits for the Schr\"{o}dinger-Newton system, {\it Differential and Integral Equations}. 26 (2013), 867-884. \bibitem{CSS} S. Cingolani, S. Secchi, M. Squassina; Semi-classical limit for Schr\"{o}dinger equations with magnetic field and Hartree-type nonlinearities, {\it Proc. Roy. Soc. Edinburgh}, Sect. A 140(5) (2010) 973-1009. \bibitem{ClapS} M. Clapp, D. Salazar; Positive and sign changing solutions to a nonlinear Choquard equation, {\it J. Math. Anal. Appl}. 407 (2013), 1-15. \bibitem{DA} J. T. Devreese, A. S. Alexandrov; Advances in polaron physics, Springer Series in Solid-State Sciences, vol. 159, Springer, 2010. \bibitem{DPS} Y. Deng, S. Peng, W. Shuai; Existence of asymptotic behavior of nodal solutions for the Kirchhoff-type problems, {\it Journal of Functional Analysis} 269 (2015), 3500-3527. \bibitem{FN} G. M. Figueiredo, R. G. Nascimento; Existence of a nodal solution with minimal energy for a Kirchhoff equation, {\it Math. Nachr.}, 288 (2015), 48-60. \bibitem{GS} M. Ghimenti, J. Van Schaftingen; Nodal solutions for the Choquard equation, {\it J. Funct. Anal.}, 271 (2016), 107-135. \bibitem{KS} S. Kim, J. Seok; On nodal solutions of nonlinear schr\"{o}dinger-poisson equations, {\it Comm. Contemp. Math.,} Vol. 14 (2012), 1-16. \bibitem{J} K. R. W. Jones; Newtonian Quantum Gravity, {\it Australian Journal of Physics}. 48 (1995), 1055-1081. \bibitem{Len} E. Lenzmann; Uniqueness of ground states for pseudo-relativistic Hartree equations, {\it Anal. PDE}. 2 (2009), 1-27. \bibitem{L} E. H. Lieb; Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, {\it Studies in Appl. Math}. 57 (1976/77), 93-105. \bibitem{LL} E. Lieb, M. Loss; Analysis, Grauates Studies in Mathematics, vol. 14. AMS, 1997. \bibitem{Lions} P.-L. Lions; The Choquard equation and related questions, {\it Nonlinear Anal.}, 4(1980), 1063-1072. \bibitem{Lions1} P.-L. Lions; The concentration-compactness principle in the calculus of variations. The locally compact case. I, {\it Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire} 1(1984), 109-145. \bibitem{M} G. P. Menzala; On regular solutions of a nonlinear equation of Choquard's type, {\it Proc. Roy. Soc. Edinburgh}, Sect. A 86 (1980), 291-301. \bibitem{MPT} I. M. Moroz, R. Penrose, P. Tod; Spherically-symmetric solution of the Schr\"{o}dinger-Newton equation, {\it Classical Quantum Gravity}. 15 (1998), 2733-2742. \bibitem{MS} V. Moroz, J. Van Schaftingen; Groundstates of nonlinear Choquard equations:existence, qualitative properties and decay asymptotics, {\it J. Funct. Anal}. 265 (2013), 153-184. \bibitem{MS1} V. Moroz, J. Van Schaftingen; Existence of groundstates for a class of nonlinear Choquard equations. {\it Trans. Amer. Math. Soc.}, 367 (2015), 6557-6579. \bibitem{MS2} V. Moroz, J. Van Schaftingen; Semi-classical states for the Choquard equation. {\it Calc. Var. Partial Differential Equations}, 52 (2015), 199-235. \bibitem{P} S. Pekar; Untersuchung \"{u}ber die Elektronentheorie der Kristalle, Akademie Berlag, Berlin, 1954. \bibitem{S} W. A. Strauss; Existence of solitary waves in higher dimensions, {\it Comm. Math. Phys}. 55 (1977), 149-162. \bibitem{WZ} Z. Wang, H. Zhou; Sign-changing solutions for the nonlinear Schr\"{o}dinger-Poisson system in $R^3$, {\it Calc. Var. PDE}, 52(2015), 927-943. \bibitem{W} M. Willem; Minimax Theorems, Birkh\"{a}user, Basel, 1996. \end{thebibliography} \end{document}