Existence of Nodal Solutions with Arbitrary Number of Nodes for Kirchhoff Type Equations

Abstract

In this paper, we are interested in the following Kirchhoff type equation

$$\begin{aligned} \left\{ \begin{aligned}&\bigg [a+\lambda \bigg (\int _{{\mathbb {R}}^3}(|\nabla u|^2+V(|x|)u^2)dx\bigg )^{\alpha }\bigg ]\bigg (-\Delta u+V(|x|)u\bigg )=|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3),\\ \end{aligned}\right. \end{aligned}$$

(0.1)

where $a,\lambda >0,\alpha \in (0,2)$ and $p\in (2\alpha +2,6).$ The potential V(|x|) is radial and bounded below by a positive number. By introducing the Gersgorin Disc’s theorem, we prove that for each positive integer k, Eq. (0.1) has a radial nodal solution $U_k^{\lambda }$ with exactly k nodes. Moreover, the energy of $U_k^{\lambda }$ is strictly increasing in k and for any sequence $\{\lambda _n\}$ with $\lambda _n\rightarrow 0^+,$ up to a subsequence, $U_k^{\lambda _n}$ converges to $U_k^0$ in $H^{1}({\mathbb {R}}^3)$, which is also a radial nodal solution with exactly k nodes to the classical Schrödinger equation

$$\begin{aligned} \left\{ \begin{aligned}&-a\Delta u+aV(|x|)u=|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3). \end{aligned}\right. \end{aligned}$$

Our results can be viewed as an extension of Kirchhoff equation concerning the existence of nodal solutions with any prescribed numbers of nodes.

1 Introduction

In this paper, we consider the following Kirchhoff type problem

$$\begin{aligned} \left\{ \begin{aligned}&\bigg [a+\lambda \bigg (\int _{{\mathbb {R}}^3}(|\nabla u|^2+V(|x|)u^2)dx\bigg )^{\alpha }\bigg ]\bigg (-\Delta u+V(|x|)u\bigg )=|u|^{p-2}u,\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3),\\ \end{aligned}\right. \nonumber \\ \end{aligned}$$

(1.1)

where $a,\lambda >0,\alpha \in (0,2),p\in (2\alpha +2,6)$ and the potential function $V\in C([0,\infty ),{\mathbb {R}})$ is radial and bounded below by a positive number. When $\alpha =1$ and $V(x)\equiv b>0$, (1.1) is reduced to the following Kirchhoff problem

$$\begin{aligned} \begin{aligned} \bigg [a+\lambda \bigg (\int _{{\mathbb {R}}^3}(|\nabla u|^2+bu^2)dx\bigg )\bigg ]\bigg (-\Delta u+bu\bigg )=|u|^{p-2}u,\quad \text{ in } {\mathbb {R}}^3, \end{aligned} \end{aligned}$$

(1.2)

which has been studied by Li et al. [17] on the existence of positive solutions, see also [3, 6] for more details about the problem (1.2).

In the last two decades, the existence of positive solutions, multiple solutions and sign-changing solutions to the following Kirchhoff type problem on an open bounded domain $\Omega \subset {\mathbb {R}}^N$ with boundary $\partial \Omega$

$$\begin{aligned} \left\{ \begin{aligned}&-\bigg [a+b\int _{\Omega }|\nabla u|^2\bigg ]\Delta u=f(x,u),&\text{ in } \Omega ,\\&u=0,&\text{ on } \partial \Omega ,\\ \end{aligned}\right. \end{aligned}$$

(1.3)

has been extensively investigated by making use of the variational method. One can refer to [4, 5, 12,13,14,15, 20,21,22, 25,26,27, 33, 34] and references therein. For the Kirchhoff type problem in the whole space ${\mathbb {R}}^N$, Li and Ye [19] considered

$$\begin{aligned} \left\{ \begin{aligned}&-\bigg [a+b\int _{{\mathbb {R}}^3}|\nabla u|^2\bigg ]\Delta u+V(|x|)u=f(x,u),\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3),\quad u>0, \end{aligned}\right. \end{aligned}$$

(1.4)

where $f(x,u)=u^{p-2}u$ with $p\in (3,6).$ Under certain assumptions on the potential V(x), they proved that (1.4) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. For related problems like (1.4), we refer to [1, 7, 9, 16, 27, 29, 32] and references cited therein.

Recently, the existence of sign-changing solutions to the Kirchhoff type problem in ${\mathbb {R}}^N$ has attracted much attention. Deng et al. [8] and Guo et al. [10] obtained the existence and asymptotic behaviors of nodal solutions with a prescribed number of nodes for problem (1.4) under some suitable assumptions on the nonlinearity f(x, u). Corresponding to the classical pure power nonlinearity model $f(x,u)=|u|^{p-2}u$, their main results in [8, 10] solve the following equation

$$\begin{aligned} \left\{ \begin{aligned}&-\bigg [a+b\int _{{\mathbb {R}}^3}|\nabla u|^2dx\bigg ]\Delta u+V(|x|)u=|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^{3}\\&u\ \in H^{1}({\mathbb {R}}^3), \end{aligned}\right. \end{aligned}$$

(1.5)

for the case $p\in (4,6)$, see [18, 24, 30, 31] for more related results. However, the presence of nonlocal term $\lambda \bigg (\int _{{\mathbb {R}}^3}(|\nabla u|^2+V(|x|)u^2)dx\bigg )^{\alpha }$ in (1.1) with $\alpha \in (0,2)$ makes this problem more complicated. Then a natural question arises: can one find nodal solutions with any prescribed number of nodes for problem (1.1)? In this paper, we shall answer this question. To the best of our knowledge, this problem still remains unsolved.

In order to illustrate our results clearly, we need the following notations. Throughout this paper, we set the radial Sobolev space $H_r^1({\mathbb {R}}^3)=\{u\in H^1({\mathbb {R}}^3):u(x)=u(|x|)\}$ and let the action space

$$\begin{aligned} H_V:=\left\{ u\in H_{r}^{1}({\mathbb {R}}^3):\int _{{\mathbb {R}}^3}(|\nabla u|^{2}+V(|x|)u^{2})dx<+\infty \right\} \end{aligned}$$

be endowed with norm $\Vert u\Vert =\left( \int _{{\mathbb {R}}^3}(|\nabla u|^{2}+V(|x|)u^{2})dx\right) ^{1/2}.$ As usual, the energy functional $I_{\lambda }:H_V\rightarrow {\mathbb {R}}$ associated with (1.1) is defined by

$$\begin{aligned} I_{\lambda }(u):=\frac{a}{2}\Vert u\Vert ^2+\frac{\lambda }{2\alpha +2}\Vert u\Vert ^{2\alpha +2}-\frac{1}{p}\int _{{\mathbb {R}}^3}|u|^p. \end{aligned}$$

(1.6)

Obviously, $I_{\lambda }\in C^2(H_V,{\mathbb {R}})$ and

$$\begin{aligned} \langle I_{\lambda }'(u),u\rangle =a\Vert u\Vert ^2+\lambda \Vert u\Vert ^{2\alpha +2}-\int _{{\mathbb {R}}^3}|u|^p. \end{aligned}$$

Then we define the usual Nehari manifold

$$\begin{aligned} {\mathcal {N}}=\left\{ u\in H_V\backslash \{0\}: \langle I_{\lambda }'(u),u\rangle =0\right\} , \end{aligned}$$

(1.7)

and the ground state energy

$$\begin{aligned} m:=\inf _{{\mathcal {N}}}I_{\lambda }(u). \end{aligned}$$

(1.8)

By [6, Theorem 1.1], there exists a ground state solution $U_0\in {\mathcal {N}}$ of (1.1) such that

$$\begin{aligned} m=I_{\lambda }(U_0)>0. \end{aligned}$$

(1.9)

For $k\in {\mathbb {N}}^*$ and $0=:r_0<r_1<\cdots<r_{k}<r_{k+1}:=+\infty$, we denote by ${{\textbf {r}}}_{k}=(r_1,\ldots ,r_k)$ and

$$\begin{aligned} \begin{aligned}&B_{1}^{{{\textbf {r}}}_k}:=\left\{ x\in {\mathbb {R}}^3:0\le |x|<r_{1}\right\} ,\\ &B_{i}^{{{\textbf {r}}}_k}:=\left\{ x\in {\mathbb {R}}^3:r_{i-1}<|x|<r_{i}\right\} ,\, i=2,\ldots ,k+1. \end{aligned} \end{aligned}$$

Obviously, $B_{1}^{{{\textbf {r}}}_k}$ is a ball, $B_{2}^{{{\textbf {r}}}_k},\ldots ,B_{k}^{{{\textbf {r}}}_k}$ are annulus and $B_{k+1}^{{{\textbf {r}}}_k}$ is the complement of a ball. Then we define the Nehari type set

$$\begin{aligned} {\mathcal {N}}_k=\left\{ u\in \ H_V:\text{ there } \text{ exists }\,{{\textbf {r}}}_{k} \text{ s.t. } u_i\ne 0 \text{ in } B_i^{{{\textbf {r}}}_k}, \langle I'_{\lambda }(u),u_{i}\rangle =0,\, i=1,\ldots ,k+1\right\} ,\nonumber \\ \end{aligned}$$

(1.10)

and the infimum level

$$\begin{aligned} c_k=\inf _{u\in {\mathcal {N}}_k}I_{\lambda }(u), \end{aligned}$$

(1.11)

where $u_i=u$ in $B_i^{{{\textbf {r}}}_k}$ and $u_i=0$ on $\partial B_i^{{{\textbf {r}}}_k}.$

Our existence result is as follows.

Theorem 1.1

For each $k\in {\mathbb {N}}^*$, problem (1.1) admits a radial nodal solution $U_k\in {\mathcal {N}}_k$ which changes sign exactly k-times and $I_{\lambda }(U_k)=c_k$.

The next result shows that the energy of $U_k$ obtained in Theorem 1.1 increases with the number of nodes.

Theorem 1.2

Under the hypotheses of Theorem 1.1, the energy of $U_k$ is strictly increasing in k. Namely,

$$\begin{aligned} I_{\lambda }(U_{k+1})>I_{\lambda }(U_k)\quad \text{ for } \text{ all } k\in {\mathbb {N}}^*. \end{aligned}$$

Moreover, $I_{\lambda }(U_{k+1})>(k+1)I_{\lambda }(U_{0}).$

Since $U_k$ obtained in Theorem 1.1 depends on $\lambda$, we denote $U_k$ by $U_k^{\lambda }$ to emphasize this dependence. The last result shows the asymptotic behavior of $U_k^{\lambda }$ as $\lambda \rightarrow 0^+$.

Theorem 1.3

Under the assumptions of Theorem 1.1, for any sequence $\{\lambda _n\}$ with $\lambda _n\rightarrow 0^+$ as $n\rightarrow \infty$, up to a subsequence, $U_k^{\lambda _n}$ converges to $U_k^0$ strongly in $H_V$ as $n\rightarrow \infty$, where $U_k^0$ is a least energy radial nodal solution among all the nodal solutions having exactly k nodes to the following equation

$$\begin{aligned} -a\Delta u+aV(|x|)u=|u|^{p-2}u. \end{aligned}$$

(1.12)

This paper is organized as follows. In Sect. 2, we give the variational framework of problem (1.1) and some preliminary lemmas. Section 3 is devoted to the proof of the existence of nodal solutions with a prescribed number of nodes. In Sect. 4, we study the energy comparison and asymptotic behaviors of those nodal solutions of (1.1).

