1 Introduction and statement of main result

Consider the Kirchhoff type equation

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}{\text {d}}x\right) \Delta u+V(x)u=f(x,u), \ \ u\in H^1(\mathbb {R}^3), \end{aligned}$$
(1.1)

where \(a,b>0\), V(x) is a smooth function. This problem is related to the stationary analogue of the equation

$$\begin{aligned} \frac{\partial ^2u}{\partial t^2}-\left( a+b\int _0^L\left( \frac{\partial u}{\partial x}\right) ^2{\text {d}}x\right) \frac{\partial ^2u}{\partial x^2}=0, \end{aligned}$$

which was proposed by Kirchhoff [12] as an extension of the classical D’Alembert’s wave equations for free vibration of elastic string. After Lions [15] introduced an abstract function analysis framework to the problem, many researchers have paid attention to it, see [1, 4] and the references therein.

There are many interesting results on the existence and multiplicity of solutions for problem (1.1), such as [6, 8,9,10, 13, 19, 21, 23] and references therein. In particular, He and Zou [10] studied the existence of positive solutions for (1.1) by the variational method, and obtained the multiplicity by means of Category theory. Moreover, they also studied the concentrated behavior of positive solutions. When \(f(x,u)=|u|^{p-1}u\), Li and Ye [13] proved (1.1) has a positive ground state solution by using a monotonicity trick and global compactness lemma. This result can be seen as a partial extension of the results of He and Zou in [10]. Wu [19] considered the existence of nontrivial solutions and infinitely many high energy solutions for (1.1) using a symmetric Mountain Pass Theorem. Guo [6] considered the positive ground state solution of (1.1) without classical Ambrosetti–Rabinowitz condition by using variational methods. The existence of sign-changing solution of (1.1) has also been studied in [16, 22].

Moreover, there are many results about the existence of nodal solutions for elliptic problems. Bartsch and Willem [2], Cao and Zhu [3] proved that for any positive integer k, there exists a pair of solutions \({u_k}^\pm \) having exact k nodes for the following equation:

$$\begin{aligned} -\Delta u +V(|x|)u=f(|x|,u),\ \ u\in H^1({\mathbb {R}^{3}}). \end{aligned}$$

By gluing method, Deng, Peng and Shuai [5] considered the existence and asymptotic behavior of nodal solutions for the following Kirchhoff equation:

$$\begin{aligned} -\left( a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}{\text {d}}x\right) \Delta u+V(|x|)u=f(|x|,u),x\in {\mathbb {R}^{3}}. \end{aligned}$$
(1.2)

Due to the existence of nonlocal items, gluing method cannot be used to solve this problem directly. To solve this difficulty, they regard this 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 vanishing on the complement. Huang, Yang and Yu [11] showed the existence of nodal solutions of Choquard equation by the same method as in [5]. Wang and Guo [20] proved the existence and nonexistence of nodal solutions for Choquard type equations with perturbation by employing the variational method, gluing approach and the Brouwer degree. Recently, Guo and Wu [7] showed the existence of nodal solutions for the Schrödinger–Poisson equations with convolution terms.

Motivated by the above results, we intend to establish infinitely many nodal solutions to the following equation:

$$\begin{aligned}&-\left( a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}{\text {d}}x\right) \Delta u+V(x)u\nonumber \\&\quad =\left( \int _{\mathbb {R}^{3}}\frac{|u(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}y\right) |u|^{p-2}u+|u|^{q-2}u, \ \ u\in H^1(\mathbb {R}^3), \end{aligned}$$
(1.3)

where V(x) is a smooth function, a, b are positive constants, \(\alpha \in (1,3)\) and \(q\in (4,6)\).

Our main result can be stated as follows.

Theorem 1.1

Suppose that V(x) satisfies

(V) \(V(x)=V(|x|)\in C([0,\infty ),\mathbb {R})\) is bounded from below by a positive constant \(V_0\).

Moreover, if \(\alpha \in (1,3)\), \(p\in (4,3+\alpha )\) and \(q\in (4,6)\). Then for any positive integer k, Eq. (1.3) has a radial solution \(u_k\) changing sign exactly k times.

The paper is organized as follows. In Sect. 2, we give some notations and preliminary results. In Sect. 3, we are devoted to the proof of our main result which mainly show the construction of least energy nodal solutions changing sign exactly k times.

2 Variational framework and some results in the matrix theory

In this section, we give some notations and preliminary results. First, we present the variational framework. The space \(H_V(\mathbb {R}^3)\) is defined by

$$\begin{aligned} H_V(\mathbb {R}^3):=\left\{ u\in H_r^1(\mathbb {R}^3)\ :\int _{\mathbb {R}^3}\left( a|\nabla u|^2+V(|x|)u^2\right) {\text {d}}x<\infty \right\} , \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u\Vert _V=\left( \int _{\mathbb {R}^3}\left( a|\nabla u|^2+V(|x|)u^2\right) {\text {d}}x\right) ^{1/2}. \end{aligned}$$

The energy functional associated with problem (1.3) is given by

$$\begin{aligned} {I_b}(u)= & {} \frac{1}{2}{\Vert u\Vert }^2_V+\frac{b}{4}\Big (\int _{\mathbb {R}^{3}}|\nabla u|^2{\text {d}}x\Big )^2- \frac{1}{2p}\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y \\&-\frac{1}{q}\int _{\mathbb {R}^{3}}|u|^q{\text {d}}x. \end{aligned}$$

By Hardy–Littlewood–Sobolev inequality ([14, Theorem 4.3]), \(I_b\) is well defined on \(H^{1}(\mathbb {R}^3)\) when \(p\in (4,3+\alpha )\).

