\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 277, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/277\hfil Schr\"odinger-Poisson type equations]
{Nodal solutions for Schr\"odinger-Poisson type equations in $\mathbb{R}^3$}

\author[J. Deng, J. Yang \hfil EJDE-2016/277\hfilneg]
{Jin Deng, Jianfu Yang}

\address{Jin Deng \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchang, Jiangxi 330022, China}
\email{jindeng\_2016@126.com}

\address{Jianfu Yang \newline
Department of Mathematics,
Jiangxi Normal University,
Nanchang, Jiangxi 330022, China}
\email{jfyang\_2000@yahoo.com}

\thanks{Submitted August 30, 2016. Published October 18, 2016.}
\subjclass[2010]{35J20, 35J25, 35J61}
\keywords{Schr\"odinger-Poisson equation; Kirchhoff type problem; 
\hfill\break\indent nodal solution}

\begin{abstract}
 In this article, we consider the existence of nodal solutions for the
 Schrodinger-Poisson type problem
 \begin{gather*}
 -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)\Delta u+V(|x|)u+\varphi u
 =|u|^{p-2}u,\quad\text{in }\mathbb{R}^3, \\
 -\Delta\varphi=u^2, \quad \lim_{|x|\to \infty}\varphi (x) =0,
 \end{gather*}
 where  $a, b$ are positive constants, $p\in(4,6)$ and $ V(x)$ is a radial
 smooth function. For each $k\in \mathbb{N}_+$, we show  the existence
 of to nodal 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 nodal solutions for
the Schr\"{o}dinger-Poisson type problem
\begin{equation}\label{eq:1.1}
\begin{gathered}
-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)\Delta u+V(|x|)u+\varphi u
 =|u|^{p-2}u,\quad\text{in }\mathbb{R}^3,\\
-\Delta\varphi=u^2,\quad  \lim_{|x|\to \infty}\varphi (x) =0,
\end{gathered}
\end{equation}
where  $a, b>0$ are a positive constants, $p\in(4,6)$ and
 $V\in C(\mathbb{R}^3,\mathbb{R})$ is a radial function.
The nonlocal operator $\big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\big)\Delta$
appears in the Kirchhoff Dirichlet problem
\begin{equation}\label{eq:1.2}
\begin{gathered}
-\Big(a+b\int_{\Omega}|\nabla u|^2\,dx\Big)\Delta u
=|u|^{p-2}u,\quad\text{in }\Omega,\\
u =0, \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
for a domain $\Omega\subset\mathbb{R}^3$. In one dimensional case, such
a problem arises in the investigation of the existence of classical D'Alembert's
wave equations for free vibration of elastic strings, see \cite{K} for details.
 After the work of Lions \cite{J}, higher dimensional problem \eqref{eq:1.2}
attracts attention of researchers. Various results have been appeared for
Kirchhoff type problems in \cite{ACM,AP, A,CKW,HZ,LY,PZ,WTXZ}
and references therein.

Taking $a = 1$, $b = 0$ in \eqref{eq:1.1}, we obtain the Schr\"{o}inger-Poisson
equation
\begin{equation}\label{eq:1.3}
\begin{gathered}
-\Delta u+V(|x|)u+\varphi u=|u|^{p-2}u,\quad \text{in }\mathbb{R}^3,\\
-\Delta\varphi=u^2,\quad  \lim_{|x|\to \infty}\varphi (x) =0.
\end{gathered}
\end{equation}
where
\[
\varphi(x)=\int_{\mathbb{R}^3} \frac{u^2(y)}{4\pi|x-y| }\,dy
\]
for all $x\in\mathbb{R}^3$. Therefore,  \eqref{eq:1.3} also involves in
 a nonlocal term. This problem with $p=5/3$ stems from the Slater
approximation of the exchange term in the Hartree-Fock model, see \cite{S}.
For the general exponent $p$, there is an interesting competition between
local and nonlocal nonlinearities. This interaction yields new phenomena,
and then problem \eqref{eq:1.3} has been extensively studied in the literature,
see \cite{AR, IV, R,R1,WZ,ZZ} and references therein.

In this paper, we intend to show the existence of radial solutions for
 \eqref{eq:1.1} with prescribed $k$ nodal for every fixed integer $k$.

In the case $k=2$, solutions with two
nodal domains were studied in \cite{AS, BS, FN, WZ} etc.
The argument generally used is to modify the method developed in \cite{N},
first one seeks a minimizer $u$ of the minimizing problem $\inf_{\mathcal{M}}J(u)$,
where
\[
\mathcal{M}= \{u\in \mathcal{N}: u^\pm\not\equiv 0, \langle J'(u),u^+ \rangle
= \langle J'(u),u^- \rangle = 0\}
\]
is a subset of the Nehari manifold $\mathcal{N}$. Then, one needs to show $u$
is a critical point of $J$. For problem \eqref{eq:1.3}, it was proved in
\cite{AS, WZ} by such an argument that the problem has a sign-changing solution.
Because problem \eqref{eq:1.3} contains a nonlocal term, the corresponding
functional $J$ does not have the decomposition
\[
J(u) = J(u^+) + J(u^-),
\]
it brings difficulties to construct a nodal solution. On the other hand, for
the Kirchhoff Dirichlet problem \eqref{eq:1.2}, besides other things,
a sign-changing solution was found in \cite{BS, FN}, some technique was
developed in treating the nonlocal operator in the problem.

For every integer $k\geq0$, it was proved in \cite{BW} and \cite{CZ} independently
that,  there is a pair of solutions $u^\pm_k$ of
\begin{equation}\label{eq:1.4}
\begin{gathered}
-\Delta u+V(|x|)u=f(|x|,u),\quad \text{in }\mathbb{R}^3,\\
u\in H^1(\mathbb{R}^N).
\end{gathered}
\end{equation}
Such solutions of \eqref{eq:1.4} 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,
such as problems \eqref{eq:1.1}-\eqref{eq:1.3}, 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.
In this way, Kim and Seok \cite{KS} found infinitely many nodal solutions for
Schr\"{o}inger-Poisson system \eqref{eq:1.3}, and then Deng et at \cite{DPS}
treated Kirchhoff problems in $\mathbb{R}^3$ in a similar way. Inspired by
\cite{DPS,KS}, we establish the existence of  infinitely many nodal solutions for  \eqref{eq:1.1}.
Since problem \eqref{eq:1.1} contains both nonlocal operator and nonlocal
nonlinear term,  the construction of nodal solutions become technically complicated.

We assume in this paper that, the potential function $V$ satisfies the
condition
\begin{itemize}
\item[(A1)] $V(r)\in C([0, +\infty), \mathbb{R})$ is bounded from below
by a positive constant $V_0$.
\end{itemize}
Our main result is as follows.

\begin{theorem}\label{thm:1.1}
Suppose condition {\rm (A1)} holds and  $4<p<6$.
For every $k\in \mathbb{N}$, there exists a radial solution $u_k$
of \eqref{eq:1.1}, which changes sign exactly $k$ times.
\end{theorem}

This theorem is proved by variational approach.
Denote by $H_r^1({\mathbb{R}}^3)$ the set of radially symmetric
functions in the Sobolev space $H^1({\mathbb{R}}^3)$. We define
\[
H=\Big\{u\in H_r^1({\mathbb{R}}^3):\int_{\mathbb{R}^3}
\big(a|\nabla u|^2+V(|x|)u^2\big)\,dx<+\infty\Big\}
\]
with the norm
 \[
 \|u\|^2=\int_{\mathbb{R}^3} \big(a|\nabla u|^2+V(|x|)u^2\big)\,dx.
 \] %\label{eq:1.2}
By assumption (A1) and the fact $a>0$, the inclusion
$H\hookrightarrow H_r^1({\mathbb{R}}^3)$ is continuous, and
$H\hookrightarrow L^q({\mathbb{R}}^3)$ is compact for $2<q<6$,
by the well known result of Strauss\cite{Str}.
Weak solutions of \eqref{eq:1.1} will be found as critical points of the functional
\begin{align*}
E(u)&=\frac{1}{2}\int_{\mathbb{R}^3}\big(a|\nabla u|^2+V(|x|)u^2\big)\,dx
 +\frac{b}{4}\Big(\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)^2\\
&\quad +\frac{1}{4}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}
 \frac{u^2(x)u^2(y)}{4\pi|x-y|}\,dxdx-\frac{1}{p}\int_{\mathbb{R}^3}|u|^p\,dx
\end{align*}
defined on $H$. The functional $E$ belongs to $C^2(H,\mathbb{R})$,
its Fr\'{e}chet derivative is given by
\begin{equation}\label{eq:1.5}
\begin{split}
\langle E'(u),\psi\rangle
&= \int_{\mathbb{R}^3}\big(a\nabla u\nabla \psi+V(|x|)u\psi\big)\,dx
 +b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\int_{\mathbb{R}^3}\nabla u\nabla\psi\,dx\\
&\quad +\int_{\mathbb{R}^3}\varphi u\psi\,dx-\int_{\mathbb{R}^3}|u|^{p-2}u\psi\,dx\\
\end{split}
\end{equation}
for any $u, \psi\in H $. Hence, critical points of $E(u)$ are weak solutions
of  \eqref{eq:1.1}, and necessarily contained in the Nehari  manifold
\[
 \mathcal{N}=\{u\in {H\ \setminus\{0\}}: \langle E'(u),\varphi\rangle=0 \}.
 \]
In constructing nodal solutions of problem \eqref{eq:1.1}, we will modify
the variational framework in accordance to fixed nodes of expected solutions.
Dividing $\mathbb{R}^3$ into $k+1$ parts, we reformulate functionals and
 Nehari sets correspondingly. The proof of Theorem \ref{thm:1.1}
consists of verifying the Nehari set is a manifold and finding a minimizer
of the related functional on the manifold.

This paper is organized as follows.
In Section 2, we present a suitable variational framework for our problem,
then we prove Theorem \ref{thm:1.1} in Section 3.