2 Preliminaries

In this section, we give some notations and recall some useful lemmas. For each $k\in {\mathbb {N}}^*,$ we define

$$\begin{aligned} \Gamma _k=\left\{ {{\textbf {r}}}_k=(r_1,\ldots ,r_{k})\in (0,\infty )^k\,\ 0=:r_0<r_1<\cdots<r_k<r_{k+1}:=\infty \right\} . \end{aligned}$$

For a fixed ${{\textbf {r}}}_k\in \Gamma _k$ and thereby a family of annulus $\{{B}_i^{{{\textbf {r}}}_k}\}_{i=1}^{k+1}$, we define a Hilbert space

$$\begin{aligned} H_{i}^{{{\textbf {r}}}_k}:=\bigg \{u\in H_{0}^{1}(B_{i}^{{{\textbf {r}}}_k}):\ u(x)=u(|x|),u(x)=0\ \text{ for } \ x\in \partial B_{i}^{{{\textbf {r}}}_k}\bigg \} \end{aligned}$$

endowed with the norm $\Vert u\Vert _{i}=\left( \int _{B_{i}^{{{\textbf {r}}}_k}}(|\nabla u|^{2}+V(|x|)u^{2})dx\right) ^{1/2}.$ Now, let the product space be

$$\begin{aligned} {\mathcal {H}}_k^{{{\textbf {r}}}_k}=H_{1}^{{{\textbf {r}}}_k}\times \cdots \times H_{k+1}^{{{\textbf {r}}}_k}, \end{aligned}$$

and we introduce an energy functional $E_{\lambda }:{\mathcal {H}}_k^{{{\textbf {r}}}_k}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \begin{aligned} E_{\lambda }(u_1,\ldots ,u_{k+1}):=\frac{a}{2}\sum _{i=1}^{k+1}\Vert u_{i}\Vert _{i}^{2}+\frac{\lambda }{2\alpha +2}\bigg ( \sum _{i=1}^{k+1}\Vert u_i\Vert _i^2\bigg )^{\alpha +1}-\frac{1}{p}\sum _{i=1}^{k+1}\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p. \end{aligned}\nonumber \\ \end{aligned}$$

(2.1)

It is obvious that

$$\begin{aligned} E_{\lambda }(u_1,\ldots ,u_{k+1})=I_{\lambda }\left( \sum _{i=1}^{k+1}u_i\right) . \end{aligned}$$

(2.2)

If $(u_1,\ldots ,u_{k+1})$ is a critical point $E_{\lambda }$, then each component $u_{i}$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\bigg [a+\lambda \bigg (\sum _{j=1}^{k+1}\Vert u_j\Vert \bigg )^{2\alpha }\bigg ]\bigg (-\Delta u_i+V(|x|)u_i\bigg )=|u_i|^{p-2}u_i\quad&x\in B_i^{{{\textbf {r}}}_k},\\&u_i=0\quad&x\notin B_i^{{{\textbf {r}}}_k}.\\ \end{aligned}\right. \end{aligned}$$

(2.3)

Note that

$$\begin{aligned} \langle E_{\lambda }'(u_1,\ldots ,u_{k+1}),u_{i}\rangle =a\Vert u_{i}\Vert _{i}^{2}+\lambda \Vert u_i\Vert _i^2\bigg (\sum _{j=1}^{k+1}\Vert u_j\Vert _j^2\bigg )^{\alpha }-\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p. \end{aligned}$$

For each ${{\textbf {r}}}_k\in \Gamma _k,$ we define another Nehari type set

$$\begin{aligned} {\mathcal {M}}_k^{{{\textbf {r}}}_k}:=\bigg \{(u_1,\ldots ,u_{k+1})\in \ {\mathcal {H}}_k^{{{\textbf {r}}}_k}\,\ u_i\ne 0,\langle E_{\lambda }'(u_1,\ldots ,u_{k+1}),u_{i}\rangle =0,\, i=1,\ldots ,k+1\bigg \}.\nonumber \\ \end{aligned}$$

(2.4)

In the following, we shall prove the non-empty of ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ by introducing two important lemmas. The first lemma is a corollary of the Gersgorin Disc’s Theorem [28].

Lemma 2.1

[11, Lemma 2.3] For any $a_{ij}=a_{ji}>0$ with $i\ne j$ and $s_i>0$ with $i=1,\ldots ,m$, if the matrix $B:=(b_{ij})_{m\times m}$ is defined by

$$\begin{aligned} b_{ij}=\left\{ \begin{aligned}&-\sum _{l\ne i}\frac{s_la_{il}}{s_i}\quad i=j,\\&a_{ij}>0\qquad \ i\ne j, \end{aligned}\right. \end{aligned}$$

then $(b_{ij})_{m\times m}$ is a negative semi-definite symmetric matrix.

Lemma 2.2

[30, Lemma 2.3] If $f\in C^2({\mathbb {R}}^m,{\mathbb {R}})$ is a strictly concave function and has a critical point $\bar{{{\textbf {s}}}}:=({\bar{s}}_1,\ldots ,{\bar{s}}_{m})$ in ${\mathbb {R}}^{m}$, then $\bar{{{\textbf {s}}}}$ is the unique critical point of f in ${\mathbb {R}}^{m}.$

Now we are ready to prove the non-empty of the set ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$.

Lemma 2.3

For each $(u_1,\ldots ,u_{k+1})\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}$ with $u_i\ne 0$ for $i=1,\ldots ,k+1$, there exists a unique $(k+1)$ tuple $(t_1,\ldots ,t_{k+1})$ of positive numbers such that $(t_1u_1,\ldots ,t_{k+1}u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}$.

Proof

For fixed $(u_1,\ldots ,u_{k+1})\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}$ with $u_i\ne 0$, $(t_1u_1,\ldots ,t_{k+1}u_{k+1})$ belongs to ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ if and only if

$$\begin{aligned} at_i^2\Vert u_i\Vert _i^2+\lambda t_i^2\Vert u_i\Vert _i^2\left( \sum _{j=1}^{k+1}t_j^2\Vert u_j\Vert _j^2\right) ^{\alpha }-t_i^p\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p=0 \end{aligned}$$

(2.5)

for each $i={1,\ldots , k+1}.$ Hence, it suffices to verify that there is a unique $(k+1)$ tuple $(t_1,\ldots ,t_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}$ satisfying (2.5).

Define a new function $g:({\mathbb {R}}_{>0})^{k+1}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \begin{aligned} g(s_1,\ldots , s_{k+1}):&=E\left( s^{\frac{1}{p}}_1u_1,\ldots ,s^{\frac{1}{p}}_{k+1}u_{k+1}\right) \\&=\frac{a}{2}\sum ^{k+1}_{i=1}s^{\frac{2}{p}}_i\Vert u_i\Vert _i^2+\frac{\lambda }{2(\alpha +1)} \bigg (\sum ^{k+1}_{i=1}s^{\frac{2}{p}}_i\Vert u_i\Vert _i^{2}\bigg )^{\alpha +1}\\&\quad -\frac{1}{p} \sum ^{k+1}_{i=1}s_i\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p. \end{aligned} \end{aligned}$$

(2.6)

According to (2.6), we see that $g(s_{1},\ldots ,s_{k+1})\rightarrow -\infty$ uniformly as $|(s_{1},\ldots ,s_{k+1})|\rightarrow \infty$, and $g(s_{1},\ldots ,s_{k+1})\rightarrow 0$ uniformly as $|(s_{1},\ldots ,s_{k+1})|\rightarrow 0.$

Some direct computations show that the partial derivatives of g satisfy

$$\begin{aligned} g'_{s_i}(s_1,\ldots , s_{k+1})&=\frac{a}{p}s^{\frac{2-p}{p}}_i\Vert u_i\Vert _i^2+\frac{\lambda }{p} s^{\frac{2-p}{p}}_i\left( \sum ^{k+1}_{j=1}s^{\frac{2}{p}}_j\Vert u_j\Vert _j^2\right) ^\alpha \Vert u_i\Vert _i^2 -\frac{1}{p}\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p,\nonumber \\ g''_{s_i s_i}(s_1,\ldots , s_{k+1})&=\frac{a(2-p)}{p}s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2+s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2 \left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1}\nonumber \\&\quad \left[ \left( \frac{2\lambda (1+\alpha )}{p^2}-\frac{\lambda }{p}\right) \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right] \nonumber \\&\qquad -\frac{2\lambda \alpha }{p^2}s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2\left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1}\left( \sum ^{k+1}_{j\ne i}s^{\frac{2}{p}}_j\Vert u_j\Vert _j^2\right) ,\nonumber \\ g''_{s_is_j}(s_1,\ldots , s_{k+1})&=\frac{2\lambda \alpha }{p^2}s^{\frac{2-p}{p}}_i\Vert u_i\Vert _i^2\left( \sum ^{k+1}_{l=1} s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1}s^{\frac{2-p}{p}}_j\Vert u_j\Vert _j^2. \end{aligned}$$

(2.7)

Let

$$\begin{aligned} A_{ii}&=\frac{a(2-p)}{p}s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2+s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2 \left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1}\\&\quad \left[ \left( \frac{2\lambda (1+\alpha )}{p^2}-\frac{\lambda }{p}\right) \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right] ,\\ B_{ii}&=-\frac{2\lambda \alpha }{p^2}s^{\frac{2-2p}{p}}_i\Vert u_i\Vert _i^2 \left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1} \left( \sum ^{k+1}_{j\ne i}s^{\frac{2}{p}}_j\Vert u_j\Vert _j^2\right) \\&=-\sum ^{k+1}_{j\ne i}\frac{s_j}{s_i}\left( \frac{2\lambda \alpha }{p^2} s^{\frac{2-p}{p}}_i\Vert u_i\Vert _i^2\left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1}s^{\frac{2-p}{p}}_j\Vert u_j\Vert _j^2\right) ,\\ A_{ij}&=0\,\ and \,\ B_{ij}=\frac{2\lambda \alpha }{p^2}s^ {\frac{2-p}{p}}_i\Vert u_i\Vert _i^2\left( \sum ^{k+1}_{l=1}s^{\frac{2}{p}}_l\Vert u_l\Vert _l^2\right) ^{\alpha -1} s^{\frac{2-p}{p}}_j\Vert u_j\Vert _j^2 \,\ while \,\ i\ne j.\\ \end{aligned}$$

Then the matrix

$$\begin{aligned} (g''_{s_is_j}(s_1,\ldots , s_{k+1}))_{(k+1)\times (k+1)}=(A_{ij})_{(k+1)\times (k+1)}+(B_{ij})_{(k+1)\times (k+1)}. \end{aligned}$$

Moreover, it follows from Lemma 2.1 that the matrix $(g''_{s_is_j}(s_1,\ldots , s_{k+1}))_{(k+1)\times (k+1)}$ is negative definite at each point $(s_1,\ldots ,s_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}$. So g is a strictly concave function in $({\mathbb {R}}_{>0})^{k+1}.$ By Lemma 2.2, we deduce that g has a unique critical point $({\bar{s}}_1,\ldots ,{\bar{s}}_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}$. Letting ${\bar{s}}_i=t_i^p$, we conclude from (2.5) and (2.7) that

$$\begin{aligned} \langle E_{\lambda }'(t_1u_1,\ldots ,t_{k+1}u_{k+1}),t_iu_{i}\rangle =pt_i^pg_{s_i}(t_1^p,\ldots ,t_{k+1}^p)=0. \end{aligned}$$

The proof is finished. $\square$

We define $\phi :({\mathbb {R}}_{\ge 0})^{k+1}\rightarrow {\mathbb {R}}$ by $\phi (c_1,\ldots ,c_{k+1})=E_{\lambda }(c_1u_1,\ldots ,c_{k+1}u_{k+1}),$ where $(u_1,\ldots ,u_{k+1})\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}.$ Then we get the following corollary.

Corollary 2.4

For fixed $(u_1,\ldots ,u_{k+1})\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}$ with $u_i\ne 0$ for $i=1,\ldots ,k+1$, $\phi$ has a unique maximum point $(t_1,\ldots ,t_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}$. Moreover, $\frac{\partial \phi }{\partial c_i}(t_1,\ldots ,t_{i-1}, c_i,t_{i+1},t_{k+1})>0$ if $c_i<t_i$ and $\frac{\partial \phi }{\partial c_i}(t_1,\ldots ,t_{i-1},c_i,t_{i+1},\ldots , t_{k+1})<0$ if $c_i>t_i$.