For any integer k, we define

$$\begin{aligned} \Gamma _k=\left\{ \varvec{\rho }_k=(\rho _1,\ldots ,\rho _k)\in \mathbb {R}^k |\quad 0:=\rho _0<\rho _1<\cdots <\rho _{k+1}:=\infty \right\} , \end{aligned}$$

and for each \(\varvec{\rho }_k\in \Gamma _k\), set

$$\begin{aligned}&{B^{\varvec{\rho }_k}_1}=\left\{ x\in \mathbb {R}^3:0\le |x|<\rho _1\right\} ,\\&\quad {B^{\varvec{\rho }_k}_i}=\left\{ x\in \mathbb {R}^3:\rho _{i-1}<|x|<\rho _i\right\} ,\quad \text{ for }\ i=1,2,\ldots ,k;\\&\quad B^{\varvec{\rho }_k}_{k+1}=\left\{ x\in \mathbb {R}^3:|x|\ge \rho _k\right\} . \end{aligned}$$

Fix \(\varvec{\rho }_k\)=\((\rho _1,\ldots ,\rho _{k})\in \Gamma _k\) and thereby a family of \(\left\{ B^{\varvec{\rho }_k}_i\right\} ^{k+1}_{i=1}\), we denote

$$\begin{aligned} H^{\varvec{\rho }_k}_i=\left\{ u\in H^1_0(B^{\varvec{\rho }_k}_i)|\ u(x)=u(|x|),u(x)=0 \quad if \ x \not \in B^{\varvec{\rho }_k}_i\right\} , \end{aligned}$$

for \(i=1,\ldots ,k+1\). Therefore, \(H^{\varvec{\rho }_k}_i\) is a Hilbert space with the norm

$$\begin{aligned} \Vert u\Vert _i=\left( \int _{B^{\varvec{\rho }_k}_i}(a|\nabla u|^2+V(|x|)u^2){\text {d}}x\right) ^{1/2}. \end{aligned}$$

Let us set

$$\begin{aligned} (u_i,u_j)_*=\int _{B^{\varvec{\rho }_k}_i}|\nabla u_i|^2{\text {d}}x\int _{B^{\varvec{\rho }_k}_j}|\nabla u_j|^2{\text {d}}x, \end{aligned}$$
(2.1)

and

$$\begin{aligned} (u_i,u_j)_\alpha =\int _{B^{\varvec{\rho }_k}_i}\int _{B^{\varvec{\rho }_k}_j}\frac{|u_i(x)|^p|u_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y. \end{aligned}$$
(2.2)

We now define the function \(J_b\): \({\mathcal {H}}^{\varvec{\rho }_k}_k\rightarrow \mathbb {R}\) by

$$\begin{aligned}&J_b(u_1,\ldots ,u_{k+1}):=I_b\left( \sum _{i=1}^{k+1}u_i\right) \\&\quad =\sum _{i=1}^{k+1}\left( \frac{1}{2}\Vert u_i\Vert ^2_i+\frac{b}{4}\left( \int _{B^{\varvec{\rho }_k}_i}|\nabla u_i|^2{\text {d}}x\right) ^2\right. \\&\qquad \left. -\frac{1}{2p}\int _{B^{\varvec{\rho }_k}_i}\int _{B^{\varvec{\rho }_k}_i}\frac{|u_i(x)|^p|u_i(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y-\frac{1}{q}\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\right) \\&\qquad +\sum _{\begin{array}{c} i,j=1\\ j\ne i \end{array}}^{k+1}\left( \frac{b}{4}\int _{B^{\varvec{\rho }_k}_i}|\nabla u_i|^2{\text {d}}x\int _{B^{\varvec{\rho }_k}_j}|\nabla u_j|^2{\text {d}}x-\right. \\&\qquad \qquad \left. \frac{1}{2p}\int _{B^{\varvec{\rho }_k}_i}\int _{B^{\varvec{\rho }_k}_j}\frac{|u_i(x)|^p|u_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y\right) \\&\quad =\sum _{i=1}^{k+1}\left( \frac{1}{2}\Vert u_i\Vert ^2_i+\frac{b}{4}(u_i,u_i)_*-\frac{1}{2p}(u_i,u_i)_\alpha -\frac{1}{q}\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\right) \\&\quad +\sum _{\begin{array}{c} i,j=1\\ j\ne i \end{array}}^{k+1}\left( \frac{b}{4}(u_i,u_j)_*-\frac{1}{2p}(u_i,u_j)_\alpha \right) , \end{aligned}$$

where \({\mathcal {H}}^{\varvec{\rho }_k}_k=H^{\varvec{\rho }_k}_1\times \cdots \times H^{\varvec{\rho }_k}_{k+1}\) and \(u_i\in H^{\varvec{\rho }_k}_i\) for \(i=1,\ldots ,k+1\). Obviously, for each \(i=1,\ldots ,k+1\)

$$\begin{aligned}&\partial _{u_i}J_b(u_1,\ldots ,u_{k+1})u_i=\Vert u_i\Vert _i^2+b(u_i,u_i)_*-(u_i,u_i)_\alpha \\&\quad -\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x+\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1} \left( b(u_i,u_j)_*-(u_i,u_j)_\alpha \right) . \end{aligned}$$

If \((u_1,\ldots ,u_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) is a critical point of \(J_b\), then every component \(u_i\) satisfies

$$\begin{aligned} \qquad \left\{ \begin{array}{ll} -\left( a+b\sum \limits _{j=1}^{k+1}\int _{B^{\varvec{\rho }_k}_j}|\nabla u_j|^2{\text {d}}x\right) \Delta u_i+V(|x|)u_i=\\ \qquad \qquad \qquad \Big (\int _{\mathbb {R}^3}\frac{|\sum \limits _{j=1}^{k+1}u_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}y\Big )|u_i|^{p-2}u_i+|u_i|^{q-2}u_i\ &{} \mathrm{}\ x\in B^{\varvec{\rho }_k}_i,\\ u_i=0 \ &{} \mathrm{}\ x\not \in B^{\varvec{\rho }_k}_i. \end{array} \right. \end{aligned}$$
(2.3)