\section{Preliminaries}

In this section, we  develop the variational construction.
Decomposing $\mathbb{R}^3$ into $k+1$ parts, we consider a system defined
on $k+1$ parts. Precisely, for $ k\in\mathbb{N}_+ $, we define
\begin{gather*}
\Gamma_k=\{\mathbf{r}_k = (r_1,\dots ,r_k)\in{\mathbb{R}^k}:
 0=r_0<r_1<\dots <r_k<r_{k+1}=\infty\}, \\
B_i=B_i^{\mathbf{r}_k}=\{x\in \mathbb{R}^3 :r_{i-1}<|x|<r_i\}
\end{gather*}
for each $i=1,\dots ,k+1$. By the definition, $ B_1 $ is a ball,
$ B_2,\dots ,B_k$ are annuli and $B_{k+1}$ is the complement of a ball.
Fix $\mathbf{r}_k = (r_1,\dots ,r_k)\in\Gamma_k$, there is family of
$ \{B_i\}_i^{k+1}$. We denote
\[
H_i=\{u\in H_0^1(B_i): u(x)= u(|x|), u(x) = 0\text{ if } x\not\in B_i \}
\]
for each $i=1,\dots,k$. $H_i$ is Hilbert space with the norm
\[
\|u\|_i^2=\int_{B_i}\Big(a|\nabla u|^2+V(|x|)u^2\Big)\,dx.
\]
Furthermore, we set
$\mathcal{H}_k=H_1\times H_2\times \dots \times H_{k+1}$
and define the functional $ I:\mathcal{H}_k\to\mathbb{R}$ by
\begin{equation}\label{eq:2.1}
\begin{split}
&I(u_1,\dots, u_{k+1})\\
&=\frac{1}{2}\sum_{i=1}^{k+1}\|u_i\|_i^2+\frac{b}{4}\sum_{i=1}^{k+1}
 \Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2+\frac{b}{4}\sum_{i=1}^{k+1}
 \int_{B_i}|\nabla u_i|^2\,dx\int_{B_j}|\nabla u_j|^2\,dx\\
&\quad +\frac{1}{4}\sum_{i=1}^{k+1}\int_{B_i}
 \int_{B_i}\frac{u_i^{2}(x)u_i^{2}(y)}{4\pi|x-y|}\,dx\,dy
 +\frac{1}{4}\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}
 \frac{u_i^{2}(x)u_j^{2}(y)}{4\pi|x-y|}\,dx\,dy\\
&\quad -\frac{1}{p}\int_{B_i}|u_i|^p\,dx,
\end{split}
\end{equation}
where $u_i\in H_i$, $i=1,\dots, k+1$. It is readily to verify that
\[
I(u_1,\dots,u_{k+1})=E\Big(\sum_{i=1}^{k+1}u_i\Big).
\]
If $(u_1,\dots,u_{k+1})$ is a critical point of $I$, then it satisfies
\begin{align*}
&\langle I'(u_1,\dots,u_{k+1}),\psi \rangle \\
&=\int_{\mathbb{R}^3}\Big(a\sum_{i=1}^{k+1}\nabla u_i\Big)
 \nabla\psi\,dx+V(|x|)\sum_{i=1}^{k+1} u_i\psi\\
&\quad +b\int_{\mathbb{R}^3}\Big|\sum_{i=1}^{k+1}\nabla u_i\Big|^2
 \int_{\mathbb{R}^3}\Big(\sum_{i=1}^{k+1}\nabla u_j\Big)\nabla \psi\,dx
 +\int_{B_i} \sum_{i=1}^{k+1}u_i\varphi\psi\,dx \\
&\quad -\int_{\mathbb{R}^3}\sum_{i=1}^{k+1}|u_i|^{p-2}u_i\psi
=0.
\end{align*}
for $\psi \in H_0^1(B_i)$, where $-\Delta\varphi=\Big(\sum_{i=1}^{k+1}u_i\Big)^2$.
That is, $(u_1,\dots,u_{k+1})$ is a solution of the system
\begin{equation}\label{eq:2.2}
\begin{gathered}
-\Big(a+b\sum_{j=1}^{k+1}\int_{B_j}|\nabla u_j|^2\,dx\Big)\Delta u_i
 +V(|x|)u_i+\varphi u_i=|u_i|^{p-2}u_i,\,\,\text{in } B_i,\\
-\Delta\varphi=\Big(\sum_{i=1}^{k+1}u_i\Big)^2, \quad
\lim_{|x|\to \infty}\varphi(x) =0.
\end{gathered}
\end{equation}
Now, we define the Nehari set
\[
\mathcal{N}_k=\{(u_1,\dots,u_{k+1})\in\mathcal{H}_k:u_i\neq 0,
\partial _{u_i}I(u_1,\dots,u_{k+1})u_i=0 \text{ for } i=1,\dots, k+1\}
\]
where
\begin{align*}
&\partial _{u_i}I(u_1,\dots,u_{k+1})u_i\\
&=\|u_i\|_i^2+b\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
 +b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}\int_{B_i}|\nabla u_j|^2\,dx\\
&\quad +\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
 +\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy
-\int_{B_i}|u_i|^p\,dx.
\end{align*}

Next, we show that the $ \mathcal{N}_k $ is nonempty manifold in
$ \mathcal{H}_k$, then we seek a minimizer of the functional $I$
constraint on $\mathcal{N}_k$. Apparently, the minimizer is a weak
solution of  \eqref{eq:2.2}. Finally, one needs to prove the minimizer
has nonzero component.
We commence with a proof of manifold for $\mathcal{N}_k$.

\begin{lemma}\label{lem:2.1}
Suppose that {\rm (A1)} holds and $p\in(4,6)$.
For $(u_1,\dots,u_{k+1})\in\mathcal{H}_k $ with $u_i\neq 0$ for
$i =1,\dots, k+1$, there exists a unique $(k+1)$-tuple
$(a_1,\dots,a_{k+1})$ with positive components such that
$ (a_1u_1,\dots,a_{k+1}u_{k+1})\in\mathcal{N}_k$.
\end{lemma}

\begin{proof}
For a fixed $(u_1,\dots,u_{k+1})\in\mathcal{H}_k$ with
$u_i\neq 0 ,(a_1u_1,\dots,a_{k+1}u_{k+1})$ is contained in $ \mathcal{N}_k $
if and only if
\begin{equation}\label{eq:2.3}
\begin{split}
&\|u_i\|_i^2+ba_i^2\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
 +b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}a_j^2\int_{B_i}\nabla u_j|^2\,dx\\
&+a_i^2\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
+\sum_{i\neq j}^{k+1}a_j^2\int_{B_i}\int_{B_j}
 \frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy\\
&-a_i^{p-2}\int_{B_i}|u_i|^p\,dx=0
\end{split}
\end{equation}
for $i =1,\dots, k+1$.
Hence, the problem is reduced to  verify that there is only one solution
 $(a_1,\dots,a_{k+1})$ of \eqref{eq:2.3} such that $a_i> 0$, $i =1,\dots, k+1$.

Fix a parameter $0\leq\alpha\leq1$, we consider the solvability of the
 system of $(k+1)$ equations
\begin{equation}\label{eq:2.4}
\begin{split}
&\|u_i\|^2+ba_i^2\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
 +\alpha b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}
 a_j^2\int_{B_i}\nabla u_j|^2\,dx\\
&+a_i^2\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
+\alpha\sum_{i\neq j}^{k+1}a_j^2\int_{B_i}
 \int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy\\
&-a_i^{p-2}\int_{B_i}|u_i|^p\,dx=0,
\end{split}
\end{equation}
$i =1,\dots, k+1$.
Define
\begin{equation}\label{eq:2.5}
 D=\big\{\alpha: 0\leq\alpha\leq 1 \text{ and  \eqref{eq:2.4} is uniquely solvable
in } (\mathbb{R}_{>0})^{k+1}\Big \}.
\end{equation}

We claim that $D = [0,\alpha]$. This will be done by showing that $D$ is not empty,
 and $D$ is both open and closed in $[0,1]$.

Firstly, we show that  $D$ contains $0$. Let
\[
f_i(t)=\|u_i\|_i^2+bt^2\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
+t^2\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
-t^{p-2}\int_{B_i}|u_i|^p\,dx,
\]
for $i=1,\dots,k+1$. Without loss of generality, we need only to prove that
there is a unique $t_0>0 $ such that $f_1(t_0)=0$.

Since $ u_1\neq 0$, we have $ f_1(t)>0$ if $t>0$ small, and  $f_1(t)<0$
if $t>0$ large. Therefore, there exists $t_0>0$ such that $f_1(t_0)=0$.

Now we show $t_0$ is unique. Indeed, were it not the case, we would
have $0<t_0<\bar{t}$ such that $f_1(t_0)=f_1(\bar{t})=0$.
That is, the equality
\[
\frac{1}{t}\|u_1\|_1^2+b\Big(\int_{B_1}|\nabla u_1|^2\,dx\Big)^2+\int_{B_1}\int_{B_1}\frac{u_1^2(x)u_1^2(y)}{4\pi|x-y|}\,dx\,dy-t^{p-4}\int_{B_1}|u_1|^p\,dx=0,
\]
holds for $t_0$ and $\bar t$. It yields
\[
0<(\frac{1}{t_0}-\frac{1}{\bar{t}})\|u_1\|_1^2=(t_0^{p-4}-\bar{t}^{p-4})\int_{B_1}|u_1|^p\,dx<0,
\]
which is contradiction. Hence, $0\in D$.

Next, we prove that $D$ is open in $[0,1]$.
Suppose that $\alpha_0\in D$ and
 $ (\bar{a}_1,\dots,\bar{a}_{k+1})\in(\mathbb{R}>0)^{k+1}$
is the unique solution of \eqref{eq:2.4} with $\alpha=\alpha_0$. Therefore,
\begin{equation}\label{eq:2.6}
\begin{split}
&\|u_i\|_i^2+bc_i\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
+\alpha_0 b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}c_j
 \int_{B_i}\nabla u_j|^2\,dx\\
&+c_i\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
+\alpha_0\sum_{i\neq j}^{k+1}c_j\int_{B_i}\int_{B_j}
 \frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy\\
&-c_i^q\int_{B_i}|u_i|^p\,dx=0
\end{split}
\end{equation}
for each $i =1,\dots, k+1$, where $c_i={\bar{a}_i}^2,q=\frac{p-2}{2}>1$.
To apply the implicit function theorem at $ \alpha_0$, we calculate the matrix
\[
M=(M_{ij})=(\partial {c_j}G_i)_{i,j=1,\dots,k+1},
 \]
 where $G_i = G_i(c_1,\dots,c_{k+1},\alpha_0)$ denotes the left-hand side
of \eqref{eq:2.6}. Hence, each component of the matrix $M$
 can be represented by
\begin{gather*}
M_{ii}=b\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
+\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
 -qc_i^{q-1}\int_{B_i}|u_i|^p\,dx, \\
M_{ij}=\alpha_0 b\int_{B_i}|\nabla u_i|^2\,dx\int_{B_i}|\nabla u_j|^2\,dx
+\alpha_0\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy.
\end{gather*}
By  \eqref{eq:2.6},
\begin{align*}
\hat M_{ii}
&:= -c_iM_{ii} \\
&=q\|u_i\|_i^2+(q-1)c_i
 \Big[b\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
 +\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy\Big]\\
&\quad +q\alpha_0b\int_{B_i}|\nabla u_i|^2dx\sum_{i\neq j}^{k+1}
\int_{B_j}|\nabla u_j|^2dx \\
&\quad +q\alpha_0\sum_{i\neq j}^{k+1}c_j
 \int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}dx\,dy
\end{align*}
and
\begin{align*}
\hat M_{ij}
&:=-c_j M_{ij}=-\alpha_0bc_j\int_{B_i}
|\nabla u_i|^2dx\int_{B_j}|\nabla u_j|^2dx\\
&\quad -\alpha_0c_j\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy.
\end{align*}
It readily verifies that
\[
\hat {M}_{ij}\leq 0,\quad
\det (\hat M_{ij})=\frac{(-1)^{k+1}}{c_1\dots c_{k+1} }\det M
\]
and
\begin{align*}
\hat M_{ii}+\sum_{i\neq j}^{k+1}\hat M_{ij}
&=q\|u_i\|_i^2+(q-1)bc_i\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2 \\
&\quad +(q-1)  c_i\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy\\
&\quad +(q-1)\alpha_0b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}
 \int_{B_i}\nabla u_j|^2\,dx\\
&\quad +(q-1)\alpha_0\sum_{i\neq j}^{k+1}c_j\int_{B_i}
 \int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy.
\end{align*}
Since $ q>1 $, we obtain
\[
\hat M_{ii}+\sum_{i\neq j}^{k+1}\hat M_{ij}>0 \quad \text{and}\quad
\det\hat M_{ii}\neq0
\]
which implies
\[
0\neq \det (\hat M_{ij})=\frac{(-1)^{k+1}}{c_1\dots c_{k+1} }\det M.
\]
The implicit function theorem yields that there exist a neighborhood
$U_0$ of $\alpha_0$ and a neighborhood $B_0 \subset (\mathbb{R}>0)^{k+1}$
 of $(\bar{a}_1,\dots,\bar{a}_{k+1})$ such that the equation of  \eqref{eq:2.4}
is uniquely solvable in $ U_0\times  B_0$.