Proof

We see that

$$\begin{aligned} \begin{aligned} \phi (c_{1},\ldots ,c_{k+1}):&=E_{\lambda }(c_1u_1,\ldots ,c_{k+1}u_{k+1})\\&=\frac{a}{2}\sum ^{k+1}_{i=1}\Vert c_i u_i\Vert _i^2+\frac{\lambda }{2(\alpha +1)}\bigg (\sum ^{k+1}_{i=1}\Vert c_i u_i\Vert _i^{2}\bigg )^{\alpha +1}\\&\quad -\frac{1}{p}\sum ^{k+1}_{i=1}\int _{B_i^{{{\textbf {r}}}_k}}|c_i u_i|^p. \end{aligned} \end{aligned}$$

(2.8)

Obviously, $\phi$ is continuous. From the proof of Lemma 2.3, $(t_1,\ldots ,t_{k+1})$ is the unique critical point of $\phi$ in $({\mathbb {R}}_{>0})^{k+1}$. Due to the fact that $p\in (2+2\alpha ,6)$, we have $\phi (c_{1},\ldots ,c_{k+1})\rightarrow -\infty$ as $|(c_{1},\ldots ,c_{k+1})|\rightarrow \infty$ and $\phi (c_{1},\ldots ,c_{k+1})\rightarrow 0$ as $|(c_{1},\ldots ,c_{k+1})|\rightarrow 0.$ This implies that $\phi$ admits a unique maximum point $(t_1,\ldots ,t_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}.$ Then we obtain that for each i, 

$$\begin{aligned} \begin{aligned}&\frac{\partial \phi }{\partial c_i}(t_1,\ldots ,c_i,\ldots ,t_{k+1})\\&\quad =a c_i\Vert u_i\Vert _i^2+\lambda \bigg (\sum ^{k+1}_{j\ne i}t_j^2\Vert u_j\Vert _j^{2}+c_i^2\Vert u_i\Vert _i^{2}\bigg )^{\alpha }c_i\Vert u_i\Vert _i^{2}-c_i^{p-1}\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p\\&\quad =c_i^{p-1}\bigg [ac_i^{2-p}\Vert u_i\Vert _i^2+\lambda \bigg (\sum ^{k+1}_{j\ne i}c_i^{\frac{2-p}{\alpha }}t_j^2\Vert u_j\Vert _j^{2}+c_i^{2+\frac{2-p}{\alpha }}\Vert u_i\Vert _i^{2}\bigg )^{\alpha }\Vert u_i\Vert _i^{2}\bigg ]\\ &\qquad -c_i^{p-1}\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p, \end{aligned} \end{aligned}$$

which implies that $\frac{\partial \phi }{\partial c_i}(t_1,\ldots ,c_i,\ldots ,t_{k+1})>0$ if $c_i<t_i$ and $\frac{\partial \phi }{\partial c_i}(t_1,\ldots ,c_i,\ldots ,t_{k+1})<0$ if $c_i>t_i$. $\square$

We define ${{\textbf {F}}}=(F_1,\ldots ,F_{k+1}):{\mathcal {H}}_k^{{{\textbf {r}}}_k}\rightarrow {\mathbb {R}}^{k+1}$ by

$$\begin{aligned} F_i(u_1,\ldots ,u_{k+1}):=\langle \partial _{u_i}E_{\lambda }'(u_1,\ldots ,u_{k+1}),u_{i}\rangle \end{aligned}$$

(2.9)

for $i=1,\ldots ,k+1$. Then we have the following lemma.

Lemma 2.5

For any $(u_1,\ldots ,u_{k+1})\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}$ with nonzero components such that $F_i(u_1,\ldots ,u_{k+1})<0$ for each $i=1,\ldots ,k+1$, the $(k+1)$ tuple $(t_1,\ldots ,t_{k+1})$ of positive numbers obtained in Lemma 2.3 satisfies $t_i\le 1$ for each i.

Proof

By Lemma 2.3, $(t_1u_1,\ldots ,t_{k+1}u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k},$ then for each $i=1,\ldots ,k+1$,

$$\begin{aligned} at_i^2\Vert u_i\Vert _i^2+\lambda t_i^2\Vert u_i\Vert _i^2\bigg (\sum _{j=1}^{k+1}t_j^2\Vert u_j\Vert _j^2\bigg )^{\alpha }=t_i^p\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p. \end{aligned}$$

(2.10)

Without loss of generality, we assume that $t_{i_0}=\max \{t_1,\ldots ,t_{k+1}\}$. Then

$$\begin{aligned} at_{i_0}^2\Vert u_{i_0}\Vert _{i_0}^2+\lambda t_{i_0}^{2+2\alpha }\Vert u_{i_0}\Vert _{i_0}^{2}\bigg (\sum _{j=1}^{k+1}\Vert u_j\Vert _j^2\bigg )^{\alpha }> t_{i_0}^p\int _{B_{i_0}^{{{\textbf {r}}}_k}}|u_{i_0}|^p. \end{aligned}$$

(2.11)

Since $F_i(u_1,\ldots ,u_{k+1})<0$, we have

$$\begin{aligned} a\Vert u_{i_0}\Vert _{i_0}^2+\lambda \Vert u_{i_0}\Vert _{i_0}^2\bigg (\sum _{j=1}^{k+1}\Vert u_j\Vert _j^2\bigg )^{\alpha }<\int _{B_{i_0}^{{{\textbf {r}}}_k}}|u_{i_0}|^p. \end{aligned}$$

(2.12)

By combining (2.11) and (2.12), we obtain

$$\begin{aligned} \bigg (\frac{a}{t_{i_0}^{2\alpha }}-a\bigg )\Vert u_{i_0}\Vert _{i_0}^2\ge (t_{i_0}^{p-2\alpha -2}-1)\int _{B_i^{{{\textbf {r}}}_k}}|u_{i_0}|^p. \end{aligned}$$

If $t_{i_0}>1$, the left side of this inequality is negative, but the right side is positive, which leads to a contradiction. Hence, we have $t_{i}\le 1$ for each i. The proof is completed. $\square$

Notice that

$$\begin{aligned} {\mathcal {M}}_k^{{{\textbf {r}}}_k}:=\bigg \{(u_1,\ldots ,u_{k+1})\in \ {\mathcal {H}}_k^{{{\textbf {r}}}_k}\,\ u_i\ne 0 \mid {{\textbf {F}}}(u_1,\ldots ,u_{k+1})=0\bigg \}, \end{aligned}$$

where ${{\textbf {F}}}(u_1,\ldots ,u_{k+1})$ is defined in (2.9). Hereafter, we say that ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ is a differentiable manifold in ${\mathcal {H}}_k^{{{\textbf {r}}}_k}$, means that the matrix

$$\begin{aligned} N:=(N_{ij})_{(k+1)\times (k+1)}=\langle \partial _{u_i} F'_j(u_1,\ldots ,u_{k+1}),u_i\rangle ,\quad i,j=1,\ldots ,k+1 \end{aligned}$$

is nonsingular at each point $(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}.$

Lemma 2.6

${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ is a differentiable manifold in ${\mathcal {H}}_k^{{{\textbf {r}}}_k}.$ Moreover, a minimizer $(u_1,\ldots ,u_{k+1})$ of $E_{\lambda }$ on ${{\mathcal {M}}_k^{{{\textbf {r}}}_k}}$ is a critical point of $E_{\lambda }$ in ${\mathcal {H}}_k^{{{\textbf {r}}}_k}$ with nonzero components.

Proof

By some calculations, we have

$$\begin{aligned} \begin{aligned} N_{ii}&=2a\Vert u_i\Vert _i^2+2\lambda \Vert u_i\Vert _i^2\bigg (\sum _{l=1}^{k+1}\Vert u_l\Vert _l^2 \bigg )^{\alpha }\\ &\quad +2\lambda \alpha \Vert u_i\Vert _i^2\bigg (\sum _{l=1}^{k+1}\Vert u_l\Vert _l^2 \bigg )^{\alpha -1}\Vert u_i\Vert _i^2-p\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p,\\ N_{ij}&=2\lambda \alpha \Vert u_i\Vert _i^2\bigg (\sum _{l=1}^{k+1}\Vert u_l\Vert _l^2\bigg )^{\alpha -1}\Vert u_j\Vert _j^2,\quad \text{ for }\quad j\ne i,\quad i,j=1,\ldots ,k+1.\end{aligned}. \end{aligned}$$

Due to the fact that $p\in (2+2\alpha ,6)$, we obtain

$$\begin{aligned} \begin{aligned} {N}_{ii}+\sum ^{k+1}_{j\ne i}{N}_{ij}&=2a\Vert u_i\Vert _i^2+2\lambda \Vert u_i\Vert _i^2\bigg (\sum _{l=j}^{k+1}\Vert u_j\Vert _j^2\bigg )^{\alpha }\\ &\quad +2\lambda \alpha \Vert u_i\Vert _i^2\bigg (\sum _{j=1}^{k+1}\Vert u_j\Vert _j^2\bigg )^{\alpha }-p\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p\\&=2a\Vert u_i\Vert _i^2+(2\alpha +2)\bigg (\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p-a\Vert u_i\Vert _i^2\bigg )-p\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p\\&=-2\alpha a \Vert u_i\Vert _i^2+(2+2\alpha -p)\int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p<0. \end{aligned} \end{aligned}$$

So

$$\begin{aligned} {N}_{ii}<-\sum _{j\ne i}^{k+1}{N}_{ij}<0\Rightarrow |{N}_{ii}|>\sum _{j\ne i}^{k+1}|{N}_{ij}|. \end{aligned}$$

Then the matrix ${N}=({N}_{ij})$ is diagonally dominant, and thereby it is nonsingular and $\det N\ne 0.$

If $(u_1,\ldots ,u_{k+1})$ is a minimizer of $E_{\lambda }|_{{\mathcal {M}}_k^{{{\textbf {r}}}_k}},$ then there is a Lagrangian multiplier $(\lambda _1,\ldots ,\lambda _{k+1})\in {\mathbb {R}}^{k+1}$ such that

$$\begin{aligned} \lambda _1F_1'(u_1,\ldots ,u_{k+1})+\cdots +\lambda _{k+1}F_{k+1}'(u_1,\ldots ,u_{k+1})=E'_{\lambda }(u_1,\ldots ,u_{k+1}). \end{aligned}$$

(2.13)

Applying $(u_1,0,\ldots ,0),(0,u_2,\ldots ,0),\ldots ,(0,\ldots ,0,u_{k+1})$ to the identity (2.13), we get

$$\begin{aligned} N_{ij}\left( \begin{matrix} \lambda _1 \\ \vdots \\ \lambda _{k+1} \end{matrix} \right) =\left( \begin{matrix} 0 \\ \vdots \\ 0 \end{matrix} \right) . \end{aligned}$$

Therefore, $\lambda _1,\ldots ,\lambda _{k+1}$ are all zeros and $(u_1,\ldots ,u_{k+1})$ is a critical point of $E_{\lambda }$ in ${\mathcal {H}}_k^{{{\textbf {r}}}_k}.$

Finally, for any $(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}$, we have

$$\begin{aligned} \begin{aligned} a\Vert u_i\Vert _i^2\le \int _{B_{i}^{{{\textbf {r}}}_k}}|u_i|^p\le C\Vert u_i\Vert _i^p \,\ and \,\ 0<\delta :=(\frac{a}{C})^{\frac{1}{p-2}}\le \Vert u_i\Vert _i. \end{aligned} \end{aligned}$$

(2.14)

Then each $u_i$ is bounded away from zero. Thus minimizers of $E_{\lambda }$ in ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ cannot have any zero components. The proof is completed. $\square$

Lemma 2.7

For fixed ${{\textbf {r}}}_k=(r_1,\ldots ,r_{k+1})\in \Gamma _k$, there exists a minimizer $(w_1,\ldots ,w_{k+1})$ of $E_{\lambda }|_{{\mathcal {M}}_k^{{{\textbf {r}}}_k}}$ such that each $(-1)^{i+1}w_i$ is positive on $B_i^{{{\textbf {r}}}_k}$ for $i=1,\ldots ,k+1$. Moreover, $(w_1,\ldots ,w_{k+1})$ satisfies (2.3).