We regard \(u_i \in H^{\varvec{\rho }_k}_i\) as an element in \(H^1(\mathbb {R}^3)\) by setting \(u\equiv 0\) in \({\mathbb {R}^3}\setminus {B^{\varvec{\rho }_k}_i}\). To find least energy radial solution which changes signs exactly k times of (1.3), let Nehari manifold

$$\begin{aligned} \begin{aligned} {\mathcal {N}}_k=\left\{ u\in H_V: \text {there\ exists}\ \varvec{\rho }_k\ \ \text {such\ that}\ (-1)^{i+1}u_i>0 \ in\ B^{\varvec{\rho }_k}_i, \right. \\ \left. u_i=0 \ on \ \partial B^{\varvec{\rho }_k}_i, \ \ \text {and} \ I^\prime _b(u)u_i=0, \quad \forall \, 1\le i\le k+1 \right\} , \end{aligned} \end{aligned}$$

and \(c_k=\inf \limits _{{\mathcal {N}}_k}J_b\). Obviously, \(u=\sum \nolimits _{i=1}^{k+1}u_i\) and \({\mathcal {N}}_k\) consists of nodal functions with precisely k nodes.

Least energy radial solution of (1.3) which changes signs exactly k times will constructed by gluing the solutions of the system (2.3). To this end, we look for critical points of \(J_b\) with nonzero component by considering the following Nehari type set

$$\begin{aligned} {\mathcal {M}}^{\varvec{\rho }_k}_k= & {} \left\{ (u_1,\ldots ,u_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k: u_i\ne 0,\ \left<\partial _{u_i}J_b(u_1,u_2,\ldots ,u_{k+1}),u_i\right>\right. \\&\quad \left. =0,i=1,\ldots ,k+1\right\} . \end{aligned}$$

Next, we will present some results of the matrix theory in order to prove that \({\mathcal {M}}^{\varvec{\rho }_k}_k\) is nonempty.

Lemma 2.1

[11] For any \((u_1,\ldots ,u_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) with \(u_i\ne 0\),\(i=1,\ldots ,k+1\), define the matrix \(A:=(a_{ij})_{(k+1)\times (k+1)}\) by \(a_{ij}=(u_i,u_j)_\alpha \). Then the matrix A is positive definite.

Lemma 2.2

(Gersgorin Disc Theorem [18, Theorem 1.1]) For any matrix \(B=(b_{ij})\in \mathbb {C}^{n\times n}\) and any eigenvalue \(\lambda \in \sigma (B):\)=\(\left\{ \mu \in \mathbb {C}: det(\mu E-B)=0\right\} \), there is a positive integer \(m\in \left\{ 1,\ldots ,n\right\} \) such that

$$\begin{aligned} |\lambda -b_{m,m}|\le \sum \limits _{\begin{array}{c} j=1\\ j\ne m \end{array}}^{k+1}|b_{m,j}|. \end{aligned}$$

By this lemma, we have the following result.

Lemma 2.3

[7] For any \(b_{ij}\)=\(b_{ji}>0\) with \(i\ne j\in \left\{ 1,\ldots ,n\right\} \) and \(s_i>0\) with \(i=1,\ldots ,n\), define the matrix \(C:=(c_{ij})_{n\times n}\) by

$$\begin{aligned} c_{ij}= \left\{ \begin{array}{ll} -\sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}s_mb_{im}/s_i\ \ \ &{} \mathrm{}\ \mathrm{for} \quad i=j,\\ b_{ij}>0 \ \ \ &{} \mathrm{}\ \mathrm{for} \quad i\ne j. \end{array} \right. \end{aligned}$$

Then the real symmetric matrix \((c_{ij})_{n\times n}\) is non-positive definite.

3 The proof of Theorem 1.1

In this section, we are devoted to the proof of Theorem 1.1. First, we give the following lemma.

Lemma 3.1

Assume that \(\varvec{\rho }_k\in \Gamma _k\) is fixed. Then for each \((v_1,\ldots ,v_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) with \(v_i\ne 0\) for \(i=1,\ldots ,k+1\), there exists a unique \((k+1)\) tuple \((t_1,\ldots ,t_{k+1})\in (\mathbb {R}>0)^{k+1}\) such that \((t_1v_1,\ldots ,t_{k+1}v_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\).

Proof

When \(q\ge p\), for each \((v_1,\ldots ,v_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) with \(v_i\ne 0\) for \(i=1,\ldots ,k+1\), we define \(G:(\mathbb {R}\ge 0)^{k+1}\rightarrow \mathbb {R}\) by

$$\begin{aligned} G(c_1,\ldots ,c_{k+1})=J_b(c^{\frac{1}{p}}_1v_1,\ldots ,c^{\frac{1}{p}}_{k+1}v_{k+1}). \end{aligned}$$

Then

$$\begin{aligned} G(c_1,\ldots ,c_{k+1})=&\sum _{i=1}^{k+1}\left[ \frac{1}{2}c^{\frac{2}{p}}_i\Vert v_i\Vert ^2_i+\frac{b}{4}c^{\frac{4}{p}}_i(v_i,v_i)_*- \frac{1}{2p}c^2_i(v_i,v_i)_\alpha -\frac{1}{q}c^{\frac{q}{p}}_i\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x\right] \\&+\sum _{\begin{array}{c} i,j=1\\ j\ne i \end{array}}^{k+1}\left( \frac{b}{4}c^{\frac{2}{p}}_ic^{\frac{2}{p}}_j(v_i,v_j)_*-\frac{1}{2p}c_ic_j(v_i,v_j)_\alpha \right) . \end{aligned}$$