Now we show that \eqref{eq:2.4} is uniquely solvable in
$ U_0\times  (\mathbb{R}_{>0})^{k+1}$, this means $U_0\subset D$.
Suppose, on the contrary, that there is $\alpha_0 \in U_0 $ such that there
exists the second solution
$(\hat{a}_1,\dots,\hat{a}_{k+1})\in (\mathbb{R}_{>0})^{k+1}\setminus B_0$
of  \eqref{eq:2.4}.  By the implicit function theorem, we can find a solution
curve $(\alpha,(\hat{a}_1,\dots,\hat{a}_{k+1}))$ in
$(\alpha_0-\varepsilon, \alpha+\varepsilon)\times
(\mathbb{R}_{>0})^{k+1}\setminus B_0$.
Assume $ \alpha_0<\alpha_1 $ for a while and extend this curve as much as possible.
Since it cannot be defined at $\alpha_0$ and enter into $ U_0\times  B_0 $,
there should have a point $\alpha_2 \in [\alpha_0,\alpha_1)$ such that
$(a_1(\alpha),\dots,a_{k+1}(\alpha))$ exists in $\alpha\in (\alpha_2,\alpha_1]$
and blows up as $\alpha \to \alpha_2^+ $. However, this is impossible,
 since if   $(a_1,\dots,a_{k+1})$ 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 D $.
The case $\alpha_0 >\alpha_1 $ can be proved in the same way.

Next, we prove that $D$ is  closed in $[0,1]$.
Let $ \{\alpha_n\} $ be a sequence in $D$ converging to $\alpha_0 \in [0,1]$
and $(a^n_1,\dots,a^n_{k+1}) \in (\mathbb{R}_{>0})^{k+1}$ be the solution
of  \eqref{eq:2.4} corresponding to $ \alpha_n $. By the preceding argument,
the sequence $(a^n_1,\dots,a^n_{k+1}) \in (\mathbb{R}_{>0})^{k+1}$
is bounded above. Thus we may assume that $(a^n_1,\dots,a^n_{k+1})$
converges to a solution $(a^0_1,\dots,a^0_{k+1}) \in (\mathbb{R}>0)^{k+1}$
of \eqref{eq:2.4} for $\alpha_0 $. Since $H_i \hookrightarrow L^p$, we obtain
\[
(a^n_i)^2\|u_i\|^2_i\leq\int_{B_i}|a^n_1u_i|^pdx\leq C (a^n_i)^p \|u_i\|^p_i,
\]
which implies that $0<C_i\leq a^n_i$ uniformly in $n$.
Thus $a^0_i\geq C_i>0$ for $ i\in\mathbb{N}$.
So $(a^0_1,\dots,a^0_{k+1})\in (\mathbb{R}_{>0})^{k+1}$.
By the implicit function theorem again,  $(a^0_1,\dots,a^0_{k+1})$
is the unique solution in $ (\mathbb{R}_{>0})^{k+1}$. Hence, $D$ is closed.
The conclusion of Lemma \ref{lem:2.1} then follows.
\end{proof}

Now, we show that $\mathcal{N}_k$ is a differentiable manifold.

\begin{lemma}\label{lem:2.2}
$\mathcal{N}_k$ is a differentiable manifold in $\mathcal{H}_k$.
 Moreover, all critical points of the restriction
$I|_{\mathcal{N}_k}$ of $I$ to $\mathcal{N}_k$ are critical point of  $I$
 with no zero component.
\end{lemma}

\begin{proof}
We observe that
\[
\mathcal{N}_k=\{(u_1,\dots,u_{k+1})\in\mathcal{H}_k:u_i\neq0,
\;\mathbf{F}(u_1,\dots,u_{k+1})={\bf 0}\},
\]
where $\mathbf{F}=(F_1,\dots,F_{k+1}):\mathcal{H}_k\to\mathbb{R}^{k+1}$ is given by
\begin{equation}\label{eq:2.7}
\begin{split}
&F_i(u_1,\dots,u_{k+1})\\
&=\|u_i\|_i^2+b\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
+b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}\int_{B_i}|\nabla u_j|^2\,dx\\
&+\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
+\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy
-\int_{B_i}|u_i|^p\,dx,
\end{split}
\end{equation}
$i=1,\dots,k+1$. To prove that $\mathcal{N}_k$ is a differentiable manifold
in $\mathcal{H}_k$, it suffices to verify that the matrix
\[
N=(N_{ij})=(\partial_{u_i}F_j(u_1,\dots,u_{k+1}),u_i )_{i, j=1,\dots, k+1},
\]
is nonsingular at each point $(u_1,\dots,u_{k+1})\in \mathcal{N}_k$,
since it implies that 0 is a regular value of $\mathbf{F}$. Straightforwardly,
we have
\begin{align*}
N_{ii}&=2\|u_i\|_i^2+4b\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2
 +2b\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}\int_{B_i}|\nabla u_j|^2\,dx\\
&\quad+4\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
 +2\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}
 \,dx\,dy \\
&\quad -p\int_{B_i}|u_i|^p\,dx
\end{align*}
and
\[
N_{ij}=2b\int_{B_i}|\nabla u_i|^2dx\int_{B_j}|\nabla u_j|^2\,dx
+2\int_{B_i}\int_{B_j}\frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy.
\]
By \eqref{eq:2.4}, we have
\begin{align*}
N_{ii}&=(2-p)\|u_i\|_i^2+(4b-bp)\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2\\
&\quad +(2b-bp)\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}
 \int_{B_i}|\nabla u_j|^2\,dx\\
&+(4-p)\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy
 +(2-p)\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}
 \frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy.
\end{align*}
Let
$\hat N_{ii}=- N_{ii}$ and $\hat N_{ij}=- N_{ij}$.
Then
\begin{align*}
\hat N_{ii}+\sum_{i\neq j}^{k+1}\hat N_{ij}
&=(p-2)\|u_i\|_i^2+b(p-4)\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2\\
&\quad +b(p-4)\int_{B_i}|\nabla u_i|^2\,dx\sum_{i\neq j}^{k+1}
 \int_{B_i}|\nabla u_j|^2\,dx\\
&\quad +(p-4)\int_{B_i}\int_{B_i}\frac{u_i^2(x)u_i^2(y)}{4\pi|x-y|}\,dx\,dy \\
&\quad +(p-4)\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}
 \frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy>0
\end{align*}
provided $4<p<6$. Hence,
$$
\det(\hat N)\neq 0,\quad \det N=(-1)^{k+1}\det(\hat N)\neq 0,
$$
and the matrix $N$ is invertible at each $(u_1,\dots,u_{k+1})\in\mathcal{N}_k$.
So $\mathcal{H}_k$ is a differential manifold.

If $ (u_1,\dots,u_{k+1}) $ is a critical point of $I|_{\mathcal{N}_k}$,
then there are Lagrange multipliers $ \mu_1,\dots, \mu_{k+1} $ satisfying
\begin{equation}\label{eq:2.8}
\mu_1F'_1(u_1,\dots,u_{k+1})+\dots+\mu_{k+1}F'_{k+1}(u_1,\dots,u_{k+1})
=I'(u_1,\dots,u_{k+1}).
\end{equation}
Inserting  $ (u_1,0,\dots,0)$, $(0,u_2,\dots,0),\,\dots,\,(0,0,\dots,0,u_{k+1})$
into  \eqref{eq:2.8}, we obtain
\[
N\begin{pmatrix} \mu_1  \\ \vdots&  \\ \mu_{k+1} \end{pmatrix}
=\begin{pmatrix} 0  \\ \vdots&  \\ 0 \end{pmatrix}.
\]
Because $ N $ is nonsingular, we find that $\mu_1=\mu_2=\dots=\mu_{k+1}=0$
and $(u_1,\dots,u_{k+1})$ is a critical point of $I$.

Next, by the Sobolev embedding $ H_i\hookrightarrow L^p$, the inequality
\begin{equation}\label{eq:2.9}
\|u_i\|^2_i\leq \int_{B_i}|u_i|^2\leq C\|u_i\|^p_i
\end{equation}
implies that $u_i\neq0$ for all $i$. Thus,
all components of critical points of $I$ in $\mathcal{N}_k$ are nontrivial.
This completes the proof.
\end{proof}

\begin{lemma}\label{lem:2.3}
For fixed $(u_1,\dots,u_{k+1})\in\mathcal{H}_k$ with nonzero component,
the vector $ (a_1,\dots,a_{k+1})$ which is obtained in Lemma \ref{lem:2.1} is
the unique maximum point of the function
$\eta:(\mathbb{R}_{>0})^{k+1}\to\mathbb{R} $ defined as
$\eta(d_1,\dots,d_{k+1})=I((d_1u_1,\dots,d_{k+1}u_{k+1})) $.
\end{lemma}

\begin{proof}
 By the proof of Lemma \ref{lem:2.1}, $(a_1,\dots,a_{k+1})$ is the unique
critical point of $\psi$ in $(\mathbb{R}_{>0})^{k+1}$.
If $|(d_1,\dots,d_{k+1})|\to\infty$, it is readily to verify that
$\psi(d_1,\dots,d_{k+1})\to-\infty$, so  it is sufficient
to show that $(a_1,\dots,a_{k+1})$ is not on the boundary of
$(\mathbb{R}_{>0})^{k+1}$.