Proof

For $(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k},$ it holds that

$$\begin{aligned} \begin{aligned} E_{\lambda }(u_1,\ldots ,u_{k+1})&=\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \sum _{i=1}^{k+1}\Vert u_i \Vert _i^2+\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) \sum _{i=1}^{k+1}\int _{B_i^{{{\textbf {r}}}_k}}|u_i|^p\\&\ge \left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \sum _{i=1}^{k+1}\Vert u_i\Vert _i^2>\delta , \end{aligned} \end{aligned}$$

(2.15)

where $\delta$ is defined in (2.14). Then there exists a minimizing sequence $\{(u_1^n,\ldots ,u_{k+1}^n)\}_{n=1}^{\infty }\subset {\mathcal {M}}_k^{{{\textbf {r}}}_k}$ such that $E_\lambda (u_1^n,\ldots ,u_{k+1}^n)\rightarrow \min \limits _{{\mathcal {M}}_k^{{{\textbf {r}}}_k}}E_{\lambda }$ as $n\rightarrow \infty .$ By combining with (1.9), we know that

$$\begin{aligned} \begin{aligned} m+1>&E_{\lambda }(u^n_1,\ldots ,u^n_{k+1})\\&=\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \sum _{i=1}^{k+1}\Vert u^n_i\Vert _i^2+\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) \sum _{i=1}^{k+1}\int _{B_i^{{{\textbf {r}}}_k}}|u^n_i|^p\\&\ge \left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \sum _{i=1}^{k+1}\Vert u^n_i\Vert _i^2. \end{aligned} \end{aligned}$$

(2.16)

Hence, $\{u_i^n\}_{n\ge 1}$ is bounded in $H_i^{{{\textbf {r}}}_k}$ for each $i=1,\ldots ,k+1.$ Up to a subsequence, there exists $(u_1^0,\ldots ,u_{k+1}^0)\in {\mathcal {H}}_k^{{{\textbf {r}}}_k}$ such that $u_i^n\rightharpoonup u_i^0$ in $H_i^{{{\textbf {r}}}_k}$ and $u_i^n\rightarrow u_i^0$ in $L^p(B_i^{{{\textbf {r}}}_k})$ with $p\in (2,6).$ Since $(u_1^n,\ldots ,u_{k+1}^n)\subset {\mathcal {M}}_k^{{{\textbf {r}}}_k},$ we have

$$\begin{aligned} \begin{aligned}&0<\delta \le a\liminf _{n\rightarrow \infty }\Vert u_i^n\Vert _i^2<\liminf _{n\rightarrow \infty }\int _{B_i^{{{\textbf {r}}}_k}}|u_i^n|^p.\\&\quad =\Vert u_i^0\Vert _i^p. \end{aligned} \end{aligned}$$

(2.17)

This implies that $u_i^0\ne 0$ for each $i=1,\ldots ,k+1.$

Now we claim that up to a subsequence, $u_i^n$ converges to $u_i^0$ strongly in $H_i^{{{\textbf {r}}}_k}.$ Notice that $u_i^n \rightharpoonup u_i^0$ weakly in $H_i^{{{\textbf {r}}}_k}.$ We may suppose on the contrary that $\Vert u_i^0\Vert _i<\liminf \limits _{n\rightarrow \infty }\Vert u_i^n\Vert _i$ for at least one $i\in \{1,\ldots ,k+1\}.$ Since each component of $(u_1^0,\ldots ,u_{k+1}^0)$ is nonzero, by Lemma 2.3, one can find $(t_1^0,\ldots ,t_{k+1}^0)\in ({\mathbb {R}}_{>0})^{k+1}$ such that $(t_1^0u_1^0,\ldots ,t_{k+1}^0u_{k+1}^0)\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}.$ However, in this situation, Corollary 2.4 implies that

$$\begin{aligned}&\inf _{(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}}E_{\lambda }(u_1,\ldots ,u_{k+1})\\&\quad \le E_{\lambda }(t_1^0u_1^0,\ldots ,t_{k+1}^0u_{k+1}^0)\\&\quad =\bigg (\frac{a}{2}-\frac{a}{2\alpha +2}\bigg )\sum _{i=1}^{k+1}\bigg ((t_i^0)^2\Vert u_i^0\Vert _i^2\bigg )\\ &\qquad - \bigg (\frac{1}{2\alpha +2}-\frac{1}{p}\bigg )\sum _{i=1}^{k+1}(t_i^0)^p\liminf _{n\rightarrow \infty }\int _{B_i^{{{\textbf {r}}}_k}}|u_i^n|^p\\&\quad <\bigg (\frac{a}{2}-\frac{a}{2\alpha +2}\bigg )\sum _{i=1}^{k+1}\bigg ((t_i^0)^2\liminf _{n\rightarrow \infty }\Vert u_i^n\Vert _i^2\bigg )\\ &\qquad - \bigg (\frac{1}{2\alpha +2}-\frac{1}{p}\bigg )\sum _{i=1}^{k+1}(t_i^0)^p\liminf _{n\rightarrow \infty }\int _{B_i^{{{\textbf {r}}}_k}}|u_i^n|^p\\&\quad \le \liminf _{n\rightarrow \infty }E_{\lambda }(u_1^n,\ldots ,u_{k+1}^n)\\&\quad =\inf _{(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}}E_{\lambda }(u_1,\ldots ,u_{k+1}). \end{aligned}$$

This is a contradiction. Thus the claim holds, and going if necessary to a subsequence, $(u_1^n,\ldots ,u_{k+1}^n)\rightarrow (u_1^0,\ldots ,u_{k+1}^0)$ in ${\mathcal {H}}_k^{{{\textbf {r}}}_k}.$

Therefore, $(u_1^0,\ldots ,u_{k+1}^0)$ is contained in ${\mathcal {M}}_k^{{{\textbf {r}}}_k}$ and is a minimizer of $E_{\lambda }|_{{\mathcal {M}}_k^{{{\textbf {r}}}_k}}.$ Obviously,

$$\begin{aligned} (w_1,\ldots ,w_{k+1}):=(|u_1^0|,-|u_2^0|,\ldots ,(-1)^{k+2}|u_{k+1}^0|) \end{aligned}$$

is also a minimizer of $E_{\lambda }|_{{\mathcal {M}}_k^{{{\textbf {r}}}_k}}$. Hence it is a critical point of $E_{\lambda }$ by Lemma 2.6 and satisfies (2.3). Then by the standard elliptic regularity theory, all $w_i\in C^2(B_i^{{{\textbf {r}}}_k})$. Furthermore, since $(-1)^{i+1}w_i\ge 0$, by applying the strong maximum principle to (2.3), it follows immediately $(-1)^{i+1}w_i>0.$ The proof is completed. $\square$

3 Existence of Nodal Solutions

In this section, we are devoted to the proof of Theorem 1.1. In view of Lemma 2.7, we can define a function $\Psi :\Gamma _k\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \begin{aligned} \Psi ({{\textbf {r}}}_k)&=\Psi (r_1,\ldots ,r_{k+1})=E_{\lambda }(w_1^{{{\textbf {r}}}_k},\ldots ,w_{k+1}^{{{\textbf {r}}}_k})\\&=\inf _{(u_1^{{{\textbf {r}}}_k},\ldots ,u_{k+1}^{{{\textbf {r}}}_k})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}} E_{\lambda }(u_1^{{{\textbf {r}}}_k},\ldots ,u_{k+1}^{{{\textbf {r}}}_k}). \end{aligned} \end{aligned}$$

(3.1)

Then we shall give the following lemma which shows some properties of $\Psi ({{\textbf {r}}}_k).$

Lemma 3.1

For any positive integer k, let ${{\textbf {r}}}_k=(r_1,\ldots ,r_k)\in \Gamma _k.$ Then the following statements are true.

• (i) If $r_i-r_{i-1}\rightarrow 0$ for some $i\in \{1,\ldots ,k\},$ then $\Psi ({{\textbf {r}}}_k)\rightarrow +\infty$.

• (ii) If $r_k\rightarrow \infty ,$ then $\Psi ({{\textbf {r}}}_k)\rightarrow +\infty$.

• (iii) $\Psi$ is continuous in $\Gamma _k.$ Moreover, there exists a minimum point $\tilde{{{\textbf {r}}}}_k\in \Gamma _k$ such that $\Psi (\tilde{{{\textbf {r}}}}_k)=\min \limits _{{{\textbf {r}}}_k\in \Gamma _k}\Psi ({{\textbf {r}}}_k).$

Proof

(i) Assume that $r_{i_0}-r_{i_0-1}\rightarrow 0$ for some $i_0\in \{1,\ldots ,k+1\}.$ Since $(w_1^{{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{{{{\textbf {r}}}}_k})\in M_k^{{{{\textbf {r}}}}_k}$, by using the Hölder inequality and Sobolev inequality, we obtain that

$$\begin{aligned} \Vert w_{i_0}^{{{{\textbf {r}}}}_k}\Vert _{i_0}^2\le \int _{B_i^{{{{\textbf {r}}}}_k}}|w_{i_0}^{{{{\textbf {r}}}}_k}| ^p\le \left( \int _{B_{i_0}^{{{\textbf {r}}}_k}}|w_{i_0}^{{{{\textbf {r}}}}_k}|^6\right) ^{\frac{p}{6}}|B_{i_0}^{{{\textbf {r}}}_k}|^{1-\frac{p}{6}}\le C\Vert w_{i_0}^{{{{\textbf {r}}}}_k}\Vert _{i_0}^p|B_{i_0}^{{{{\textbf {r}}}}_k}|^{1-\frac{p}{6}}, \end{aligned}$$

(3.2)

where $C>0$ is a positive constant. Note that $2\alpha +2<p<6$. Then $\Vert w_{i_0}^{{{{\textbf {r}}}}_k}\Vert _{i_0}\rightarrow \infty .$ We see that

$$\begin{aligned} \begin{aligned}&E_\lambda (w_1^{{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{{{{\textbf {r}}}}_k})\\ &\quad =\sum _{i=1}^{k+1} \bigg [\left( \frac{a}{2}-\frac{a}{p}\right) \Vert w_i^{{{{\textbf {r}}}}_k}\Vert _i^2+\left( \frac{\lambda }{2\alpha +2}- \frac{\lambda }{p}\right) \Vert w_i^{{{{\textbf {r}}}}_k}\Vert _i^2\left( \sum _{j=1}^{k+1}\Vert w_j^{{{{\textbf {r}}}}_k}\Vert _j^2\right) ^{\alpha }\bigg ]\\&\quad \ge \sum _{i=1}^{k+1}\left( \frac{a}{2}-\frac{a}{p}\right) \Vert w_i^{{{{\textbf {r}}}}_k}\Vert _i^2\\&\quad \ge \left( \frac{a}{2}-\frac{a}{p}\right) \Vert w_{i_0}^{{{{\textbf {r}}}}_k}\Vert _{i_0}^2. \end{aligned} \end{aligned}$$

(3.3)

This combined with (3.2), implies that

$$\begin{aligned} \begin{aligned} \Psi ({{\textbf {r}}}_k)\rightarrow +\infty ,\quad \text{ if } r_i-r_{i-1}\rightarrow 0. \end{aligned} \end{aligned}$$

Thus (i) follows.