It is clearly that G is continuous and \(G(c_1,\ldots ,c_{k+1})\rightarrow 0\) as \(|(c_1,\ldots ,c_{k+1})|\rightarrow 0\) and \(G(c_1,\ldots ,c_{k+1})\rightarrow -\infty \) as \(|(c_1,\ldots ,c_{k+1})|\rightarrow \infty \), due to \(p\in (4,\alpha +3)\) and \(q\in (4,6)\). Thus, G possesses a global maximum point \(({\bar{c}}_1,\ldots ,{\bar{c}}_{k+1})\in (\mathbb {R}\ge 0)^{k+1}\).

We claim that \({\bar{c}}_i>0\) for all \(i=1,\ldots ,k+1\). Otherwise, there exists \(i_0\in \left\{ 1,\ldots ,{k+1}\right\} \) such that \({\bar{c}}_i=0\). Without loss of generality, we assume \({\bar{c}}_1=0\). Then since

$$\begin{aligned}&G(\tau ,{\bar{c}}_2,\ldots ,{\bar{c}}_{k+1})\\&\quad = \frac{\tau ^{\frac{2}{p}}}{2}\Vert v_1\Vert ^2_1+\frac{b}{4}\tau ^{\frac{4}{p}}(v_1,v_1)_*- \frac{\tau ^2}{2p}(v_1,v_1)_\alpha -\frac{\tau ^{\frac{q}{p}}}{q}\int _{B^{\varvec{\rho }_k}_1}|v_1|^q{\text {d}}x\\&\qquad +\sum _{j=2}^{k+1}\left[ \frac{b}{2}\tau ^{\frac{2}{p}}{\bar{c}}^{\frac{2}{p}}_j(v_1,v_j)_*-\frac{1}{p}\tau {\bar{c}}_j(v_1,v_j)_\alpha \right] \\&\qquad +\sum _{i=2}^{k+1}\left[ \frac{1}{2}{\bar{c}}^{\frac{2}{p}}_i\Vert v_i\Vert ^2_i+\frac{b}{4}{\bar{c}}^{\frac{4}{p}}_i(v_i,v_i)_*- \frac{{\bar{c}}^2_i}{2p}(v_i,v_i)_\alpha -\frac{{\bar{c}}^{\frac{q}{p}}_i}{q}\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x\right] \\&\qquad + \sum _{\begin{array}{c} i,j=2\\ j\ne i \end{array}}^{k+1}\left[ \frac{b}{4}{\bar{c}}^{\frac{2}{p}}_i{\bar{c}}^{\frac{2}{p}}_j(v_i,v_j)_*-\frac{1}{2p}{\bar{c}}_i{\bar{c}}_j(v_i,v_j)_\alpha \right] \end{aligned}$$

is increasing with respect to \(\tau >0\) when \(\tau \) is small enough. Thus, \((0,{\bar{c}}_2,\ldots , {\bar{c}}_{k+1})\) is not a maximum point of G. This contradicts the assumption above. Therefore, the claim follows.

Next, we prove that this global maximum point is unique in \((\mathbb R> 0)^{k+1}\). In fact, by direct computation, we have

$$\begin{aligned} \frac{\partial G}{\partial c_i}=&\frac{1}{p}c^{\frac{2}{p}-1}_i\Vert v_i\Vert ^2_i+\frac{b}{p}c^{\frac{4}{p}-1}_i(v_i,v_i)_*-\frac{1}{p}c_i(v_i,v_i)_\alpha - \frac{1}{p}c^{\frac{q}{p}-1}_i\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x\\&+\frac{b}{p}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}c^{\frac{2}{p}-1}_ic^{\frac{2}{p}}_j(v_i,v_j)_*-\frac{1}{p}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}c_j(v_i,v_j)_\alpha ,\\ \frac{\partial ^2G}{\partial c^2_i}=&\frac{2-p}{p^2}c^{\frac{2}{p}-2}_i\Vert v_i\Vert ^2_i+\frac{b(4-p)}{p^2}c^{\frac{4}{p}-2}_i(v_i,v_i)_*-\frac{1}{p}(v_i,v_i)_\alpha \\&-\frac{q-p}{p^2}c^{\frac{q}{p}-2}_i\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x+\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}\frac{b(2-p)}{p^2}c^{\frac{2}{p}-2}_ic^{\frac{2}{p}}_j(v_i,v_j)_*,\\ \frac{\partial ^2G}{\partial c_i\partial c_j}=&\frac{2b}{p^2}c^{\frac{2}{p}-1}_ic^{\frac{2}{p}-1}_j(v_i,v_j)_*-\frac{1}{p}(v_i,v_j)_\alpha \quad for \quad i\ne j. \end{aligned}$$