Choose $(d^0_1,\dots,d^0_{k+1})\in\partial(\mathbb{R}_{>0})^{k+1}$,
without loss of generality, we may assume that $d^0_1=0$. Since
\begin{align*}
  &\eta(t,d_2^0\dots,d_{k+1}^0)\\
  &=I((tu_1,d_2^0u_2\dots,d_{k+1}^0u_{k+1}))\\
&=\frac{t^2}{2}\|u_1\|^2_1+\frac{bt^4}{4}\Big(\int_{B_1}|\nabla u_1|^2\,dx\Big)^2
 +\frac{bt^2}{2}\sum_{j= 2}^{k+1}{d^0_j}^2\int_{B_1}|\nabla u_i|^2\,dx
 \int_{B_j}|\nabla u_j|^2\,dx \\
&\quad +\frac{t^4}{4}\int_{B_1}\int_{B_1}\frac{u_1^2(x)u_1^2(y)}{4\pi|x-y|}\,dx\,dy
 +\frac{t^2}{2}  \sum_{j=2}^{k+1}{d^0_j}^2\int_{B_1}
\int_{B_j}\frac{u_1^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy  \\
&\quad -\frac{t^p}{p}\int_{B_1}|u_1|^p\,dx
 +\sum_{i= 2}^{k+1}{d^0_i}^2\|u_i\|^2
 +\frac{b}{4}\sum_{i,j=2}^{k+1}{(d^0_id^0_j)}^2\int_{B_1}|\nabla u_i|^2\,dx
 \int_{B_j}|\nabla u_j|^2\,dx  \\
&\quad +\frac{1}{4}\sum_{i=1}^{k+1}{(d^0_id^0_j)}^2\int_{B_i}\int_{B_j}
 \frac{u_i^2(x)u_j^2(y)}{4\pi|x-y|}\,dx\,dy
  -\frac{1}{p}\sum_{i=2}^{k+1}{d^0_i}^p\int_{B_i}|u_i|^p\,dx
\end{align*}
is increasing if $t>0$ is small enough, $(0,\dots,d^0_{k+1})$ is not a
maximum point of $\eta$ in ${\mathbb{R}_{>0}}^{k+1}$.
The proof is complete.
\end{proof}

Next, we have existence results.
 \begin{lemma}\label{lem:2.4} 
For fixed $\mathbf{r}_k=(r_1,\dots,r_{k+1})\in \boldsymbol{\Gamma}_k $, there is a minimizer $(v_1,\dots,v_{k+1})$ of its corresponding energy $I|_{\mathcal{N}_k}$ on $ \mathcal{N}_k$ such that $(-1)^{i+1}v_i $ is positive on $B_i$  for $i=1,\dots ,k+1$. Moreover, $(v_1,\dots,v_{k+1})$ satisfies  \eqref{eq:2.2}.
\end{lemma}

\begin{proof} For  $(u_1,\dots,u_{k+1})\in\mathcal{N}_k$, we have
 \begin{equation}\label{eq:2.10}
 \begin{split}
 I(u_1,\dots, u_{k+1})
&=\Big(\frac{1}{2}-\frac{1}{p}\Big)\sum_{i=1}^{k+1}\|u_i\|_i^2
 +(\frac{b}{4}-\frac{b}{p})\sum_{i=1}^{k+1}\Big(\int_{B_i}|\nabla u_i|^2\,dx\Big)^2\\
&\quad +\Big(\frac{b}{4}-\frac{b}{p}\Big)\sum_{i=1}^{k+1}
\int_{B_i}|\nabla u_i|^2\,dx  \int_{B_j}|\nabla u_j|^2\,dx \\
&\quad +\Big(\frac{1}{4}-\frac{1}{p}\Big)\sum_{i=1}^{k+1}\int_{B_i}\int_{B_i}
 \frac{u_i^{2}(x)u_j^{2}(y)}{4\pi|x-y|}\,dx\,dy\\
&\quad +\Big(\frac{1}{4} -\frac{1}{p}\Big)\sum_{i\neq j}^{k+1}\int_{B_i}
 \int_{B_j}\frac{u_i^{2}(x)u_j^{2}(y)}{4\pi|x-y|}\,dx\,dy \\
&\geq\frac{1}{4}\sum_{i=1}^{k+1}\|u_i\|_i^2.
\end{split}
 \end{equation}
Let $\{(u^n_1,\dots,u^n_{k+1})\}\subset\mathcal{N}_k $ be a minimizing
sequence of $I|_{\mathcal{N}_k }$; that is,
 \[\lim_{n\to\infty}I(u^n_1,\dots,u^n_{k+1})
=\inf_{(u_1,\dots,u_{k+1})\in\mathcal{N}_k}I(u_1,\dots,u_{k+1}).
 \]
By \eqref{eq:2.10}, $\{ u^n_i\}$ is bounded. Hence, we may assume that
  \[
  (u^n_1,\dots,u^n_{k+1})\rightharpoonup (u^0_1,\dots,u^0_{k+1})
\quad\text{weakly in } \mathcal{H}_k.
  \]
We claim that $u^0_i\neq 0$ in $\mathcal{H}_i$ for $i=1,\dots,k+1$.
Indeed, if $ u^n_i \to u^0_i$ in $\mathcal{H}_i$, we may show as
\eqref{eq:2.9} that
\[
  \|u^n_i\|^2_i\leq\int_{B_i}|u_i^n|^p\,dx\leq C \|u_i^n\|^p_i,
\]
 which implies $\|u^n_i\|_i\geq C_i>0$, and the strongly convergence yields
$u^0_i\neq 0$.

If $u^n_i\rightharpoonup  u^0_i$ in $\mathcal{H}_i$, but
$u^n_i\not\to  u^0_i$ in $\mathcal{H}_i$, we have
 \begin{equation}\label{eq:2.14}
 \|u^0_i\|_i< \lim _{n\to\infty}\inf  \|u^n_i\|_i.
  \end{equation}
Therefore,
\begin{align*}
  \|u^0_i\|_i^2
&\leq\|u^0_i\|^2_i+b\Big(\int_{B_i}|\nabla u^0_i|^2\,dx\Big)^2
  +b\int_{B_i}|\nabla u^0_i|^2\,dx\sum_{i\neq j}^{k+1}
 \int_{B_j}|\nabla u^0_j|^2\,dx\\
&\quad +\int_{B_i}\int_{B_i}\frac{(u_i^0)^2(x)(u^0_i)^2(y)}{4\pi|x-y|}\,dx\,dy
+\sum_{i\neq j}^{k+1}\int_{B_i}\int_{B_j}
 \frac{(u_i^0)^2(x)(u^0_j)^2(y)}{4\pi|x-y|}\,dx\,dy\\
&<\int_{B_i}|u^0_i|^p\,dx.\\
\end{align*}
This and the Sobolev embedding
 $\mathcal{H}^1_r({\mathbb{R}^3})\hookrightarrow L^p({\mathbb{R}^3})$ yield
 \[
 \|u^0_i\|^2_i<\int_{B_i}|u^0_i|^p\,dx\leq C\|u^0_i\|^p_i.
 \]
Thus, $u^0_i\neq 0$ for each $i$.

  Next we prove that $(u^n_1,\dots,u^n_{k+1})\to (u^0_1,\dots,u^0_{k+1})$
in $\mathcal{H}_k$ as $n\to\infty $ .
Suppose on the contrary that $(u^n_1,\dots,u^n_{k+1})$ converges weakly
to $(u^0_1,\dots,u^0_{k+1})$ but does not  converge strongly
in $\mathcal{H}_k$ as $n\to\infty$. Thus, there exists at least one $i$ such that
\[
\|u^0_i\|_i< \lim_{n\to\infty}\inf  \|u^n_i\|_i
\]
and $\|u^0_i\|\neq0$. By Lemma \ref{lem:2.1},  there exists a unique $(a^0_1,\dots,a^0_{k+1})\neq(1,\dots,1)\in (\mathbb{R}_{>0})^{k+1}$ such that $(a^0_1u^0_1,\dots,a^0_{k+1}u^0_{k+1})\in\mathcal{N}_k$.

If $(a_1,\dots,a_{k+1})=(1,\dots,1)$, we have
\begin{align*}
\inf I(u_1,\dots,u_{k+1})
&= \lim _{n\to\infty}\inf I(u^n_1,\dots,u^n_{k+1}) \\
&\geq I(u^0_1,\dots,u^0_{k+1}) \\
&\geq\inf I(u_1,\dots,u_{k+1}),
\end{align*}
which implies
\[
I(u^0_1,\dots,u^0_{k+1})=\inf I(u_1,\dots,u_{k+1}).
\]

On the other hand, Lemma \ref{lem:2.3} leads to
  \begin{align*}
 &\inf_{(u_1,\dots,u_{k+1})\in\mathcal{N}_k} I(u_1,\dots,u_{k+1}) \\
&\leq I(a^0_1u^0_1,\dots,a^0_{k+1}u^0_{k+1})\\
&<\frac{1}{2}\sum_{i=1}^{k+1}(a^0_i)^2\liminf_{n\to\infty}  \|u^n_i\|^2_i+
 \frac{b}{4}\sum_{i=1}^{k+1}(a^0_i)^4\liminf_{n\to\infty}
 \Big(\int_{B_i}|\nabla u^n_i|^2\,dx\Big)^2
 \\
&\quad +\frac{b}{4}\sum_{i\neq j}^{k+1}(a^0_i)^2(a^0_j)^2
 \liminf_{n\to\infty}\int_{B_i}|\nabla u^n_i|^2dx\int_{B_j}|\nabla u^n_j|^2\,dx\\
&\quad +\frac{1}{4}\sum_{i=1}^{k+1}(a^0_i)^4\liminf_{n\to\infty}\int_{B_i}
 \int_{B_i}\frac{(u_i^n)^2(x)(u^n_i)^2(y)}{4\pi|x-y|}\,dx\,dy
  \\
&\quad + \frac{1}{4}\sum_{i\neq j}^{k+1}(a^0_i)^2(a^0_j)^2\liminf_{n\to\infty}
 \int_{B_i}\int_{B_j}\frac{(u_i^n)^2(x)(u^n_j)^2(y)}{4\pi|x-y|}\,dx\,dy\\
&\quad -\frac{1}{p}\sum_{i=1}^{k+1}(a^0_i)^p\liminf_{n\to\infty}
 \int_{B_i}|u^n_i|^p\,dx
 \\
&< \liminf_{n\to\infty} I(u^n_1,\dots,u^n_{k+1})
=\inf_{(u_1,\dots,u_{k+1})\in\mathcal{N}_k} I(u_1,\dots,u_{k+1}).
  \end{align*}
This is a contradiction. Therefore, 
$(u^n_1,\dots,u^n_{k+1})\to(u^0_1,\dots,u^0_{k+1})$
in $\mathcal{H}_k$, and $(u^0_1,\dots,u^0_{k+1})$ is a minimizer of
$I|_{\mathcal{N}_k}$.

Let $(v_1,\dots,v_{k+1})=(|u^0_1|,-|u^0_2|,\dots,(-1)^{k+1}|u^0_{k+1}|)$,
we can check that
\[
(v_1,\dots,v_{k+1}) \in\mathcal{N}_k\quad\text{and}\quad
I(v_1,\dots,v_{k+1})=I(u^0_1,\dots,u^0_{k+1}).
\]
Hence $(v_1,\dots,v_{k+1})$ is a minimizer of $ I|_{\mathcal{N}_k}$.
 By the Lemma \ref{lem:2.1},
$(v_1,\dots,v_{k+1})$  is a critical point of $ I|_{\mathcal{N}_k}$,
and  it satisfies  \eqref{eq:2.2}.  By the strong maximum principle,
 each $(-1)^{i+1}|v^0_i|$ is positive in $B_i$, $i = 1,\dots, k+1$.
Hence, $(v_1,\dots,v_{k+1})$ is the solution we want.
\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 $v^{\mathbf{r}_k}=(v^{\mathbf{r}_k}_1,\dots,v^{\mathbf{r}_k}_{k+1})$
of \eqref{eq:2.2} which consists of sign changing components.
We shall find a $\tilde{\mathbf{r}}_k=(\tilde r_1,\dots,\tilde r_k)
\in \boldsymbol{\Gamma}_k $
such that $v^{\tilde{\mathbf{r}}_k}=(v^{\tilde{\mathbf{r}}_k}_1,\dots,
v^{\tilde{\mathbf{r}}_k}_{k+1})$
is a solution of \eqref{eq:2.2} 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{eq:1.1}
that changes sign exactly $k$ times.
 Denote by $B_i^{\tilde{\mathbf{r}}_k}$ the nodal domain and by
 $I^{\tilde{\mathbf{r}}_k}$ the functional related to
 $\tilde{\mathbf{r}}_k$. Note that $v^{\tilde{\mathbf{r}}_k}_i$ is
$\mathcal{C}^2(B_i^{\tilde{\mathbf{r}}_k})$ by standard elliptic
regularity results. Hence it is enough to match the first derivative with
respect to the radial variable, of adjacent components
$v^{\tilde{\mathbf{r}}_k}_i$ and $v^{\tilde{\mathbf{r}}_k}_{i+1}$ at the point
 $r_i$ to ensure the existence of a solution of  \eqref{eq:1.1} with $k$
times sign changing.