(ii) Recall the Strauss inequality [23], for any $u\in H_r^1({\mathbb {R}}^3)$, there exists a constant $C>0$ such that $|u(x)|\le C\frac{\Vert u\Vert }{|x|}, \text{ a.e } \text{ in } {\mathbb {R}}^3.$ Then we obtain

$$\begin{aligned} \begin{aligned} \Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^2\le \int _{B_{k+1}^{{{{\textbf {r}}}}_k}}|w_{k+1}^{{{{\textbf {r}}}}_k}|^p&\le C\int _{B_{k+1}^{{{{\textbf {r}}}}_k}}\frac{\Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^{p-2}|w_{k+1}^{{{{\textbf {r}}}}_k}|^2}{|x|^{p-2}}dx\\&\le C\frac{\Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^{p-2}}{r_k^{p-2}}\Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^{2}\\&= C{r}_k^{2-p}\Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^p. \end{aligned} \end{aligned}$$

(3.4)

This yields that ${r}_k^{p-2}\le C\Vert w_{k+1}^{{{{\textbf {r}}}}_k}\Vert _{k+1}^{p-2}.$ Therefore, the conclusion follows from (3.3).

(iii) Take a sequence $\{{{\textbf {r}}}_k^n\}^\infty _{n=1}=\{(r_1^n,\ldots ,r_k^n)\}^\infty _{n=1}\subset \Gamma _k$ converging to $\bar{{{\textbf {r}}}}_k=({\bar{r}}_1,\ldots ,{\bar{r}}_k)\in \Gamma _k.$ It suffices to prove that $\Psi ({{\textbf {r}}}_k^n)\rightarrow \Psi (\bar{{{\textbf {r}}}}_k)$. By Lemma 2.7, we assume that $(w_1^{{{\textbf {r}}}^n_k},\ldots ,w_{k+1}^{{{\textbf {r}}}^n_k})$ and $(w_1^{\bar{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\bar{{{\textbf {r}}}}_k})$ are minimizers of $E_{\lambda }|_{{\mathcal {M}}_k^{{{\textbf {r}}}^n_k}}$ and $E_{\lambda }|_{{\mathcal {M}}_k^{\bar{{{\textbf {r}}}}_k}}$, respectively. In the sequel, we shall prove that

$$\begin{aligned} \Psi (\bar{{{\textbf {r}}}}_k)\ge \limsup _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n)\quad \text{ and }\quad \Psi (\bar{{{\textbf {r}}}}_k)\le \liminf _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n). \end{aligned}$$

(3.5)

First, we prove that $\Psi (\bar{{{\textbf {r}}}}_k)\ge \limsup _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n)$. Define $v_{i}^{{{\textbf {r}}}_k^n}:[r^n_{i-1},r^n_i]\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \begin{aligned}&v_{i}^{{{\textbf {r}}}_k^n}(r)=\alpha _i^nw_i^{\bar{{{\textbf {r}}}}_k}(\frac{{\bar{r}}_i-{\bar{r}}_{i-1}}{r_i^n-r_{i-1}^n}(r-r_{i-1}^n)+{\bar{r}}_{i-1}),\quad \text{ for }\quad i=1,\ldots ,k,\\&v_{k+1}^{{{\textbf {r}}}_k^n}(r)=\alpha _{k+1}^nw_{k+1}^{\bar{{{\textbf {r}}}}_k}(\frac{{\bar{r}}_k}{r_k^n}r), \end{aligned} \end{aligned}$$

where $r_0^n=0,r_{k+1}^n=\infty$ and each $(\alpha _1^n,\ldots ,\alpha _{k+1}^n)$ is a unique (k+1)-tuple of positive real number such that $(v_{1}^{{{\textbf {r}}}_k^n},\ldots ,v_{k+1}^{{{\textbf {r}}}_k^n})\in M_k^{{{\textbf {r}}}_k^n}.$ Then by the definition of $(w_1^{{{\textbf {r}}}^n_k},\ldots ,w_{k+1}^{{{\textbf {r}}}^n_k})$, we have

$$\begin{aligned} E_\lambda (v_{1}^{{{\textbf {r}}}_k^n},\ldots ,v_{k+1}^{{{\textbf {r}}}_k^n})\ge E_\lambda (w_1^{{{\textbf {r}}}^n_k},\ldots ,w_{k+1}^{{{\textbf {r}}}^n_k})=\Psi ({{\textbf {r}}}_k^n). \end{aligned}$$

If n is large enough, we can calculate that for each $i,j=1,\ldots ,k+1,$

$$\begin{aligned} \begin{aligned}&\Vert v_{i}^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2\\&\quad =\int ^{r_i^n}_{r^n_{i-1}}|\nabla v_{i}^{{{\textbf {r}}}_k^n}|^2\beta (N)r^2dr+\int ^{r_i^n}_{r^n_{i-1}} V(v_{i}^{{{\textbf {r}}}_k^n})^2\beta (N)r^2dr\\&\quad =(\alpha _i^n)^2\int ^{r_i^n}_{r^n_{i-1}}|\nabla w_i^{\bar{{{\textbf {r}}}}_k}(\frac{{\bar{r}}_ i-{\bar{r}}_{i-1}}{r_i^n-r_{i-1}^n}(r-r_{i-1}^n)+{\bar{r}}_{i-1})|^2\beta (N)r^2dr\\&\qquad +(\alpha _i^n)^2\int ^{r_i^n}_{r^n_{i-1}}V| w_i^{\bar{{{\textbf {r}}}}_k}\left( \frac{{\bar{r}}_i -{\bar{r}}_{i-1}}{r_i^n-r_{i-1}^n}(r-r_{i-1}^n)+{\bar{r}}_{i-1}\right) |^2\beta (N)r^2dr\\&\quad =\beta (N)(\alpha _i^n)^2\frac{{\bar{r}}_i-{\bar{r}}_{i-1}}{r_i^n-r_{i-1}^n}\int ^{r_i^n} _{r^n_{i-1}}|\nabla w_i^{\bar{{{\textbf {r}}}}_k}(t)|^2\left( \frac{r_i^n-r_{i-1}^n}{{\bar{r}}_i-{\bar{r}}_{i-1}} (t-{\bar{r}}_{i-1})+r_{i-1}^n\right) ^2\left( \frac{r_i^n-r_{i-1}^n}{{\bar{r}}_i-{\bar{r}}_{i-1}}\right) dt\\&\qquad +\beta (N)(\alpha _i^n)^2\int ^{r_i^n}_{r^n_{i-1}}V| w_i^{\bar{{{\textbf {r}}}}_k}(t)|^ 2\left( \frac{r_i^n-r_{i-1}^n}{{\bar{r}}_i-{\bar{r}}_{i-1}}(t-{\bar{r}}_{i-1})+r_{i-1}^n\right) ^2 (\frac{r_i^n-r_{i-1}^n}{{\bar{r}}_i-{\bar{r}}_{i-1}})dt\\&\quad =(\alpha _i^n)^2\Vert w_i^{\bar{{{\textbf {r}}}}_k}\Vert _{B_i^{\bar{{{\textbf {r}}}}_k}}^2+o(1),\\ \end{aligned} \end{aligned}$$

where $\beta (N)$ indicates the surface area of the unit sphere in ${\mathbb {R}}^N.$ Similarly,

$$\begin{aligned} \begin{aligned} \Vert v_{i}^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2\bigg (\sum _{j=1}^{k+1}\Vert v_{j}^ {{{\textbf {r}}}_k^n}\Vert _{B_j^{{{\textbf {r}}}_k^n}}^2\bigg )^{\alpha }=(\alpha _i^n)^2\Vert w_i^{\bar{{{\textbf {r}}}}_k} \Vert _{B_i^{\bar{{{\textbf {r}}}}_k}}^2\bigg (\sum _{j=1}^{k+1}(\alpha _j^n)^2\Vert w_j^{\bar{{{\textbf {r}}}}_ k}\Vert _{B_j^{\bar{{{\textbf {r}}}}_k}}^2\bigg )^{\alpha }+o(1) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \int _{B_i^{{{\textbf {r}}}_k^n}}|v_{i}^{{{\textbf {r}}}_k^n}|^p=(\alpha _i^n)^p\int _ {B_i^{\bar{{{\textbf {r}}}}_k}}|w_i^{\bar{{{\textbf {r}}}}_k}|^p+o(1). \end{aligned}$$

This combined with the fact that $(v_{1}^{{{\textbf {r}}}_k^n},\ldots ,v_{k+1}^{{{\textbf {r}}}_k^n})\in M_k^{{{\textbf {r}}}_k^n}$, yields

$$\begin{aligned} a(\alpha _i^n)^2\Vert w_i^{\bar{{{\textbf {r}}}}_k}\Vert _{B_i^{\bar{{{\textbf {r}}}}_k}}^2+\lambda (\alpha _i^n)^2\Vert w_i^{\bar{{{\textbf {r}}}}_k}\Vert _ {B_i^{\bar{{{\textbf {r}}}}_k}}^2 \bigg (\sum _{j=1}^{k+1}(\alpha _j^n)^2\Vert w_j^ {\bar{{{\textbf {r}}}}_k}\Vert _{B_j^{\bar{{{\textbf {r}}}}_k}}^2\bigg )^{\alpha } -(\alpha _i^n)^p\int _{B_i^{\bar{{{\textbf {r}}}}_k}}|w_i^{\bar{{{\textbf {r}}}}_k}|^p=o(1) \end{aligned}$$

for each $i=1,\ldots ,k+1.$ In addition,

$$\begin{aligned} a\Vert w_i^{\bar{{{\textbf {r}}}}_k}\Vert _{B_i^{\bar{{{\textbf {r}}}}_k}}^2 +\lambda \Vert w_i^{\bar{{{\textbf {r}}}}_k}\Vert _{B_i^{\bar{{{\textbf {r}}}}_k}}^2 \bigg (\sum _{j=1}^{k+1}\Vert w_j^{\bar{{{\textbf {r}}}}_k}\Vert _{B_j^{\bar{{{\textbf {r}}}}_k}}^2\bigg )^{\alpha } -\int _{B_i^{\bar{{{\textbf {r}}}}_k}}|w_i^{\bar{{{\textbf {r}}}}_k}|^p=0 \end{aligned}$$

for each i, and this gives that $\lim \limits _{n\rightarrow \infty }\alpha ^n_i=1$ for all i. Therefore, we get

$$\begin{aligned} \begin{aligned} \Psi (\bar{{{\textbf {r}}}}_k)=\limsup _{n\rightarrow \infty }E_\lambda (v_{1}^{{{\textbf {r}}}_k^n},\ldots ,v_{k+1}^{{{\textbf {r}}}_k^n}) \ge \limsup _{n\rightarrow \infty }E_\lambda (w_1^{{{\textbf {r}}}_k^n},\ldots ,w_{k+1}^{{{\textbf {r}}}_k^n}) =\limsup _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n). \end{aligned} \end{aligned}$$