Let the matrix

$$\begin{aligned} \left( \frac{\partial ^2G}{\partial c_i\partial c_j}\right) _{(k+1)\times (k+1)}= & {} \frac{1}{p^2}(a_{ij})_{(k+1)\times (k+1)}+\frac{2b}{p^2}(b_{ij})_{(k+1)\times (k+1)}\\&+\frac{1}{p}(c_{ij})_{(k+1)\times (k+1)}, \end{aligned}$$

where

$$\begin{aligned} a_{ij}= & {} \left\{ \begin{array}{ll} (2-p)c^{\frac{2}{p}-2}_i\Vert v_i\Vert ^2_i+(4-p)bc^{\frac{4}{p}-2}_i(v_i,v_i)_*\\ +(4-p)\sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}bc^{\frac{2}{p}-2}_ic^{\frac{2}{p}}_m(v_i,v_m)_*- \\ (q-p)c^{\frac{2}{p}-2}_i\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x,\ \ \ &{} \mathrm{}\ i=j,\\ 0 \ \ \ &{} \mathrm{}\ i\ne j \end{array}\right. \\ b_{ij}= & {} \left\{ \begin{array}{ll} -\sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}c^{\frac{2}{p}-2}_ic^{\frac{2}{p}}_m(v_i,v_m)_*\ \ \ &{} \mathrm{}\ i=j,\\ c^{\frac{2}{p}-1}_ic^{\frac{2}{p}-1}_j(v_i,v_j)_*\ \ \ &{} \mathrm{}\ i\ne j, \end{array} \right. \quad \quad c_{ij}= \left\{ \begin{array}{ll} -(v_i,v_i)_\alpha \ \ \ &{} \mathrm{}\ i=j,\\ -(v_i,v_j)_\alpha \ \ \ &{} \mathrm{}\ i\ne j. \end{array} \right. \end{aligned}$$

Note the fact that \(p>4\) and \(q\ge p\), thus \((a_{ij})\) is negative definite. By Lemma 2.3, \((b_{ij})\) is non-positive definite. By Lemma 2.1, \((c_{ij})\) is negative definite. Thus, \(\left( \frac{\partial ^2G}{\partial _{c_i}\partial _{c_j}}\right) \) is negative definite and G is strictly concave in \((\mathbb R>0)^{k+1}\). Therefore, G has a unique maximum point in \((\mathbb R>0)^{k+1}\).

When \(q<p\), for each \((v_1,\ldots ,v_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) with \(v_i\ne 0\) for \(i=1,\ldots ,k+1\), we define \(G:(\mathbb {R}\ge 0)^{k+1}\rightarrow \mathbb {R}\) by

$$\begin{aligned} G(c_1,\ldots ,c_{k+1})=J_b(c^{\frac{1}{q}}_1v_1,\ldots ,c^{\frac{1}{q}}_{k+1}v_{k+1}). \end{aligned}$$

Then

$$\begin{aligned}&G(c_1,\ldots ,c_{k+1})\\&\quad = \sum _{i=1}^{k+1}\left[ \frac{1}{2}c^{\frac{2}{q}}_i\Vert v_i\Vert ^2_i+\frac{b}{4}c^{\frac{4}{q}}_i(v_i,v_i)_*- \frac{1}{2p}c^{\frac{2p}{q}}_i(v_i,v_i)_\alpha -\frac{1}{q}c_i\int _{B^{\varvec{\rho }_k}_i}|v_i|^q{\text {d}}x\right] \\&\qquad +\sum _{\begin{array}{c} i,j=1\\ j\ne i \end{array}}^{k+1}\left( \frac{b}{4}c^{\frac{2}{q}}_ic^{\frac{2}{q}}_j(v_i,v_j)_*-\frac{1}{2p}c^{\frac{p}{q}}_ic^{\frac{p}{q}}_j(v_i,v_j)_\alpha \right) . \end{aligned}$$

By the same arguments as above, the conclusion follows. \(\square \)

We define \(\psi _b:({\mathbb {R}}>0)^{k+1}\rightarrow \mathbb {R}\) by

$$\begin{aligned} \psi _b(t_1,\ldots ,t_{k+1})=J_b(t_1v_1,\ldots ,t_{k+1}v_{k+1}), \end{aligned}$$

where \((v_1,\ldots ,v_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\). Then we have the following corollary.

Corollary 3.2

Let \(\varvec{\rho }_k\in \Gamma _k\). Then for any \((v_1,\ldots ,v_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k\) with \(v_i\ne 0\) for \(i=1,\ldots ,k+1\), there exists a unique global maximal point \(({\bar{t}}_1,\ldots ,{\bar{t}}_{k+1})\in \mathbb {R}_+^{k+1}\) of \(\psi _b\) such that

$$\begin{aligned} \psi _b({\bar{t}}_1,\ldots ,{\bar{t}}_{k+1})=\sup _{\mathbb {R}_+^{k+1}}\psi _b(t_1,\ldots ,t_{k+1}), \end{aligned}$$

and \(({\bar{t}}_1v_1,\ldots ,{\bar{t}}_{k+1}v_{k+1}) \in {\mathcal {M}}^{\varvec{\rho }_K}_k\).

Lemma 3.3

For \(p\in (4,3+\alpha )\), \(q\in (4,6)\), \({\mathcal {M}}^{\varvec{\rho }_k}_k\) is a differentiable manifold in \({\mathcal {H}}^{\varvec{\rho }_k}_k\). Moreover, all critical points of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\) are critical points of \(J_b\) in \({\mathcal {H}}^{\varvec{\rho }_k}_k\) with no zero component.