To find a least energy radial solution of \eqref{eq:2.2} among elements
in $\boldsymbol{\Gamma}_k $ with nonzero components, we need to estimate
the energy of the solution $(v^{\mathbf{r}_k}_1,\dots,v^{\mathbf{r}_k}_{k+1})$
of \eqref{eq:2.2}. To this end, we first define the function
 $\chi: \boldsymbol{\Gamma}_k \to\mathbb{R}$ by
\begin{equation} \label{eq:3.1}
\chi(\mathbf{r}_k)=\chi(r_1,\dots,r_k)=I^{\mathbf{r}_k}(v^{\mathbf{r}_k}_1,\dots ,
 v^{\mathbf{r}_k}_{k+1})=\inf_{(u^{\mathbf{r}_k}_1,\dots,
u^{\mathbf{r}_k}_{k+1})\in\mathcal{N}_k} I(u^{\mathbf{r}_k}_1,
\dots,u^{\mathbf{r}_k}_{k+1}).
\end{equation}


 \begin{lemma}\label{lem:3.1}
For any positive integer $k$, let
 $\mathbf{r}_k =(r_1,\dots,r_{k})\in{\boldsymbol{\Gamma}_k }$. Then
\begin{itemize}
\item[(i)] if $r_i-r_{i-1}\to 0$ for some $i\in\{1,\dots,k\}$,
then $\chi( \mathbf{r}_k)\to  +\infty$.

\item[(ii)] if $r_k\to\infty$, then $\chi( \mathbf{r}_k)\to +\infty$.

\item[(iii)] $\chi$ is continuous in ${\boldsymbol{\Gamma}_k }$.

\end{itemize}
In particular,  there is a ${\tilde{\mathbf{r}}_k}
=(\tilde{\mathbf{r}}_1,\dots,\tilde{\mathbf{r}}_k)\in {\boldsymbol{\Gamma}_k }$
such that
\[
\chi(\tilde{\mathbf{r}}_k) = \inf_{\mathbf{r}_k\in {\boldsymbol{\Gamma}_k}}
\chi(\mathbf{r}_k)
\]
\end{lemma}

\begin{proof}
(i) Suppose $r_{i_0}-r_{{i_0-1}}\to 0$ for some $i_0\in\{1,\dots,k\}$,
by the H\"{o}lder inequality and Sobolev inequality, we obtain
\[
\|v^{\mathbf{r}_k}_{i_0}\|^2_{i_0}
\leq\int_{B^{\mathbf{r}_k}_{i_0}}|v^{\mathbf{r}_k}_{i_0}|^p\,dx
\leq \Big(\int_{B^{\mathbf{r}_k}_{i_0}}|v^{\mathbf{r}_k}_{i_0}|^6\,dx
 \Big)^{p/6}
 |B^{\mathbf{r}_k}_{i_0}|^{1-\frac{p}{6}}
\leq C\|v^{\mathbf{r}_k}_{i_0}\|^p_{i_0} |B^{\mathbf{r}_k}_{i_0}|^{1-\frac{p}{6}};
\]
that is,
\[
|B^{\mathbf{r}_k}_{i_0}|^{\frac{p}{6}-1}\leq C\|v^{\mathbf{r}_k}_{i_0}\|^{p-2}_{i_0}.
\]
Since $4<p<6$, $\|v^{\mathbf{r}_k}_{i_0}\|_{i_0}\to\infty$. Inequality
\eqref{eq:2.10} implies
 \[
 I(v^{\mathbf{r}_k}_1,\dots,v^{\mathbf{r}_k}_{k+1})
\geq\big(\frac{1}{2}-\frac{1}{p}\big)\sum_{i=1}^{k+1}\|v^{\mathbf{r}_k}_i\|^2_i
\to\infty,
\]
and then
\[
\chi(\mathbf{r}_k)=\chi(r_1,\dots,r_k)
=I^{\mathbf{r}_k}_b(v^{\bf\rho_k}_1,\dots,v^{\mathbf{r}_k})\to\infty.
\]
Thus the first item holds.

(ii) By the Strauss inequality \cite{BS}, that is, for $u \in H^1_r (\mathbb{R}^3)$,
 there exists $C>0$, such that
\[
|u(x)|\leq c \frac{\|u\|}{|x|},\quad\text{a.e. in } \mathbb{R}^3,
\]
we obtain
\[
\|v^{\mathbf{r}_k}_{k+1}\|^2_{k+1}
\leq\int_{B^{\mathbf{r}_k}_{k+1}}|v^{\mathbf{r}_k}_{k+1}|^p\,dx
\leq C\int_{B^{r_k}_{k+1}}\frac{\|v^{\mathbf{r}_k}_{k+1}\|^p_{k+1}}{|x|^p}
\leq C\|v^{\mathbf{r}_k}_{k+1}\|^p_{k+1}r^{3-p}_k.
\]
Therefore,
\[
r^{p-3}_k\leq C\|v^{\mathbf{r}_k}_{k+1}\|^{p-2}_{k+1}
\]
yielding $\chi(\mathbf{r}_k)\to\infty$. The conclusion follows.

(iii) Take a sequence
 $\{r^n_k\}=\{(r^n_1,\dots,r^n_k)\}\subset\boldsymbol{\Gamma}_k $ such that
\[
r^n_k\to{\bar{\mathbf{r}}_k }=(\bar{r}_1,\dots,\bar{r}_k)\in \boldsymbol{\Gamma}_k .
\]
The assertion follows by showing that
\begin{equation} \label{eq:3.2}
\chi({\bar{\mathbf{r}}_k})\geq  \lim_{n\to\infty}\sup\chi({\bf{r_k^n}}),\quad
\chi(\bf\bar{r}_k)\leq  \lim _{n\to\infty}\inf\chi(\mathbf{r}_k^n).
\end{equation}
We first prove $\chi({\bf\bar{r}_k})\geq  \lim_{n\to\infty}\sup\chi({\mathbf{r}_k^n})$.
Define $\xi^{{\mathbf{r}_k^n}}_i :[r^n_{i-1},r^n_i]\to\mathbb{R}$ by
\[
\xi^{{\mathbf{r}_k^n}}_i=a^n_iv^{\bar{\mathbf{r}}_k}_i
\Big(\frac{\bar{r}_i-\bar{r}_{i-1}}
{r^n_i-r^n_{i-1}}(t-r^n_{i-1})+\bar{r}_{i-1}\Big)
\]
for $i=1,\dots,k$, and
\[
\xi^{{\mathbf{r}_k^n}}_{k+1}=a^n_{k+1}v^{{\bf\bar{r}_k}}_{k+1}
\Big(\frac{\bar{r}}{r^n_k}t\Big),
\]
where $r^n_0=0,\, r^n_{k+1}=\infty$ and each $(a^n_1,\dots,a^n_{k+1})$
is the unique positive vector such that
$(\xi^{\mathbf{r}_k^n}_1,\dots,\xi^{\bf{r_k^n}}_{k})\in{\mathcal{N}_k}^{\mathbf{r}^n_k}$.
By the definition of $(v^{\mathbf{r}_k^n}_1,\dots,v^{\bf{r_k^n}}_{k+1})$, we have
\[
I^{\mathbf{r}_k^n}(\xi^{\mathbf{r}_k^n}_1,\dots,\xi^{\mathbf{r}_k^n}_{k+1})
\geq I^{\mathbf{r}_k^n}(v^{\mathbf{r}_k^n}_1,\dots,v^{\mathbf{r}_k^n}_{k+1})=\chi(\mathbf{r}_k^n).
\]
Therefore, for $n$ large enough we have
\begin{gather*}
\|\xi^{{\mathbf{r}_k^n}}_i\|^2_{B^{\mathbf{r}_k^n}_i}
=(a^n_i)^2\|v^{\bf\bar{r}_k}_i\|_{B^{\mathbf{r}_k^n}_i}+o(1),
 \\
\begin{split}
&\int_{B^{\mathbf{r}_k^n}_i}|\nabla \xi^{\mathbf{r}_k^n}_i|^2\,dx
\int_{B^{\mathbf{r}_k^n}_j}|\nabla \xi^{\mathbf{r}_k^n}_j|^2\,dx \\
&=(a^n_i)^2(a^n_j)^2\int_{B^{\mathbf{r}_k^n}_i}|\nabla v^{\bf\bar{r}_k}_i|^2\,dx
\int_{B^{\bf\bar{r}_k}_j}|\nabla v^{\mathbf{r}_k^n}_j|^2\,dx+o(1),
\end{split}\\
\begin{split}
&\int_{B^{\mathbf{r}_k^n}_i} \int_{B^{\mathbf{r}_k^n}_j}
 \frac{(\xi^{\mathbf{r}_k^n}_i)^2(x)(\xi^{\mathbf{r}_k^n}_j)^2(y)}{4\pi|x-y|}\,dx\,dy\\
&=(a^n_i)^2(a^n_j)^2\int_{B^{\bar{\mathbf{r}}_k}_i}
\int_{B^{\bf\bar{r}_k}_j} \frac{(v^{\bf\bar{r}_k}_i)^2(x)(v^{\bar{\mathrm{r}}_k}_j
)^2(y)}{4\pi|x-y|}\,dx\,dy+o(1),
\end{split} 
\\
\int_{B^{\mathbf{r}_k^n}_i}|\xi^{\mathbf{r}_k^n}_i|^p\,dx
=(a^n_i)^p\int_{B^{\bar{\mathbf{r}}_k}_i}| 
v^{\bar{\mathbf{r}}_k}_i|^p\,dx+o(1).
\end{gather*}
Since $(\xi^{\mathbf{r}_k^n}_1,\dots,\xi^{\mathbf{r}_k^n}_{k+1})
\in{\mathcal{N}_k}^{\mathbf{r}^n_k}$,
 we have
\begin{equation} \label{eq:3.3}
\begin{aligned}
&(a^n_i)^2\|v^{{\bar{\mathbf{r}}_k}}_i\|^2_{B^{\mathbf{r}_k^n}_i}
+b(a^n_i)^2\int_{B^{\bf{\bar r_k}}_i}|\nabla v^{\bar{\mathbf{r}}_k}_i|^2\,dx
\sum_{i=1}^{k+1}(a^n_j)^2\int_{B^{\bf\bar{r}_k}_j}|\nabla v^{\bf\bar{r}_k}_j|^2\,dx
\\
&+\sum_{i\neq j}^{k+1}(a^n_i)^2(a^n_j)^2\int_{B^{\bf\bar{r}_k}_i}
\int_{B^{\bf\bar{r}_k}_j}\frac{( v^{\bar{\mathbf{r}}_k}_i)^2(x)( v^{\bar{\mathbf{r}}_k}_j)^2(y)}
{4\pi|x-y|}\,dx\,dy-(a^n_k)^{p}\int_{B^{\bar{\mathbf{r}}_k}_i}
|v^{\bar{\mathbf{r}}_k}_i|^p\,dx\\
&=o(1)
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq:3.4}
\begin{aligned}
&\|v^{{\bar{\mathbf{r}}_k}}_i\|^2_{B^{\mathbf{r}_k^n}_i}
+b\int_{B^{\bf{\bar r}_k}_i}|\nabla v^{\bar{\mathbf{r}}_k}_i|^2\,dx
\sum_{j=1}^{k+1}\int_{B^{\bar{\mathbf{r}}_k}_j}|\nabla v^{\bar{\mathbf{r}}_k}_j|^2\,dx
\\
& +\sum_{i\neq j}^{k+1}\int_{B^{\bar{\mathbf{r}}_k}_i}\int_{B^{\bar{\mathbf{r}}_k}_j}
\frac{( v^{\bar{\mathbf{r}}_k}_i)^2(x)( v^{\bar{\mathbf{r}}_k}_j)^2(y)}{4\pi|x-y|}\,dx\,dy
-\int_{B^{\bf\bar{\rho}_k}_i}|v^{\bar{\mathbf{r}}_k}_i|^p\,dx=o(1)
\end{aligned}
\end{equation}
for $i=1,\dots,k+1$.
From \eqref{eq:3.3} and \eqref{eq:3.4}, we deduce that $\lim_{n\to\infty}a^n_i=1$.
Thus,
\begin{align*}
\chi({\bar{\mathbf{r}}_k})
&=I^{\bar{\mathbf{r}}_k}(v^{\bar{\mathbf{r}}_k}_1,\dots,v^{\bar{\mathbf{r}}_k}_{k+1})
=\limsup _{n\to\infty} I^{\bar{\mathbf{r}}_k}(v^{\bar{\mathbf{r}}_k}_1,\dots,
v^{\bar{\mathbf{r}}_k}_{k+1})
\\
&\geq\limsup_{n\to\infty}  I^{\bar{\mathbf{r}}_k}(\xi^{\bar{\mathbf{r}}_k}_1,\dots,
\xi^{\bar{\mathbf{r}}_k}_{k+1})
=\limsup_{n\to\infty} \chi({\mathbf{r}^n_k}).
\end{align*}
This also implies
\begin{equation} \label{eq:3.5}
\begin{gathered}
\limsup_{n\to\infty} \|v^{\mathbf{r}_k^n}_i\|^2_{B^{\mathbf{r}_k^n}_i}<\infty, \\
\limsup_{n\to\infty} \int_{B^{\mathbf{r}_k^n}_i}|\nabla \nu^{\mathbf{r}_k^n}_i|^2\,dx
\int_{B^{\bf\rho_k^n}_j}|\nabla v^{\mathbf{r}_k^n}_j|^2\,dx<\infty, \\
\limsup_{n\to\infty} \int_{B^{\mathbf{r}_k^n}_i}
\int_{B^{\mathbf{r}_k^n}_j}\frac{(v^{\mathbf{r}_k^n}_i)^2(x)(v^{\mathbf{r}_k^n}_j)^2(y)}
{4\pi|x-y|}\,dx\,dy<\infty.
\end{gathered}
\end{equation}