On the other hand, we prove $\Psi (\bar{{{\textbf {r}}}}_k)\le \liminf \limits _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n).$ Similarly as the former case, define $u_{i}^{{{\textbf {r}}}^n_k}:[{\bar{r}}_{i-1},{\bar{r}}_i]\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \begin{aligned}&u_{i}^{{{\textbf {r}}}^n_k}(t)=t_i^nw_i^{{{\textbf {r}}}_k^n}\left( \frac{r_i^n-r_{i-1}^n}{{\bar{r}}_i -{\bar{r}}_{i-1}}(t-{\bar{r}}_{i-1})+r_{i-1}^n\right) ,\quad \text{ for }\quad i=1,\ldots ,k,\\&u_{k+1}^{{{\textbf {r}}}^n_k}(t)=t_{k+1}^nw_{k+1}^{{{\textbf {r}}}_k^n}\left( \frac{r_k^n}{{\bar{r}}_k}t\right) . \end{aligned}\end{aligned}$$

where $r_0^n=0,r_{k+1}^n=\infty$ and each $(t_1^n,\ldots ,t_{k+1}^n)$ is a unique (k+1)-tuple of positive real number such that $(u_{1}^{{{\textbf {r}}}^n_k},\ldots ,u_{k+1}^{{{\textbf {r}}}^n_k})\in M_k^{\bar{{{\textbf {r}}}}_k}.$ Then it also follows that

$$\begin{aligned} a(t_i^n)^2\Vert w_i^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2+\lambda (t_i^n)^2\Vert w_i^{{{\textbf {r}}}_k^n} \Vert _{B_i^{{{\textbf {r}}}_k^n}}^2\left( \sum _{j=1}^{k+1}(t_j^n)^2\Vert w_j^{{{\textbf {r}}}_k^n}\Vert _{B_j^{{{\textbf {r}}}_k^n}}^2\right) ^{\alpha } -(t_i^n)^p\int _{B_i^{{{\textbf {r}}}_k^n}}|w_i^{{{\textbf {r}}}_k^n}|^p=o(1) \end{aligned}$$

and

$$\begin{aligned} a\Vert w_i^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2+\lambda \Vert w_i^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2 \left( \sum _{j=1}^{k+1}\Vert w_j^{{{\textbf {r}}}_k^n}\Vert _{B_j^{{{\textbf {r}}}_k^n}}^2\right) ^{\alpha }-\int _{B_i^{{{\textbf {r}}}_k^n}}|w_i ^{{{\textbf {r}}}_k^n}|^p=0 \end{aligned}$$

for each i. Since $\liminf \limits _{n\rightarrow \infty }\Vert w_i^{{{\textbf {r}}}_k^n}\Vert _{B_i^{{{\textbf {r}}}_k^n}}^2$ is strictly positive, we conclude that $t_i^n\rightarrow 1$ as $n\rightarrow \infty$ for all i. Thus

$$\begin{aligned} \begin{aligned} \Psi (\bar{{{\textbf {r}}}}_k)\le \liminf _{n\rightarrow \infty }E_\lambda (v_{1}^{{{\textbf {r}}}^n_k},\ldots ,v_{k+1}^{{{\textbf {r}}}^n_k}) =\liminf _{n\rightarrow \infty }E_\lambda (w_1^{{{\textbf {r}}}_k^n},\ldots ,w_{k+1}^{{{\textbf {r}}}_k^n})=\liminf _{n\rightarrow \infty }\Psi ({{\textbf {r}}}_k^n). \end{aligned} \end{aligned}$$

This completes the proof of (iii). Finally, by (i)–(iii), we can conclude that there exists a minimum point $\tilde{{{\textbf {r}}}}_k=({\tilde{r}}_1,\ldots ,{\tilde{r}}_k)\in \Gamma _k$ of $\Psi .$ $\square$

According to Lemmas 2.7 and 3.1, there exists $(w_1^{\tilde{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\tilde{{{\textbf {r}}}}_{k}})$ satisfying (2.3) and

$$\begin{aligned} \begin{aligned} E_{\lambda }(w_1^{\tilde{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\tilde{{{\textbf {r}}}}_{k}}) =\inf \limits _{{{\textbf {r}}}_k\in \Gamma _k}\Psi ({{\textbf {r}}}_k). \end{aligned} \end{aligned}$$

(3.6)

Now we are in position to show that $\sum _{i=1}^{k+1}w_i^{\tilde{{{\textbf {r}}}}_k}$ is a desired nodal solution of (1.1) which changes sign exactly k times.

Proof of Theorem 1.1

We shall argue it by contradiction. Suppose on the contrary that $\sum _{i=1}^{k+1}w_i^{\tilde{{{\textbf {r}}}}_k}$ is not the solution of (1.1). In other words, suppose that there is $l\in \{1,\ldots ,k\}$ such that

$$\begin{aligned} w_-:=\lim _{t\rightarrow {\tilde{r}}_{l-}}\frac{dw_l^{\tilde{{{\textbf {r}}}}_k}(t)}{dt}\ne \lim _{t\rightarrow {\tilde{r}}_{l+}} \frac{dw_{l+1}^{\tilde{{{\textbf {r}}}}_k}(t)}{dt}=:w_+ \end{aligned}$$

We define a $(k+1)$-tuple of function $({\tilde{z}}_1,\ldots ,{\tilde{z}}_{k+1})$ as follows. Given a small positive number $\delta$, set

$$\begin{aligned} {\tilde{y}}(t)=\left\{ \begin{aligned}&w^{\tilde{{{\textbf {r}}}}_k}_l(t),&\text{ for }\ t\in ({\tilde{r}}_{l-1},{\tilde{r}}_l-\delta ),\\&w^{\tilde{{{\textbf {r}}}}_k}_l({\tilde{r}}_l-\delta )+\frac{w^{\tilde{{{\textbf {r}}}}_k}_{l+1}({\tilde{r}}_l+\delta ) -w^{\tilde{{{\textbf {r}}}}_k}_l({\tilde{r}}_l-\delta )}{2\delta }(t-{\tilde{r}}_l+\delta ),&\text{ for }\ t\in ({\tilde{r}}_l-\delta ,{\tilde{r}}_l+\delta ),\\&w^{\tilde{{{\textbf {r}}}}_k}_{l+1}(t),&\text{ for }\ t\in ({\tilde{r}}_l+\delta ,{\tilde{r}}_{l+1}).\\ \end{aligned}\right. \end{aligned}$$

Then ${\tilde{y}}$ has a unique zero point ${\tilde{s}}_l$ in $({\tilde{r}}_{l-1},{\tilde{r}}_{l+1})$. Let

$$\begin{aligned} \begin{aligned} {\tilde{z}}_l(t)&={\tilde{y}}(t) \ \text{ in }\ ({\tilde{r}}_{l-1},{\tilde{s}}_l),\quad {\tilde{z}}_{l+1}(t)={\tilde{y}}(t) \ \text{ in }\ ({\tilde{s}}_l,{\tilde{r}}_{l+1}) \ \text{ and }\\ {\tilde{z}}_i(t)&=w_i^{\tilde{{{\textbf {r}}}}_k}(t) \ \text{ for }\ ({\tilde{r}}_{i-1},{\tilde{r}}_i),\quad i\ne l, l+1. \end{aligned} \end{aligned}$$

By Lemma 2.3, there exists $(a_1,\ldots ,a_{k+1})\in ({\mathbb {R}}_{>0})^{k+1}$ such that $(z_1,\ldots ,z_{k+1})$ $:=(a_1{\tilde{z}}_1,\ldots ,a_{k+1}{\tilde{z}}_{k+1})\in M_k^{\bar{{{\textbf {r}}}}_k}$ with ${{\bar{r}}}_k=({\tilde{r}}_1,\ldots ,{\tilde{r}}_{l-1},{\tilde{s}}_l,{\tilde{r}}_{l+1},\ldots ,{\tilde{r}}_{k+1}).$ In addition, we have

$$\begin{aligned} (a_1,\ldots ,a_{k+1})\rightarrow (1,\ldots ,1)\quad \text{ as }\quad \delta \rightarrow 0 \end{aligned}$$

(3.7)

and

$$\begin{aligned} {I_\lambda }(W)=E_\lambda (w_1^{\tilde{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\tilde{{{\textbf {r}}}}_k})\le E_\lambda (z^{\tilde{{{\textbf {r}}}}_k}_1,\ldots ,z^{\tilde{{{\textbf {r}}}}_k}_{k+1})={I_\lambda }(Z), \end{aligned}$$

(3.8)

where $W(t)=\sum _{i=1}^{k+1}w^{\tilde{{{\textbf {r}}}}_k}_i(t)$ and $Z(t)=\sum _{i=1}^{k+1}z^{\tilde{{{\textbf {r}}}}_k}_i(t).$

On the other hand, since $Z,W>0$, we can check that

$$\begin{aligned} \frac{1}{p}|Z|^p\ge \frac{1}{p}|W|^p+\frac{Z^2-W^2}{2}|W|^{p-2}. \end{aligned}$$

(3.9)

Then there holds that

$$\begin{aligned} \begin{aligned}&I_\lambda (Z)-I_\lambda (W)\\&\quad \le \bigg (\int _0^{{\tilde{r}}_l-\delta }+\int _{{\tilde{r}}_l+\delta }^{\infty }\bigg ) \bigg (\frac{a}{2}Z'^2+\frac{a}{2}V(t)Z^2-\frac{1}{p}|W|^p-\frac{Z^2-W^2}{2}|W|^{p-2}\bigg )t^2dt\\&\qquad -\bigg (\int _0^{{\tilde{r}}_l-\delta }+\int _{{\tilde{r}}_l+\delta }^{\infty }\bigg )\bigg (\frac{a}{2}W'^2 +\frac{a}{2}V(t)W^2-\frac{1}{p}|W|^p\bigg )t^2dt\\&\qquad +\int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a}{2}Z'^2+\frac{a}{2}V(t)Z^2 -\frac{1}{p}|Z|^p\bigg )t^2dt\\&\qquad -\int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a}{2}W'^2 +\frac{a}{2}V(t)W^2-\frac{1}{p}|W|^p\bigg )t^2dt\\&\qquad +\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(Z'^2+V(t)Z^2)t^2dt\bigg )^{\alpha +1}\\&\qquad -\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^{\alpha +1}.\\ \end{aligned} \end{aligned}$$

Furthermore, by the definition of W, we have

$$\begin{aligned} & \int _0^{\infty }\bigg (aW'^2+aV(t)W^2\bigg )t^2dt+\lambda \bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg ) ^{\alpha +1}\nonumber \\ & \quad =\int _0^{\infty }|W|^pt^2dt. \end{aligned}$$

(3.10)

Set $A:=\bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^{\alpha }.$ Then we conclude from (3.9)–(3.10) that

$$\begin{aligned} \begin{aligned}&I_\lambda (Z)-I_\lambda (W)\\&\quad \le \underbrace{\bigg (\int _0^{{\tilde{r}}_l-\delta }+\int _{{\tilde{r}}_l +\delta }^{\infty }\bigg )\bigg (\frac{a}{2}Z'^2+\frac{a}{2}V(t)Z^2-\frac{Z^2}{2}|W|^{p-2}+\frac{\lambda A}{2}W'^2+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt}_{{{\textbf {A}}}}\\&\qquad \underbrace{+\int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a}{2}Z'^2 +\frac{a}{2}V(t)Z^2-\frac{1}{p}|Z|^p+\frac{1}{p}|W|^p+\frac{\lambda A}{2}W'^2+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt}_{{{\textbf {B}}}}\\&\qquad \underbrace{+\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(Z'^2+V(t)Z^2) t^2dt\bigg )^{\alpha +1}-\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^{\alpha +1}.}_{{{\textbf {C}}}}\\ \end{aligned}\nonumber \\ \end{aligned}$$

(3.11)

We consider the first part (A). Notice that W satisfies

$$\begin{aligned} & \bigg [a+\lambda \bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^ {\alpha }\bigg ]\bigg (-(t^2W')'+V(t)Wt^2\bigg )\nonumber \\ & \quad =|W|^{p-2}Wt^2,\quad {\tilde{r}}_{l-1}\le t\le {\tilde{r}}_l. \end{aligned}$$

(3.12)

Since $W({\tilde{r}}_l)=0$, by Taylor formula, we have $W({\tilde{r}}_l)=W({\tilde{r}}_l-\delta )+W'({\tilde{r}}_l-\delta )\delta +o(\delta )$ and $W({\tilde{r}}_l-\delta )=-\delta \cdot w_-+o(\delta ).$ Moreover, by (3.12) and Taylor formula again, we have $(t^2W')'({\tilde{r}}_l)=0$ and

$$\begin{aligned} ({\tilde{r}}_l-\delta )^2W'({\tilde{r}}_l-\delta )-{\tilde{r}}_l^2W'({\tilde{r}}_l) =-\delta ({\tilde{r}}_l^2W'({\tilde{r}}_l))'+o(\delta ). \end{aligned}$$

So

$$\begin{aligned} \quad ({\tilde{r}}_l-\delta )^2W'({\tilde{r}}_l-\delta )={\tilde{r}}_l^2w_-+o(\delta ). \end{aligned}$$

(3.13)