Proof

Note that

$$\begin{aligned} {\mathcal {M}}^{\varvec{\rho }_k}_k=\left\{ (u_1,\ldots ,u_{k+1})\in {\mathcal {H}}^{\varvec{\rho }_k}_k: u_i\ne 0,F(u_1,\ldots ,u_{k+1})=0,i=1,\ldots ,k+1\right\} , \end{aligned}$$

where \(F=(F_1,\ldots ,F_{k+1}):{\mathcal {H}}^{\varvec{\rho }_k}_k\rightarrow \mathbb {R}\) is given by

$$\begin{aligned} F_i=\partial _{u_i}J_b(u_1,\ldots ,u_{k+1})u_i. \end{aligned}$$

Then

$$\begin{aligned} F_i= & {} \Vert u_i\Vert ^2_i+b(u_i,u_i)_*-(u_i,u_i)_\alpha -\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\\&+\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}b(u_i,u_j)_*-\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_\alpha . \end{aligned}$$

When \(q\ge p\), by direct computation, we have that at each point \((u_1,\ldots ,u_{k+1}) \in {\mathcal {M}}^{\varvec{\rho }_k}_k\), there holds that

$$\begin{aligned} M_{ii}&:=\left<\partial _{u_i}F_i(u_1,\ldots ,u_{k+1}),u_i\right>\\&=2\Vert u_i\Vert ^2_i+4b(u_i,u_i)_*-2p(u_i,u_i)_\alpha -q\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\\&\quad +2b\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_*-p\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_\alpha \\&= (2-p)\Vert u_i\Vert ^2_i+(4b-bp)(u_i,u_i)_*-p(u_i,u_i)_\alpha \\&\quad -(q-p)\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x+(2b-bp)\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_*\end{aligned}$$

for \(i=1,\ldots ,k+1\), and

$$\begin{aligned} M_{ij}:= & {} \left<\partial _{u_i}F_j(u_1,\ldots ,u_{k+1}),u_i\right>\\= & {} 2b(u_i,u_j)_*-p(u_i,u_j)_\alpha ,\quad \text{ for }\ i\ne j \quad \text{ and }\quad i,j=1,\ldots ,k+1. \end{aligned}$$

By the same arguments as Lemma 3.1, when \(q\ge p\), the matrix

$$\begin{aligned} \left( M_{ij}\right) _{(k+1)\times (k+1)}=(a_{ij})_{(k+1)\times (k+1)}+(b_{ij})_{(k+1)\times (k+1)}+2b(c_{ij})_{(k+1)\times (k+1)} \end{aligned}$$

is negative definite and, therefore, det\(\left( M_{ij}\right) \ne 0\), where

$$\begin{aligned} a_{ij}= \left\{ \begin{array}{ll} (2-p)\Vert u_i\Vert ^2+(4-p)b(u_i,u_i)_*-(q-p)\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\\ +(4-p)b\sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}(u_i,u_m)_*,\ &{} \mathrm{if}\ i=j,\\ 0, \ \ \ &{} \mathrm{if}\ i\ne j, \end{array} \right. \\ b_{ij}= \left\{ \begin{array}{ll} -p(u_i,u_i)_\alpha ,\ \ \ &{} \mathrm{if}\ i=j,\\ -p(u_i,u_j)_\alpha , \ \ \ &{} \mathrm{if}\ i\ne j, \end{array} \right. \quad \quad c_{ij}= \left\{ \begin{array}{ll} -\sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}(u_i,u_m)_*,\ \ \ &{} \mathrm{if}\ i=j,\\ (u_i,u_j)_*,\ \ \ &{} \mathrm{if}\ i\ne j. \end{array} \right. \end{aligned}$$

When \(q<p\), by direct computation, we have that at each point \((u_1,\ldots ,u_{k+1}) \in {\mathcal {M}}^{\varvec{\rho }_k}_k\), there holds that

$$\begin{aligned} M_{ii}:=&\left<\partial _{u_i}F_i(u_1,\ldots ,u_{k+1}),u_i\right>\\ =&2\Vert u_i\Vert ^2_i+4b(u_i,u_i)_*-2p(u_i,u_i)_\alpha -q\int _{B^{\varvec{\rho }_k}_i}|u_i|^q{\text {d}}x\\&\quad +2b\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_*-p\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_\alpha \\ =&(2-q)\Vert u_i\Vert ^2_i+b(4-q)(u_i,u_i)_*-(2p-q)(u_i,u_i)_\alpha \\&\quad +b(2-q)\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_*-(p-q)\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{k+1}(u_i,u_j)_\alpha , \end{aligned}$$

for \(i=1,\ldots ,k+1\), and

$$\begin{aligned} M_{ij}=2b(u_i,u_j)_*-p(u_i,u_j)_\alpha ,\quad \text{ for }\ i\ne j \quad \text{ and }\quad i,j=1,\ldots ,k+1. \end{aligned}$$

By the same arguments as above, when \(q<p\), the matrix

$$\begin{aligned} \left( M_{ij}\right) _{(k+1)\times (k+1)}=(a_{ij})_{(k+1)\times (k+1)}+(b_{ij})_{(k+1)\times (k+1)}+2b(c_{ij})_{(k+1)\times (k+1)}, \end{aligned}$$

is also negative definite and, therefore, det\(\left( M_{ij}\right) \ne 0\), where

$$\begin{aligned} a_{ij}= \left\{ \begin{array}{ll} (2-q)\Vert u_i\Vert ^2+(4-q)b(u_i,u_i)_*-(p-q)(u_i,u_i)_\alpha \\ +b(4-q)\sum \limits _{\begin{array}{c} j=1\\ m\ne i \end{array}}^{k+1}(u_i,u_m)_*,\ \ &{} \mathrm{if}\ i=j,\\ 0, \ \ \ &{} \mathrm{if}\ i\ne j, \end{array} \right. \end{aligned}$$
$$\begin{aligned} b_{ij}= \left\{ \begin{array}{ll} -p(u_i,u_i)_\alpha ,\ \ \ &{} \mathrm{if}\ i=j,\\ -p(u_i,u_j)_\alpha , \ \ \ &{} \mathrm{if}\ i\ne j, \end{array} \right. \quad \quad c_{ij}= \left\{ \begin{array}{ll} - \sum \limits _{\begin{array}{c} m=1\\ m\ne i \end{array}}^{k+1}(u_i,u_m)_*,\ \ \ &{} \mathrm{if}\ i=j,\\ (u_i,u_j)_*,\ \ \ &{} \mathrm{if}\ i\ne j. \end{array} \right. \end{aligned}$$

Thus, \((M_{ij})\) is nonsingular at each point \((u_1,\ldots ,u_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\). So \({\mathcal {M}}^{\varvec{\rho }_k}_k\) is differentiable in \({\mathcal {H}}^{\varvec{\rho }_k}_k\).