Next, we prove $\chi({\bar{\mathbf{r}}_k})\leq  \liminf _{n\to\infty}\chi({\mathbf{r}_k^n})$.
As above, we define
 $\xi^{\mathbf{r}_k^n}_i :[\bar r_{i-1},\bar r_i]\to\mathbb{R}$ by
\[
\xi^{\mathbf{r}_k^n}_i=a^n_iv^{\mathbf{r}_k^n}_i
\Big(\frac{r^n_i-r^n_{i-1}}{\bar r_i-\bar r_{i-1}}(t-\bar r_{i-1})+r^n_{i-1}\Big)
\]
for $i=1,\dots,k+1$ and
\[
\xi^{\mathbf{r}_k^n}_{k+1}=a^n_{k+1}v^{\mathbf{r}^n_k}_{k+1}
\Big(\frac{r^n_k}{\bar r_k}\Big)t,
\]
where $r^n_0=0, r^n_{k+1}=\infty$ and each $(a^n_1,\dots,a^n_{k+1})$
is the unique $(k+1)$-tuple of positive real numbers such that
$(\xi^{\mathbf{r}_k^n}_1,\dots,\xi^{\mathbf{r}_k^n}_{k+1})
\in{\mathcal{N}_k}^{\mathbf{r}^n_k}$.
Then,  from \eqref{eq:3.5} it also follows that
\begin{equation} \label{eq:3.6}
\begin{aligned}
&(a^n_i)^2\|v^{\mathbf{r}^n_k}_i\|^2_{B^{\mathbf{r}_k^n}_i}+b(a^n_i)^2
\int_{B^{\mathbf{r}_k^n}_i}|\nabla v^{\mathbf{r}_k^n}_i|^2\,dx\sum_{j=1}^{k+1}
(a^n_j)^2\int_{B^{\mathbf{r}_k^n}_j}|\nabla v^{\mathbf{r}_k^n}_j|^2\,dx
\\
&+\sum_{i\neq j}^{k+1}(a^n_i)^2(a^n_j)^2\int_{B^{\mathbf{r}_k^n}_i}
\int_{B^{\mathbf{r}^n_k}_j}\frac{(v^{\mathbf{r}^n_k}_i)^2(x)
( v^{\mathbf{r}_k^n}_j)^2(y)}{4\pi|x-y|}\,dx\,dy-(a^n_i)^p
\int_{B^{\mathbf{r}^n_k}_i}|v^{\mathbf{r}^n_k}_i|^p\,dx
\\
&=o(1)
\end{aligned}
\end{equation}
as well as
\begin{equation} \label{eq:3.7}
\begin{aligned}
&\|v^{\mathbf{r}^n_k}_i\|^2_{B^{\mathbf{r}_k^n}_i}
+b\int_{B^{\mathbf{r}_k^n}_i}|\nabla v^{\mathbf{r}_k^n}_i|^2\,dx
\sum_{j=1}^{k+1}\int_{B^{\mathbf{r}_k^n}_j}|\nabla v^{\mathbf{r}_k^n}_j|^2\,dx
\\
& +\sum_{i\neq j}^{k+1}\int_{B^{\mathbf{r}_k^n}_i}
 \int_{B^{\mathbf{r}^n_k}_j}\frac{( v^{\mathbf{r}^n_k}_i)^2(x)
( v^{\mathbf{r}_k^n}_j)^2(y)}{4\pi|x-y|}\,dx\,dy
-\int_{B^{\mathbf{r}_k^n}_i}|v^{\mathbf{r}^n_k}_i|^p\,dx=o(1)
\end{aligned}
\end{equation}
for $i=1,\dots,k+1$.


From \eqref{eq:3.6}, \eqref{eq:3.7} and
$\liminf_{n\to\infty}\|v^{\mathbf{r}^n_k}_i\|_{B^{\mathbf{r}_k^n}_i}>0$
we deduce that $\lim_{n\to\infty}a^n_i=1$ for all $i$.
So we have
\begin{align*}
\chi({\bar{\mathbf{r}}_k})
&=I^{\bar{\mathbf{r}}_k}(v^{\bar{\mathbf{r}}_k}_1,\dots,v^{\bar{\mathbf{r}}_k}_{k+1})
\leq\liminf _{n\to\infty}I^{\mathbf{r}^n_k}(v^{\mathbf{r}^n_k}_1,
\dots,v^{\mathbf{r}^n_k}_{k+1})
\\
&=\liminf_{n\to\infty} I^{\mathbf{r}^n_k}(\xi^{\mathbf{r}^n_k}_1,\dots,
\xi^{\mathbf{r}^n_k}_{k+1})
=\liminf_{n\to\infty}\chi({\mathbf{r}^n_k}).
\end{align*}
We completed the proof of (iii).

As a result, from (i)--(iii) we deduce that there is a minimum point
 $\tilde{\mathbf{r}_k} = (\tilde r_1,\dots,\tilde r_{k})\in{\boldsymbol{\Gamma}_k}$
of $\chi$.
\end{proof}

Next, we show that $(v^{\tilde{\mathbf{r}}_k}_1,\dots ,
v^{\tilde{\mathbf{r}}_k}_{k+1})$ found in the previous lemma,  corresponding
to $(\tilde r_1,\dots,\tilde r_k)\in\boldsymbol{\Gamma}_k $ for every
$k\in\mathbb{N}$, is a $ k$-times sign-changing  and radial solution
of  \eqref{eq:1.1}.