By multiplying both sides of (3.12) by W, we obtain

$$\begin{aligned} \begin{aligned} \int _0^{{\tilde{r}}_{l}-{\delta }}|W|^pt^2dt&=\int _0^{{\tilde{r}}_{l}-{\delta }}(a+\lambda A)(-(t^2W')'W+V(t)W^2t^2)dt\\&=(a+\lambda A)\left( \int _0^{{\tilde{r}}_{l}- {\delta }}(-(t^2W')'W+\int _0^{{\tilde{r}}_{l}-{\delta }}V(t)W^2t^2\right) \\&=(a+\lambda A)\left( -t^2W'W\bigg |_0^{{\tilde{r}}_{l}-{\delta }}+\int _0^{{\tilde{r}}_{l}-{\delta }}\bigg (t^2{W'}^2+V(t)W^2t^2\bigg )\right) . \end{aligned} \end{aligned}$$

(3.14)

Thus,

$$\begin{aligned} \begin{aligned} \int _0^{{\tilde{r}}_{l}-{\delta }}\frac{|W|^p}{2}t^2dt&=-\frac{(a+\lambda A)}{2}({{\tilde{r}}_{l}-{\delta }})^2W'({{\tilde{r}}_{l}-{\delta }})W({{\tilde{r}}_{l}-{\delta }})\\&\quad +\int _0^{{\tilde{r}}_{l}-{\delta }}\bigg (\frac{(a+\lambda A)}{2}{W'}^2+\frac{(a+\lambda A)}{2}V(t)W^2\bigg )t^2dt. \end{aligned} \end{aligned}$$

(3.15)

This combined with (3.13), implies that

$$\begin{aligned}&\int _0^{{\tilde{r}}_l-\delta }\bigg (\frac{a}{2}Z'^2+\frac{a}{2}V(t)Z^2-\frac{Z^2}{2}|W|^{p-2}+\frac{\lambda A}{2}W'^2+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt\nonumber \\&\quad =(1+o(1))\int _0^{{\tilde{r}}_l-\delta }\bigg (\frac{a}{2}W'^2+\frac{a}{2}V(t)W^2 -\frac{|W|^p}{2}+\frac{\lambda A}{2}W'^2+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt\nonumber \\&\quad =(1+o(1))\frac{(a+\lambda A)}{2}({\tilde{r}}_l-\delta )^2W'({\tilde{r}}_l-\delta )W({\tilde{r}}_l-\delta )\nonumber \\&\quad =-\frac{(a+\lambda A)}{2}{\tilde{r}}_l^2(w_-)^2\delta +o(\delta ). \end{aligned}$$

(3.16)

By the same method, we obtain

$$\begin{aligned} \begin{aligned}&\int _{{\tilde{r}}_l+\delta }^{\infty }\bigg (\frac{a}{2}Z'^2+\frac{a}{2}V(t)Z^2-\frac{Z^2}{2}|W|^{p-2} +\frac{\lambda A}{2}W'^2+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt\\&\quad =-\frac{(a+\lambda A)}{2}{\tilde{r}}_l^2(w_+)^2\delta +o(\delta ). \end{aligned} \end{aligned}$$

(3.17)

Next, we consider the second part (B). Indeed,

$$\begin{aligned} \int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a}{2}V(t)Z^2-\frac{1}{p}|Z|^p+\frac{1}{p}|W|^p+\frac{\lambda A}{2}V(t)W^2\bigg )t^2dt=o(\delta ), \end{aligned}$$

(3.18)

and

$$\begin{aligned} \begin{aligned} \int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a}{2}Z'^2+\frac{\lambda A}{2}W'^2\bigg )t^2dt&=(1+o(1))\int _{{\tilde{r}}_l-\delta }^{{\tilde{r}}_l+\delta }\bigg (\frac{a+\lambda A}{2}W'^2\bigg )t^2dt\\&=\frac{a+\lambda A}{4}(w_++w_-)^2{\tilde{r}}_l^2\delta +o(\delta ). \end{aligned} \end{aligned}$$

(3.19)

Finally, we consider the third part (C). Notice that

$$\begin{aligned} \begin{aligned}&\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(Z'^2+V(t)Z^2)t^2dt\bigg )^{\alpha +1} -\frac{\lambda }{2\alpha +2}\bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^{\alpha +1}\\&\quad =o(1)\bigg (\int _0^{\infty }(W'^2+V(t)W^2)t^2dt\bigg )^{\alpha +1}=o(\delta ). \end{aligned} \end{aligned}$$

(3.20)

Consequently, we conclude from (3.16)–(3.20) that

$$\begin{aligned} I_\lambda (Z)-I_\lambda (W)\le -\frac{(a+\lambda A)}{4}(w_+-w_-)^2{\tilde{r}}_l^2\delta +o(\delta ). \end{aligned}$$

(3.21)

By taking $\delta >0$ small enough, we have $I_\lambda (Z)-I_\lambda (W)<0$, which is a contradiction with (3.8). This completes the proof. $\square$

4 Energy Comparison and Asymptotic Behaviors

In this section, we are going to prove Theorems 1.2 and  1.3 by establishing subtle energy estimates.

Proof of Theorem 1.2

By applying Theorem  1.1, we can assume that for any fixed positive integer k, equation (1.1) has a radial nodal solution $U_{k+1}$ with exactly $k+1$ nodes $0<{\bar{r}}_1<\cdots<{\bar{r}}_{k+1}<+\infty$, and $I_\lambda (U_{k+1})= E_\lambda (w_1^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}})=\inf \limits _{{{\textbf {r}}}_{k+1}\in \Gamma _{k+1}}\Psi ({{\textbf {r}}}_{k+1})$. Set

$$\begin{aligned} \tilde{{{\textbf {r}}}}_{k+1}:=({\bar{r}}_1,{\bar{r}}_2,\ldots ,{\bar{r}}_{k+1}) \end{aligned}$$

and

$$\begin{aligned} w_i^{\tilde{{{\textbf {r}}}}_{k+1}}:=\chi _{B_i^{\tilde{{{\textbf {r}}}}_{k+1}}}U_{k+1}, \end{aligned}$$

where $\chi _{B_i^{\tilde{{{\textbf {r}}}}_{k+1}}}$ is the characteristic function on $B_i^{\tilde{{{\textbf {r}}}}_{k+1}}$. Obviously, $(w_1^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}})$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\bigg [a+\lambda \bigg (\sum _{j=1}^{k+1}\Vert w_j^{\tilde{{{\textbf {r}}}}_{k+1}}\Vert _j^2\bigg )^{\alpha }\bigg ]\bigg (-\Delta w_i^{\tilde{{{\textbf {r}}}}_{k+1}}+V(|x|)w_i^{\tilde{{{\textbf {r}}}}_{k+1}}\bigg ) =|w_i^{\tilde{{{\textbf {r}}}}_{k+1}}|^{p-2}w_i^{\tilde{{{\textbf {r}}}}_{k+1}}, \ x\in B_i^{\tilde{{{\textbf {r}}}}_{k+1}},\\&\quad w_i^{\tilde{{{\textbf {r}}}}_{k+1}}=0, \quad x\notin B_i^{\tilde{{{\textbf {r}}}}_{k+1}}.\\ \end{aligned}\right. \end{aligned}$$

(4.1)

Next, let $\hat{{{\textbf {r}}}}_k:=({\bar{r}}_2,\ldots ,{\bar{r}}_{k+1}).$ Clearly, $\hat{{{\textbf {r}}}}_k\in \Gamma _k$. By using Lemma 2.7, there is a minimizer $(w_1^{\hat{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\hat{{{\textbf {r}}}}_k})$ of its corresponding energy $E_{\lambda }|_{{\mathcal {M}}_k^{\hat{{{\textbf {r}}}}_k}}$, i.e.

$$\begin{aligned} E_{\lambda }(w_1^{\hat{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\hat{{{\textbf {r}}}}_k}) =\inf _{(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}_k^{\hat{{{\textbf {r}}}}_k}}E_{\lambda }(u_1,\ldots ,u_{k+1}). \end{aligned}$$

(4.2)

Then, by Lemma 2.3, there exists a unique $(k+1)-$tuple $(t_1,t_3,\ldots ,t_{k+2})$ of positive numbers such that

$$\begin{aligned} (t_1w_1^{\tilde{{{\textbf {r}}}}_{k+1}},t_3w_3^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,t_{k+2}w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}})\in {\mathcal {M}}_k^{\hat{{{\textbf {r}}}}_k}. \end{aligned}$$

This combined with (4.2), implies that

$$\begin{aligned} E_{\lambda }(w_1^{\hat{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^{\hat{{{\textbf {r}}}}_k})\le E_{\lambda }(t_1w_1^{\tilde{{{\textbf {r}}}}_{k+1}},t_3w_3^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,t_{k+2}w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}}). \end{aligned}$$

(4.3)

By letting $s>0$ be small enough, we obtain

$$\begin{aligned} & E_{\lambda }(t_1w_1^{\tilde{{{\textbf {r}}}}_{k+1}},t_3w_3^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots , t_{k+2}w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}})\nonumber \\ & \quad <E_{\lambda }(t_1w_1^{\tilde{{{\textbf {r}}}}_{k+1}}, sw_2^{\tilde{{{\textbf {r}}}}_{k+1}},t_3w_3^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,t_{k+2}w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}}). \end{aligned}$$

(4.4)

On the other hand, Corollary 2.4 gives that

$$\begin{aligned} & E_{\lambda }(t_1w_1^{\tilde{{{\textbf {r}}}}_{k+1}},{sw_2^{\tilde{{{\textbf {r}}}}_{k+1}}},t_3w_3^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,t_{k+2}w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}})\nonumber \\ & \quad <E_{\lambda }(w_1^{\tilde{{{\textbf {r}}}}_{k+1}},w_2^{\tilde{{{\textbf {r}}}}_{k+1}},\ldots ,w_{k+2}^{\tilde{{{\textbf {r}}}}_{k+1}}) =I_\lambda (U_{k+1}). \end{aligned}$$

(4.5)

Since $E_\lambda (w_1^{\tilde{{{\textbf {r}}}}_{k}},\ldots ,w_{k+1}^{\tilde{{{\textbf {r}}}}_{k}})=\inf \limits _{{{\textbf {r}}}_{k}\in \Gamma _{k}}\Psi ({{\textbf {r}}}_{k})$, we deduce from Lemma 3.1 that

$$\begin{aligned} I_{\lambda }(U_k)=E_{\lambda }(w_1^{\tilde{{{\textbf {r}}}}_k},\ldots ,w_{k+1}^ {\tilde{{{\textbf {r}}}}_k})<E_{\lambda }(w_1^{\hat{{{\textbf {r}}}}_{k}},\ldots ,w_{k+1}^{\hat{{{\textbf {r}}}}_{k}}) \end{aligned}$$

(4.6)

Then, it follows from (4.3)–(4.5) that

$$\begin{aligned} I_{\lambda }(U_k)<I_{\lambda }(U_{k+1}). \end{aligned}$$

Thus $I_{\lambda }(U_k)$ is strictly increasing with respect to k.