If \((u_1,\ldots ,u_{k+1})\) is a critical point of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\), by the Lagrange multiplier principle, there exist \(\eta _1,\ldots ,\eta _{k+1}\) such that

$$\begin{aligned} J^\prime _b(u_1,\ldots ,u_{k+1})=\eta _1F^\prime _1(u_1,\ldots ,u_{k+1})+\cdots +\eta _{k+1}F^\prime _{k+1}(u_1,\ldots ,u_{k+1}). \end{aligned}$$

Applying \((u_1,0,\ldots ,0)\),\((0,u_2,\ldots ,0)\),\((0,\ldots ,0,u_{k+1})\) into the identity above, we get

$$\begin{aligned} \left( M_{ij}\right) \begin{bmatrix} \eta _1\\ \vdots \\ \eta _{k+1} \end{bmatrix}=\begin{bmatrix} 0\\ \vdots \\ 0 \end{bmatrix}. \end{aligned}$$

Since \(det\left( M_{ij}\right) \ne 0\), we see that \(\eta _i=0\) for all \(i=1,\ldots ,k+1\). Thus \((u_1,\ldots ,u_{k+1})\) is a critical point of \(J_b\). \(\square \)

Consider the infimum level

$$\begin{aligned} d(\varvec{\rho }_k):=\inf _{(u_1,\ldots ,u_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k}J_b(u_1,\ldots ,u_{k+1}). \end{aligned}$$

Then we have the following result.

Lemma 3.4

For any \(p\in (4,3+\alpha )\), \(q\in (4,6)\) and \(\varvec{\rho }_k\in \Gamma _k\), there is a minimizer \((\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho }_k}_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\) of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\) with \((-1)^{i+1}\xi ^{\varvec{\rho }_k}_i>0\) in \(B^{\varvec{\rho }_k}_i\), for \(i=1,\ldots ,k+1\) such that

$$\begin{aligned} J_b(\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho }_k}_{k+1})=d(\varvec{\rho }_k). \end{aligned}$$
(3.1)

Moreover, \((\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho }_k}_{k+1})\) satisfies (2.3).

Proof

For \((u_1,\ldots ,u_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\), denote by \(u=\sum \limits _{i=1}^{k+1}u_i\), then

$$\begin{aligned} 0=&\sum _{i=1}^{k+1}\partial _{u_i}J_b(u_1,\ldots ,u_{k+1})u_i=I^\prime _b\left( \sum _{i=1}^{k+1}u_i\right) \left( \sum _{i=1}^{k+1}u_i\right) \\ =&\Vert u\Vert ^2_V+b\left( \int _{\mathbb {R}^3}|\nabla u|^2{\text {d}}x\right) ^2-\int _{\mathbb {R}^3}\int _{\mathbb {R}^3}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}y{\text {d}}x-\int _{\mathbb {R}^3}|u|^q{\text {d}}x. \end{aligned}$$

By Hardy–Littlewood–Sobolev inequality and Sobolev embedding theorem, we can see

$$\begin{aligned} \Vert u\Vert ^2_V\le \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}y{\text {d}}x+\int _{\mathbb {R}^3}|u|^q{\text {d}}x\le c\Vert u\Vert ^{2p}_V+c\Vert u\Vert ^q_V. \end{aligned}$$

Since \(p\in (4,3+\alpha )\), \(q\in (4,6)\), we have \(\Vert u\Vert _V\ge c_1>0\) for some \(c_1>0\).

When \(q\ge 2p\), we have

$$\begin{aligned}&J_b(u_1,\ldots ,u_{k+1})=I_b\left( \sum _{i=1}^{k+1}u_i\right) =I_b(u)-\frac{1}{2p}I^\prime _b(u)u \nonumber \\&\quad =\left( \frac{1}{2}-\frac{1}{2p}\right) \Vert u\Vert ^2_V+b\left( \frac{1}{4}-\frac{1}{2p}\right) \left( \int _{\mathbb {R}^3}|\nabla u|^2{\text {d}}x\right) ^2\nonumber \\&\qquad +\left( \frac{1}{2p}-\frac{1}{q}\right) \int _{\mathbb {R}^3}|u|^q{\text {d}}x\nonumber \\&\quad \ge \left( \frac{1}{2}-\frac{1}{2p}\right) \Vert u\Vert ^2_V\ge c_2>0. \end{aligned}$$
(3.2)

When \(q< 2p\), we have

$$\begin{aligned}&J_b(u_1,\ldots ,u_{k+1})=I_b\left( \sum _{i=1}^{k+1}u_i\right) =I_b(u)-\frac{1}{q}I^\prime _b(u)u \nonumber \\&\quad =\left( \frac{1}{2}-\frac{1}{q}\right) \Vert u\Vert ^2_V+b\left( \frac{1}{4}-\frac{1}{q}\right) \left( \int _{\mathbb {R}^3}|\nabla u|^2{\text {d}}x\right) ^2\nonumber \\&\qquad +\left( \frac{1}{q}-\frac{1}{2p}\right) \int _{\mathbb {R}^3}\int _{\mathbb {R}^3}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}y{\text {d}}x\nonumber \\&\quad \ge \left( \frac{1}{2}-\frac{1}{q}\right) \Vert u\Vert ^2_V\ge c_2>0. \end{aligned}$$
(3.3)

Thus, \(d(\varvec{\rho }_k)\ge c_2>0\). We can choose a minimizing sequence \(\{(u^n_1,\ldots ,u^n_{k+1})\}_{n=1}^\infty \subset {\mathcal {M}}^{\varvec{\rho }_k}_k\) of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\). From (3.2), (3.3), we know that \(\left\{ u_i^n\right\} _{n=1}^\infty \) is bounded in \({\mathcal {H}}^{\varvec{\rho }_k}_k\). Up to a subsequence, \((u^n_1,\ldots ,u^n_{k+1})\) converges to an element \((u^0_1,\ldots ,u^0_{k+1})\) weakly in \({\mathcal {H}}^{\varvec{\rho }_k}_k\).