\begin{proof}[Proof of Theorem \ref{thm:1.1}]
  Suppose to the contrary that $\sum_{i=1}^{k+1}v^{\tilde{\mathbf{r}}_k}_i$
is not a solution of \eqref{eq:1.1}, there would exist $l \in\{1,\dots,k\}$
such that
\[
v_-= \lim _{r\to \tilde r^-_l}\frac{dv^{\tilde{\mathbf{r}}_k}_l(r)}{dr}
\neq\lim_{r\to \tilde r^+_l}\frac{d v^{\tilde{\mathbf{r}}_k}_l(r)}{dr}= v_+ ,
\]
Fix a  positive number $\delta$ small enough and set
\begin{equation*}
{\tilde y}(r)=
\begin{cases}
v_l(r),& \text{for } r\in(\tilde r_{l-1}, \tilde r_l-\delta),\\
v_l(\tilde r_l-\delta)+\frac{v_{l+1}(\tilde r+\delta)
-v_l(\tilde r+\delta)}{2\delta},&\text{for }
 r\in(\tilde r_l-\delta, \tilde r_{l+1}+\delta)\\
v_{l+1}(r),& \text{for }  r\in(\tilde r_l+\delta,\tilde r_{l+1}).
\end{cases}
\end{equation*}
There exists ${\tilde s}_l\in(\tilde r_{l-1},\tilde r_{l+1})$ such that
\[
\tilde{y}(r)\big|_{r=\tilde{s}_l}=0
\]
Define the $k+1$-tuple of functions $(\tilde z_l,\dots , \tilde z_{k+1})$,
\begin{gather*}
\tilde z_l(r)=\tilde y(r),\quad \text{for }  r\in(\tilde r_{l-1},{\hat s}_l),\\
\tilde z_i(r)=v_i(r),\quad \text{for } r\in(\tilde r_{i-1},\tilde r_{i+1}) ,
\quad i\neq l, l+1\\
\tilde z_{l+1}(r)=\tilde y(r),\quad \text{for }  r\in({\hat s}_l,\tilde r_{l+1}).
\end{gather*}
By Lemma \ref{lem:2.1}, there exists
$(\tilde a_l,\dots,\tilde a_{k+1})\in{(\mathbb{R}_{>0})}^{k+1}$ such that
\[
(z_1,\dots,z_{k+1})=(\tilde a_1\tilde z_1,\dots,
\tilde a_{k+1}\tilde z_{k+1})\in\mathcal{N}^{\bar{\mathbf{r}}_k }
\]
with $\bar{\mathbf{r}}_k =(\tilde r_1,\dots,\tilde r_l,
\tilde s_l,\tilde r_{l+1}\dots,\tilde r_k)$. On
the other hand, we can verify that
\begin{equation} \label{eq:3.8}
(\tilde a_l,\dots,\tilde a_{k+1})\to(1,\dots,1)
\end{equation}
as $\delta\to0$. Let $W(r)=\sum_{i=1}^{k+1}v_i(r)\in H$ and
$Z(r)=\sum_{i=1}^{k+1}z_i(r)\in H$. Then
\begin{equation} \label{eq:3.9}
I(W)=I^{\tilde{\mathbf{r}}_k}(v^{\tilde{\mathbf{r}}_k}_l,\dots,
v^{\tilde{\mathbf{r}}_k}_{k+1})\leq I^{\bar{\mathbf{r}}_k }
(z^{\bar{\mathbf{r}}_k }_1,\dots,z^{\bar{\mathbf{r}}_k }_{k+1})=I(Z).
\end{equation}
On the other hand, for any $u\in H$, the solution of $-\Delta\varphi=u^2$
is radial and it can be expressed as
\[
\varphi(t)=\frac{1}{t}\int_{0}^{\infty}u^2(s)s\min\{s,t\}\,ds
\]
for $t>0$. Therefore,
\begin{align*}
I(Z)-I(W)&=\Big(\int_{0}^{\tilde r_l-\delta}+\int_{\tilde r_l
+\delta}^{\infty}\Big)\Big(\frac{a}{2}z'^2+\frac{1}{2}V(r)z^2
-\frac{1}{p}|z|^p\Big)r^2\,dr
\\
&\quad +\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}\Big(\frac{a}{2}z'^2
+\frac{1}{2}V(r)z^2 -\frac{1}{p}|z|^p\Big)r^2\,dr
\\
&\quad -\int_{0}^{\infty}\Big(\frac{a}{2}v'^2+\frac{1}{2}V(r)v^2\Big)r^2\,dr
 +\frac{b}{4}\Big(\int_{0}^{\infty}z'r^2\,dr\Big)^2
\\
&\quad -\frac{b}{4}\Big(\int_{0}^{\infty}v'r^2\,dr\Big)^2
 +\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}z^2(s)z^2(t)st\min\{s, t\}\,ds\,dt
\\
&\quad -\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}v^2(s)v^2(t)st
 \min\{s, t\}\,ds\,dt.
\end{align*}
Since  $\frac{1}{p}|u|^p $ is convex,
\begin{equation}\label{eq:3.10}
\frac{1}{p}|z|^p\geq\frac{1}{p}|v|^p+\frac{z^2-\nu^2}{2}vu^{p-2}
\end{equation}
for $zv>0$, and we have
\begin{equation} \label{eq:3.11}
\begin{aligned}
&\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}(\frac{a}{2}z'^2+\frac{1}{2}V(r)z^2
-\frac{1}{p}|z|^p)r^2\,dr\\
&\leq \int_{\tilde r_l-\delta}^{\tilde r_l+\delta}\Big(\frac{a}{2}z'^2
+\frac{1}{2}V(r)z^2 +\frac{1}{2}|v|^p-\frac{1}{2}|v|^p-\frac{1}{p}|z|^p\Big)r^2\,dr.
\end{aligned}
\end{equation}
By the definition of $W$, we have
\begin{equation} \label{eq:3.12}
\begin{aligned}
&\int_{0}^{\infty}(av'^2+V(r)vu^2)r^2\,dr
 +b(\int_{0}^{\infty}v'r^2\,dr)^2 \\
&+\int_{0}^{\infty}\int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}\,ds\,dt\\
&=\int_{0}^{\infty}|v|^pr^2\,dr.
\end{aligned}
\end{equation}
Set
$A=\int_{0}^{\infty}v'^2r^2\,dr$.
By \eqref{eq:3.10}--\eqref{eq:3.11}, we have
\begin{equation} \label{eq:3.13}
\begin{aligned}
I(Z)-I(W)
&\leq A_1 + A_2 + A_3 +\frac{1}{4}\int_{0}^{\infty}
\int_{0}^{\infty}z^2(s)z^2(t)st\min\{s, t\}\,ds\,dt
\\
&\quad +\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}v^2(s)v^2(t)st
\min\{s, t\}\,ds\,dt,
\end{aligned}
\end{equation}
where
\begin{gather*}
A_1 = \Big(\int_{0}^{\tilde r_l-\delta}+\int_{\tilde r_l+\delta}^{\infty}\Big)
\Big(\frac{a+bA}{2}z'^2+\frac{1}{2}V(r)z^2-\frac{1}{2}|z|^2|v|^{p-2}\Big)r^2\,dr,
\\
A_2 = \int_{\tilde r_l-\delta}^{\tilde r_l+\delta}\Big(\frac{a+bA}{2}{z'}^2
+\frac{1}{2}V(r)z^2 -\frac{1}{p}|z|^p+\frac{1}{p}|v|^p\Big)r^2\,dr,
\\
A_3 = \frac{b}{4}\Big(\int_{0}^{\infty}z'r^2\,dr\Big)^2
-\frac{b}{2}\int_{0}^{\infty}v'r^2\,dr\int_{0}^{\infty}z'r^2\,dr
+\frac{b}{4}\Big(\int_{0}^{\infty}v'r^2\,dr\Big)^2.
\end{gather*}
From
\[
v_-= \lim_{\delta\to 0}\frac{v(\tilde r_l-\delta)-v(\tilde r_l)}{-\delta}
\]
we deduce that
$v(\tilde r_l-\delta)=-\delta v_-+o(\delta)$,
and using  that $v$ is a solution of  \eqref{eq:1.1}, that is $v$ satisfies
\begin{equation}\label{eq:3.14}
\begin{gathered}
-\Big(a+b\int_{0}^{\infty}v'^2r^2\,dr\Big)\Big(v''+\frac{2}{r}v'\Big)+(V(r)v
+\varphi v)r^2=r^2|v|^{p-2}v, \\
-\Delta\varphi=v^2, \lim_{|x|\to \infty}\varphi (x) =0, \\
\end{gathered}
\end{equation}
we obtain
\[
(\tilde r_l-\delta)^2v'(\tilde r_l-\delta)=\tilde r^2v_-+o(\delta).
\]
Integrating by part, we find
\begin{equation}
\begin{aligned}\label{eq:3.15}
&\int_{0}^{\tilde r_l-\delta}\frac{a+bA}{2}v'^2r^2\,dr\\
&=\frac{a+bA}{2}\Big(v'vr^2\Big|_{0}^{\tilde r_l-\delta}
-\int_{0}^{\tilde r_l-\delta}v\,d(v'r^2)\Big)
\\
&=\frac{a+bA}{2}v'(\tilde r_l-\delta)v(\tilde r_l-\delta)(\tilde r_l-\delta)^2
-\frac{a+bA}{2}\int_{0}^{\tilde r_l-\delta}v[r^2v''+2rv']\,dr
\\
&=\frac{a+bA}{2}v'(\tilde r_l-\delta)v(\tilde r_l-\delta)
 (\tilde r_l-\delta)^2-\frac{1}{2}\int_{0}^{\tilde r_l-\delta}V(r)v^2r^2\,dr
\\
&\quad -\frac{1}{2}\int_{0}^{\tilde r_l-\delta}\int_{0}^{\infty}v^2(s)v^2(t)st
\min\{s, t\}\,ds\,dt
+\frac{1}{2}\int_{0}^{\tilde r_l-\delta}|v|^{p-2}vr^2\,dr.
\end{aligned}
\end{equation}
We deduce from  \eqref{eq:3.13}--\eqref{eq:3.15} that
\begin{equation} \label{eq:3.16}
\begin{aligned}
&\int_{0}^{\tilde r_l-\delta}\Big(\frac{a+bA}{2}v'^2
+\frac{1}{2}V(r)v^2-\frac{1}{2}v^2|v|^{p-2}\Big)r^2\,dr
\\
&=\frac{a+bA}{2}v'(\tilde r_l-\delta)v(\tilde r_l-\delta)(\tilde r_l-\delta)^2 \\
&\quad -\frac{1}{2}\int_{0}^{\tilde r_l-\delta}\int_{0}^{\infty}v^2(s)v^2(t)st
\min\{s, t\}\,ds\,dt
\\
&=-\frac{a+bA}{2}\tilde r^2_lv^2_-\delta+ o(\delta)
 -\frac{1}{2}\int_{0}^{\tilde r_l-\delta}\int_{0}^{\infty}v^2(s)v^2(t)st
\min\{s, t\}\,ds\,dt.
\end{aligned}
\end{equation}
By \eqref{eq:3.8}, we obtain
\begin{equation} \label{eq:3.17}
\begin{aligned}
&\int_{0}^{\tilde r_l-\delta}\Big(\frac{a+bA}{2}z'^2+\frac{1}{2}V(r)z^2
-\frac{1}{2}z^2|z|^{p-2}\Big)r^2\,dr
\\
&=(1+o(1))\int_{0}^{\tilde r_l-\delta}\Big(\frac{a+bA}{2}v'^2
+\frac{1}{2}V(r)v^2-\frac{1}{2}v^2|v|^{p-2}\Big)r^2\,dr
\\
&=-\frac{a+bA}{2}\tilde r^2_lv^2_-\delta+o(\delta) \\
&\quad -\Big(\frac{1}{2}+o(1)\Big)\int_{0}^{\tilde r_l-\delta}
\int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}\,ds\,dt.
\end{aligned}
\end{equation}
In the same way, we have
\begin{equation}\label{eq:3.18}
\begin{aligned}
&\int_{\tilde r_l+\delta}^{\infty}\Big(\frac{a+bA}{2}z'^2
+\frac{1}{2}V(r)z^2-\frac{1}{2}z^2|z|^{p-2}\Big)r^2\,dr\\
&=-\frac{a+bA}{2}\tilde r^2_lv^2_+\delta+o(\delta) \\
&\quad -\Big(\frac{1}{2}+o(1)\Big)\int_{\tilde r_l+\delta}^{\infty}
\int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}\,ds\,dt.
\end{aligned}
\end{equation}
Equations \eqref{eq:3.17} and \eqref{eq:3.18} lead to
\begin{equation}\label{eq:3.19}
\begin{split}
A_1 &= -\frac{a+bA}{2}\tilde r^2_l\delta(v^2_+ + v^2_-) + o(\delta)\\
&\quad-\Big(\frac{1}{2}+o(1)\Big)\Big(\int_{0}^{\tilde r_l-\delta}
+ \int_{\tilde r_l+\delta}^{\infty}\Big)\int_{0}^{\infty}v^2(s)v^2(t)st
\min\{s, t\}\,ds\,dt.
\end{split}
\end{equation}