Finally, we claim that $I_{\lambda }(U_k)>(k+1)I_{\lambda }(U_{0}).$ In fact, since $\langle I_{\lambda }'(U_k),w_i^{\tilde{{{\textbf {r}}}}_k}\rangle =0$, we have

$$\begin{aligned} \langle I_{\lambda }'(w_i^{\tilde{{{\textbf {r}}}}_k}),w_i^{\tilde{{{\textbf {r}}}}_k}\rangle =a\Vert w_i^{\tilde{{{\textbf {r}}}}_k}\Vert _i^2+\lambda \Vert w_i^{\tilde{{{\textbf {r}}}}_k}\Vert _i^{2\alpha +2} -\int _{B_i^{\tilde{{{\textbf {r}}}}_k}}|w_i^{\tilde{{{\textbf {r}}}}_k}|^pdx<0. \end{aligned}$$

By Lemma 2.3, there is a unique ${\bar{\delta }}_i\in (0,1)$ such that ${\bar{\delta }}_iw_i^{\tilde{{{\textbf {r}}}}_k}\in {\mathcal {N}},$ where ${\mathcal {N}}$ is defined in (1.7). Hence, $I_{\lambda }({\bar{\delta }}_iw_i^{\tilde{{{\textbf {r}}}}_k})\ge I_{\lambda }(U_{0})$ and

$$\begin{aligned} \begin{aligned} (k+1)I_{\lambda }(U_{0})&\le \sum _{i=1}^{k+1}\left( I_{\lambda }({\bar{\delta }}_iw_i^{\tilde{{{\textbf {r}}}}_k}) -\frac{1}{2\alpha +2}\langle I_{\lambda }'({\bar{\delta }}_iw_i^{\tilde{{{\textbf {r}}}}_k}),{\bar{\delta }} _iw_i^{\tilde{{{\textbf {r}}}}_k}\rangle \right) \\&=\sum _{i=1}^{k+1}\bigg (\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) {\bar{\delta }}_i^2\Vert w_i^{\tilde{{{\textbf {r}}}} _k}\Vert _i^2+\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) {\bar{\delta }}_i^p\int _{B_i^{\tilde{{{\textbf {r}}}}_k}}| w_i^{\tilde{{{\textbf {r}}}}_k}|^p\bigg )\\&<\sum _{i=1}^{k+1}\bigg (\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \Vert w_i^{\tilde{{{\textbf {r}}}}_k}\Vert _i^2 +\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) \int _{B_i^{\tilde{{{\textbf {r}}}}_k}}|w_i^{\tilde{{{\textbf {r}}}}_k}|^p\bigg )\\&=I_{\lambda }\left( \sum _{i=1}^{k+1}w_i^{\tilde{{{\textbf {r}}}}_k}\right) -\frac{1}{2\alpha +2}\left\langle I_{\lambda }' \left( \sum _{i=1}^{k+1}w_i^{\tilde{{{\textbf {r}}}}_k}\right) ,\sum _{i=1}^{k+1}w_i^{\tilde{{{\textbf {r}}}}_k}\right\rangle \\&=I_{\lambda }(U_k). \end{aligned} \end{aligned}$$

The claim hods and we complete the proof. $\square$

Hereafter, we denote $U_k$ by $U_k^{\lambda }$ in order to emphasize the dependence on $\lambda .$ Analogically, set ${{\textbf {r}}}_{k,\lambda }=({\bar{r}}_{1,\lambda },\ldots ,{\bar{r}}_{k,\lambda })$ and $U_k^{\lambda }=\sum _{i=1}^{k+1}w_i^{{{\textbf {r}}}_{k,\lambda }}\in H_V$ obtained in Theorem 1.1. In the following, we shall show the asymptotic behaviors of $U_k^{\lambda }$ as $\lambda \rightarrow 0^+$.

Proof of Theorem 1.3

We divide the whole proof into three steps.

Step 1. We claim that for any sequence $\{\lambda _n\}$ with ${\lambda _n}\rightarrow 0^+$ as $n\rightarrow \infty ,$ $\{U_k^{\lambda _n}\}_n$ is bounded in $H_V.$ In fact, for fixed ${{\textbf {r}}}_k\in \Gamma _k$, we take nonzero radial functions $\varphi _i\in C_c^{\infty }(B_i^{{{\textbf {r}}}_k}),$ $i=1,\ldots ,k+1$. Then for any $\lambda \in [0,1],$ there exists a $k+1$ tuple $(b_1,\ldots , b_{k+1})$ of positive numbers such that

$$\begin{aligned} F_i(b_1\varphi _1,\ldots ,b_{k+1}\varphi _{k+1})<0,\quad \text{ for }\quad i=1,\ldots ,k+1. \end{aligned}$$

By Lemmas 2.3 and  2.5, there is a $k+1$ tuple $(a_1(\lambda ),\ldots ,a_{k+1}(\lambda ))\in (0,1]^{k+1}$ depending on $\lambda$ such that

$$\begin{aligned} ({\bar{\varphi }}_1,\ldots ,{\bar{\varphi }}_{k+1}):=(a_1(\lambda )b_1\varphi _i,\ldots ,a_{k+1}(\lambda ) b_{k+1}\varphi _i)\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}. \end{aligned}$$

Then there is $C_0>0$ such that for n large enough,

$$\begin{aligned} \begin{aligned}&I_{\lambda _n}(U_k^{\lambda _n})-\frac{1}{2\alpha +2}\langle I_{\lambda _n}'(U_k^{\lambda _n}),U_k^{\lambda _n}\rangle \\&\quad =\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \Vert U_k^{\lambda _n}\Vert ^2+\left( \frac{a}{2}-\frac{1}{2\alpha +2}\right) \int _{{\mathbb {R}}^3}|U_k^{\lambda _n}|^p\\&\quad \le I_{\lambda _n}\left( \sum _{i=1}^{k+1}{\bar{\varphi }}_i(x)\right) = I_{\lambda _n}\left( \sum _{i=1}^{k+1} {\bar{\varphi }}_i(x)\right) -\frac{1}{2\alpha +2}\langle I_{\lambda _n}' \left( \sum _{i=1}^{k+1}{\bar{\varphi }}_i(x)\right) ,{\bar{\varphi }}_i(x)\rangle \\&\quad =\sum _{i=1}^{k+1}\bigg (\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \Vert {\bar{\varphi }}_i(x)\Vert ^2+\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) \int _{B_i^{{{\textbf {r}}}_k}}|{\bar{\varphi }}_i(x)|^p\bigg )\\&\quad \le \sum _{i=1}^{k+1}\bigg (\left( \frac{a}{2}-\frac{a}{2\alpha +2}\right) \Vert b_i\varphi _i(x)\Vert ^2+\left( \frac{1}{2\alpha +2}-\frac{1}{p}\right) \int _{B_i^{{{\textbf {r}}}_k}}|b_i\varphi _i(x)|^p\bigg )\\&\quad =:C_0. \end{aligned} \end{aligned}$$

(4.7)

This implies that $\{U_k^{\lambda _n}\}_n$ is bounded in $H_V$. So the claim follows.

Step 2. According to Step 1, there exists a subsequence $\{\lambda _{n_j}\}$ of $\{\lambda _n\}$ and $U_k^0\in H_V$ such that $U_k^{\lambda _{n_j}}\rightharpoonup U_k^0$ and $(U_k^{\lambda _{n_j}})_i\rightharpoonup (U_k^0)_i$ weakly in $H_V$ as $n_j\rightarrow +\infty$. Then $U_k^0$ is a weak solution of (1.12). It suffices prove that $U_k^0$ is a radial nodal solution of (1.12) with exactly $k+1$ nodal domains.

In fact, by the compact embedding $H_V\hookrightarrow L^s({\mathbb {R}}^3)$ with $2<s<6$, it follows that

$$\begin{aligned} \begin{aligned}&\Vert U_k^{\lambda _{n_j}}-U_k^0\Vert ^2\\&\quad =\langle I_{\lambda _{n_j}}'(U_k^{\lambda _{n_j}}) -I'_0(U_k^0),U_k^{\lambda _{n_j}}-U_k^0\rangle \\&\qquad +\int _{{\mathbb {R}}^3} \bigg (|U_k^{\lambda _{n_j}}|^{p-2}U_k^{\lambda _{n_j}}-|U_k^0|^{p-2}U_k^0\bigg ) (U_k^{\lambda _{n_j}}-U_k^0)dx\\&\qquad -\lambda _{n_j}\Vert U_k^{\lambda _{n_j}}\Vert ^{2\alpha }\int _{{\mathbb {R}}^3}\bigg (\nabla U_k ^{\lambda _{n_j}}\nabla (U_k^{\lambda _{n_j}}-U_k^0)+V(|x|)U_k^{\lambda _{n_j}}(U_k^{\lambda _{n_j}}-U_k^0)\bigg )\\&\qquad \rightarrow 0, \qquad as\ j\rightarrow \infty . \end{aligned} \end{aligned}$$

Then $U_k^{\lambda _{n_j}}\rightarrow U_k^0$ strongly in $H_V$.

Next, we prove $(U_k^0)_i\ne 0$. Since $\langle I_{\lambda _{n_j}}'(U_k^{\lambda _{n_j}}),(U_k^{\lambda _{n_j}})_i\rangle =0$, there is a number $\delta >0$ such that

$$\begin{aligned} \liminf _{j\rightarrow \infty } \Vert (U_k^{\lambda _{n_j}})_i\Vert _i\ge \delta >0. \end{aligned}$$

This together with the compact embedding $H_V\hookrightarrow L^s({\mathbb {R}}^3)$, gives that

$$\begin{aligned} \delta ^2\le \Vert (U_k^{\lambda _{n_j}})_i\Vert _i^2\le \int _{{\mathbb {R}}^3} |(U_k^{\lambda _{n_j}})_i|^p\rightarrow \int _{{\mathbb {R}}^3} |(U_k^0)_i|^p, \end{aligned}$$

which shows that $(U_k^0)_i\ne 0$. Thus, $U_k^0$ is a radial nodal solution of (1.12) with exactly $k+1$ nodal domains.

Step 3. We prove that $U_{k}^0$ is a least energy radial solution of (1.12) among all the radial solutions changing sign exactly k times.

In fact, according to [2, Theorem 2.1], we assume that there is $\bar{{{\textbf {r}}}}_{k}\in \Gamma _k$ and $V_k:=\sum _{i=1}^{k+1}v_{i}$ is a least energy radial solution of (1.12) among all the nodal solutions changing sign exactly k times, where $v_i$ is supported on annuli $B_i^{\bar{{{\textbf {r}}}}_{k}}$. We assume that $U_k^{\lambda _n}:=w_1^{\lambda _n}+\cdots +w_{k+1}^{\lambda _n}.$

By Lemma 2.3, for each $\lambda _n>0$, there is a unique $(k+1)-$tuple $(t_1(\lambda _n),\ldots ,t_{k+1}(\lambda _n))$ of positive numbers such that

$$\begin{aligned} (t_1(\lambda _n)v_{1},\ldots ,t_{k+1}(\lambda _n)v_{k+1})\in {\mathcal {M}}_k^{{{\textbf {r}}}_k}. \end{aligned}$$

Then, for $i=1,\ldots ,k+1$, we have

$$\begin{aligned} a(t_i(\lambda _n))^2\Vert v_{i}\Vert _i^2+\lambda _n(t_i(\lambda _n))^2\Vert v_{i}\Vert _i^2 \bigg (\sum _{j=1}^{k+1}(t_j(\lambda _n))^2\Vert v_{j}\Vert _i^2\bigg )^{\alpha }=\int _{B_i^{{{\textbf {r}}}_k}} (t_i(\lambda _n))^p|v_{i}|_i^p.\nonumber \\ \end{aligned}$$

(4.8)

Recall that $v_{i}$ satisfies $a\Vert v_{i}\Vert _i^2=\int _{B_i^{{{\textbf {r}}}_k}}|v_{i}|_i^p.$ One can easily check that

$$\begin{aligned} (t_1(\lambda _n),\ldots ,t_{k+1}(\lambda _n))\rightarrow (1,\ldots ,1),\quad \text{ as }\quad n\rightarrow \infty . \end{aligned}$$

(4.9)

From (4.8)–(4.9), we have

$$\begin{aligned} \begin{aligned} I_0(V_k)\le I_0(U_k^0)=\lim _{n\rightarrow \infty }I_{\lambda _n}(U_k^{\lambda _n})&=\lim _{n\rightarrow \infty }E_{\lambda _n}(w_1^{\lambda _n},\ldots ,w_{k+1}^{\lambda _n})\\&\le \lim _{n\rightarrow \infty }E_{\lambda _n}(t_1(\lambda _n)v_{1}+\cdots +t_{k+1}(\lambda _n)v_{k+1})\\&=E_0(v_{1}+\cdots +v_{k+1})\\ &=I_0(V_k). \end{aligned} \end{aligned}$$

Therefore, $U_k^0$ is a least energy radial solution of (1.12) which changes sign exactly k times. $\square$