Next, we estimate $A_2$. It is readily to verify that
\begin{align}\label{eq:3.20}
\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}(\frac{1}{2}V(r)z^2
-\frac{1}{p}|z|^p+\frac{1}{p}|v|^p)r^2\, dr=o(\delta).
\end{align}
If $r\in[\tilde r_l-\delta,\tilde r_l+\delta]$, we have
\[
\tilde {y}'(t)=\frac{{v(\tilde r_l+\delta)-v(\tilde r_l-\delta)}}{2\delta}.
\]
Therefore,
\begin{equation} \label{eq:3.21}
\begin{aligned}
&\frac{a+bA}{2}\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}{\tilde y}'(t)r^2\,dr\\
&=\frac{a+bA}{2}\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}
\frac{[{v(\tilde r_l+\delta)-v(\tilde r_l-\delta)}]^2}{4\delta^2}r^2\,dr
\\
&=\frac{a+bA}{8}\delta\frac{[v(\tilde r_l+\delta)
-v(\tilde r_l+\delta)]^2}{\delta^2}\frac{(\tilde r_l+\delta)^3
-(\tilde r_l-\delta)^3}{3\delta}.
\end{aligned}
\end{equation}
Since
\[
v_+= \lim_{\delta\to 0}\frac{v(\tilde r_l+\delta)-v(\tilde r_l)}{\delta}
\quad\text{and}\quad
v_-= \lim_{\delta\to 0}\frac{v(\tilde r_l)-v(\tilde r_l-\delta)}{\delta},
\]
we have
\begin{equation}\label{eq:3.22}
(v_+ + v_-)^2=\lim _{\delta\to 0}
\frac{[v(\tilde r_l+\delta)-v(\tilde r_l-\delta)]^2}{\delta^2}.
\end{equation}
Obviously,
\begin{equation} \label{eq:3.23}
\lim_{\delta\to 0}\frac{(\tilde r_l+\delta)^3-(\tilde r_l-\delta)^3}{3\delta}
=\lim_{\delta\to 0}
\frac{3(\tilde r_l+\delta)^2+3(\tilde r_l-\delta)^2}{3}=2{\tilde r_l}^2.
\end{equation}
Hence, by \eqref{eq:3.21}, \eqref{eq:3.22} and \eqref{eq:3.23},  for $\delta\to 0$
we obtain
$$
\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}\frac{a+bA}{2}y'(r)r^2\,dr
=\frac{a+bA}{4}\delta{\tilde r_l}^2(v_+ + v_-)^2+o(\delta).
$$
By \eqref{eq:3.8},
\begin{align}\label{eq:3.24}
\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}\frac{a+bA}{2}z'(r)r^2dr
=\frac{a+bA}{4}\delta{\tilde r_l}^2(v_+ + v_-)^2+o(\delta).
\end{align}
Finally, we claim that
\[
\tilde a_i=1+o(\delta^\frac{1}{2})
\]
as $\delta\to 0$ for $i=1,\dots,k+1$. Suppose by a contradiction,
for some $i_0$, that
\[
\lim_{\delta\to 0^+}|\delta^{-\frac{1}{2}}(\tilde a_{i_0}-1)|
=\max_{1\leq i\leq k+1}\lim_{\delta\to 0^+}|\delta^{-\frac{1}{2}}(\tilde a_{i_0}-1)|
=B\neq 0.
\]
Since $(v_1,\dots,v_{k+1})\in\mathcal{N}^{\tilde{\mathbf{r}}_k}_k$,
$(\tilde a_1\tilde z_1,\dots, \tilde a_{k+1}
\tilde z_{k+1})\in\mathcal{N}^{\bar{\mathbf{r}}_k }_k$ and
$\bar{\mathbf{r}}_k \to \tilde{\mathbf{r}}_k $ as $\delta\to0^+$, we have
\begin{align*}
0&=\lim_{\delta\to 0^+}\delta^{-\frac{1}{2}}(F_{i_0}
(\tilde a_1\tilde z_1,\dots,\tilde a_{k+1}\tilde z_{k+1})-F_{i_0}
(\tilde z_1,\dots,\tilde z_{k+1}))
\Big(\delta^{-\frac{1}{2}}(\tilde a_{i_0}-1)\Big)^{-1}
\\
&\leq 2\|v_{i_0}\|^2_{i_0}+4b\Big(\int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}
|\nabla v_{i_0}|^2\,dx\Big)^2+4b\int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}
|\nabla v_{i_0}|^2\,dx\sum_{j\neq {i_0}}^{k+1}
\int_{B^{\tilde{\mathbf{r}}_k}_{j}}|\nabla v_j|^2\,dx
\\
&\quad +4\int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}
 \int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}\frac{v_{i_0}^2(x)v_{i_0}^2(y)}{4\pi|x-y|}\,dx\,dy
+4\sum_{j\neq {i_0}}^{k+1}\int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}
\int_{B^{\tilde{\mathbf{r}}_k}_j}\frac{v_{i_0}^2(x)v_j^2(y)}{4\pi|x-y|} \\
&\quad -p\int_{B^{\tilde{\mathbf{r}}_k}_{i_0}}|v_{i_0}|^{(p-1)}\,dx,
\end{align*}
where $F_{i_0}$ is given by \eqref{eq:2.7}. This leads to a contradiction
since $(v_1,\dots,v_{k+1})\in\mathcal{N}^{{\hat \rho}}_k$ and $p>4$.
Thus,
\begin{equation} \label{3.25}
\begin{aligned}
A_3&=\frac{b}{4}\Big(\int_{0}^{\infty}v'r^2\,dr-\int_{0}^{\infty}z'r^2\,dr\Big)^2
\\
&=\frac{b}{4}\Big[o(\delta^\frac{1}{2})
\Big(\int_{0}^{\tilde r_l-\delta}+\int_{\tilde r_l+\delta}^{\infty}\Big)v'^2r^2\,dr
+\frac{1}{2}(v_+ + v_-)^2 \tilde r^2_l\delta\\
&\quad -{v_-}^2\tilde r^2_l\delta-{v_+}^2
\tilde r^2_l\delta\Big]^2=o(\delta).
\end{aligned}
\end{equation}
By the estimates on $A_1, A_2$ and $A_3$, we obtain
\begin{align*}
&I(Z)-I(W)\\
&\leq-\frac{a+bA}{4}(v_+ - v_-)^2 \tilde r^2_l\delta+o(\delta)
 + \frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}z^2(s)z^2(t)st\min\{s, t\}\,ds\,dt\\
&\quad +\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}ds\,dt
\\
&\quad -\Big(\frac{1}{2}+o(1)\Big)(\int_{0}^{\tilde r_l-\delta}
 +\int_{\tilde r_l+\delta}^{\infty})v'^2r^2dr)
 \int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}\,ds\,dt
\\
&=-\frac{a+bA}{4}(v_+ - v_-)^2 \tilde r^2_l\delta+o(\delta) \\
&\quad -( \frac{1}{4}+o(1))\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}
 \int_{0}^{\infty}z^2(s)z^2(t)st\min\{s, t\}\,ds\,dt
\\
&\quad -(\frac{1}{4}+o(1))\int_{\tilde r_l-\delta}^{\tilde r_l+\delta}
\int_{0}^{\infty}v^2(s)v^2(t)st\min\{s, t\}\,ds\,dt.
\end{align*}
So we obtain $I(Z)<I(W)$ if $\delta>0$ is sufficiently small, which is
a contradiction to the definition of $W$. Consequently,  $v_-=v_+$, and
then $v_k=\sum_{i=1}^{k+1}v_i^{\tilde r_k}$ is a solution of \eqref{eq:1.1}
changing sign exactly $k$ times. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This work ws supported by the  NNSF of China, No: 11271170 and 11371254;
GAN PO 555 program of Jiangxi.

\begin{thebibliography}{00}

\bibitem{ACM} C. O. Alves, F. J, S. A. Correa, T. F. Ma;
Positive Solutions for a quasilinear of Kirchhoff-type,
{\it Comput. Math. Appl.}, 49 (2005), 85-93.

\bibitem{AR} A. Ambrosetti, D. Ruiz;
 Multiple bound states for the Schr\"{o}dinger-Poisson problem,
 {\it Commun. Contemp. Math.}, 10 (2008), 391-404.

\bibitem{AS} C. O. Alves, M. A. S. Souto;
Existence of least energy nodal solution for a Schrdinger-Poisson system in
bounded domains, {\it Z. Angew. Math. Phys.} 65 (2014), 1153-1166.

\bibitem{AP} A. Arosio, S. Panizzi;
 On the well-podedness of the Kirchhoff string, {\it Trans. Amer. Math. Soc.},
 348 (1996), 305-330.

\bibitem{A} A. Azzollini;
 The Kirchhoff equation in $\mathbb{R}^N$ perturbed by a local nonlinearity,
{\it Differential Intergral Equations}, 25 (2012), 543-554.

\bibitem{BS} C. J. Batkam, J. R. Santos Jnior;
Schr\"{o}dinger-Kirchhoff-Poisson type systems, {\it Commun. Pure Appl. Anal.},
15 (2016),  429-444.

\bibitem{BW} T. Bartsch, M. Willem;
 Infinitely many radial solution of a semilinear elliptic problem on
$\mathbb{R}^N$, {\it Arch. Ration. Mech. Anal.}, 124 (1993), 283-293.

\bibitem{CKW} C. Chen, Y. Kou, T. Wu;
 The Nehari manfold for a Kirchhoff type problem involving sign-changing weight
functions, {\it J. Differential Equations}, 250 (2011), 1876-1908.

\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{DPS} Y. Deng, S. Peng, W. Shuai;
 Existence of asympototic behavior of nodal solutions for the Kirchhoff-type problems,
 {\it Journal of Functional Analysis}, 269 (2015), 3500-3527.

\bibitem{FN} G. M. de Figueiredo, R. G. Nascimento;
Existence of a nodal solution with
minimal energy for a Kirchhoff equation, {\it Math. Nachr.}, 288 (2015), 48-60.

\bibitem{HZ}  X. He, W. Zou;
Existence and concentration behavior of positive solutions for a Kirchhoff Equation
in  $\mathbb{R}^3$, {\it J. Differential Equations}, 2 (2012), 1813-1834.

\bibitem{IV} I. Ianni, G. Vaira;
 On concentration of positive bound states for the Schr\"{o}dinger-Poisson problem
with potentials, {\it Adv. Nonlinear Stud.}, 8(2008), 573-595.

\bibitem{K} G. Kirchhoff; \emph{Mechanik}, Teubner, Leipzig, 1883.

\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} J. L. Lions;
\emph{On some questions in boundary value problems of mathematical physics}, in:
Contempo-rary Developments in Continuum Mechanics and Partial
 Differential Equations, in: North-Holland Math. Stud., vol.30, North-Holland,
 Amsterdam, New York, 1978, pp. 284-346.

\bibitem{LY} G. B. Li, H. Y. Ye;
 Existence of positive ground state solutions for the nonlinear Kirchhoff type
equtions in $\mathbb{R}^3$, {\it J. Differential Equation}, 257 (2014), 566-600.

\bibitem{N} Z. Nehari;
Characteristic values associated with a class of non-linear second-order
differential equations, {\it Acta Math.}, 105 (1961), 141-175.

\bibitem{PZ} K. Perera, Z. Zhang;
 Nontrivial solutions of  Kirchhoff-type  problems via the Young index,
{\it J. Differential Equations}, 211 (2006), 246-255.

\bibitem{R} D. Ruiz;
 The Schr\"{o}dinger-Poisson equation under the effect of a nonlinear local term,
{\it J. Funct. Anal.}, 237 (2006), 655-674.

\bibitem{R1} Ruiz, D.;
 On the Schr\"{o}dinger-Poission-Slater system: behavior of minimizers,
radial and nonradial cases, {\it Arch. Rational Mech. Anal.}, 198, 349-368 (2010).

\bibitem{S} J. C. Slater;
 A simplification of the Hartree-Fock method, {\it Phys. Review},
81 (1951), 385-390.

\bibitem{Str} W. A. Strauss;
 Existence of solitary waves in higer dimensions, {\it Comm. Math. Phys.},
55 (1977), 149-162.

\bibitem{WTXZ} J. Wang, L. Tian, J. Xu, F. Zhang;
 Multiplicity and concentration of positive solutions for a Kirchhoff type
problem with critical growth, {\it J. Differential Equation}, 253 (2012), 2314-2351.


\bibitem{WZ}  Z. Wang, H.-S. Zhou;
 Sign-changing solutions for the nonlinear Schr\"{o}inger-Poisson system in $R^3$,
{\it Calc. Var. PDE}, 52(2015), 927-943.

\bibitem{ZZ} L. Zhao, F. Zhao;
On the existence of solutions for the Schr\"{o}dinger-Poisson
equations, {\it J. Math. Anal. Appl.}, 346 (2008), 155-169.

\end{thebibliography}

\end{document}