We claim that for all \(i=1,\ldots ,k+1\), \(u^0_i\ne 0\). If \(u_i^n\rightarrow u_i^0\) strongly in \(H^{\varvec{\rho }_k}_i\), for any \(i=1,\ldots ,k+1\),

$$\begin{aligned} \Vert u^n_i\Vert ^2_i&\le \int _{\mathbb {R}^3}\int _{B^{\varvec{\rho }_k}_i}\frac{\left| \sum \limits _{j=1}^{k+1}u^n_j(y)\right| ^p|u^n_i(x)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y+ \int _{B^{\varvec{\rho }_k}_i}|u^n_i|^q{\text {d}}x\\&\le c\left( \left\| \sum _{i=1}^{k+1}u^n_i\right\| ^p\Vert u^n_i\Vert ^p_i+\Vert u^n_i\Vert ^q_i\right) \le c\left( \Vert u^n_i\Vert ^p_i+\Vert u^n_i\Vert ^q_i\right) . \end{aligned}$$

Hence,

$$\begin{aligned} \liminf _{n\rightarrow \infty }\Vert u^n_i\Vert _i>0. \end{aligned}$$
(3.4)

This implies that \(u^0_i\ne 0\) for all \(i=1,\ldots ,k+1\). Thus, the claim follows.

If \(u_i^n\rightharpoonup u_i^0\) weakly but not strongly in \(H^{\varvec{\rho }_k}_i\), then there exists \(i\in \left\{ 1,\ldots ,k+1\right\} \) such that \(\Vert u_i^0\Vert _i<\liminf \limits _{n\rightarrow \infty }\Vert u_i^n\Vert _i\), we have

$$\begin{aligned}&\Vert u_i^0\Vert _i^2+b\int _{\mathbb {R}^3}\left| \nabla \left( \sum _{i=1}^{k+1}u^0_i(x)\right) \right| ^2{\text {d}}x\int _{B^{\varvec{\rho }_k}_i}|\nabla u^0_i|^2{\text {d}}x\\ <&\int _{\mathbb {R}^3}\int _{B^{\varvec{\rho }_k}_i}\frac{\left| \sum \limits _{j=1}^{k+1}u^0_j(y)\right| ^p|u^0_i(x)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y + \int _{B^{\varvec{\rho }_k}_i}|u^0_i(x)|^q{\text {d}}x. \end{aligned}$$

By Hardy–Littlewood–Sobolev inequality and Sobolev embedding theorem, the claim also follows.

We further claim that \((u^n_1,\ldots ,u^n_{k+1})\rightarrow (u^0_1,\ldots ,u^0_{k+1})\) in \({\mathcal {H}}^{\varvec{\rho }_k}_k\). Suppose by contradiction that the claim does not hold. There exists \(i\in \left\{ 1,\ldots ,k+1\right\} \) such that \(\Vert u_i^0\Vert _i<\liminf \limits _{n\rightarrow \infty }\Vert u_i^n\Vert _i\). By Lemma 3.1, there is \((t^0_1,\ldots ,t^0_{k+1})\ne (1,\ldots ,1)\) satisfying \((t^0_1u^0_1,\ldots ,t^0_{k+1}u^0_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\), then

$$\begin{aligned} d(\varvec{\rho }_k)&\le J_b(t^0_1u^0_1,\ldots ,t^0_{k+1}u^0_{k+1})\\&<\liminf _{n\rightarrow \infty }\left\{ \frac{1}{2}\sum _{i=1}^{k+1}(t^0_i)^2\Vert u^n_i\Vert ^2_i+\frac{b}{4}\left( \sum _{i=1}^{k+1}(t^0_i)^2\int _{\mathbb {R}^3} |\nabla u^n_i(x)|^2{\text {d}}x\right) ^2\right. \\&\quad -\frac{1}{2p}\sum \limits _{i,j=1}^{k+1}(t^0_i)^p(t^0_j)^p\int _{B^{\varvec{\rho }_k}_i}\int _{B^{\varvec{\rho }_k}_j} \frac{(|u^n_i(y)|^p)(|u^n_j(x)|^p)}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y\\&\quad \left. -\sum _{i=1}^{k+1}(t^0_i)^q\int _{B^{\varvec{\rho }_k}_i}|u^n_i(x)|^q{\text {d}}x\right\} \\&\le \liminf _{n\rightarrow \infty }J_b(u^n_1,\ldots ,u^n_{k+1})=d(\varvec{\rho }_k), \end{aligned}$$

which is a contradiction. Thus, the claim follows and \((u^0_1,\ldots ,u^0_{k+1})\in {\mathcal {M}}^{\varvec{\rho }_k}_k\) is a minimizer of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\).

It is easy to check that

$$\begin{aligned} (\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho }_k}_{k+1}):=(|u^0_1|,-|u^0_2|,\ldots ,(-1)^k|u^0_{k+1}|) \end{aligned}$$

belongs to \({\mathcal {M}}^{\varvec{\rho }_k}_k\) and is a minimizer of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }_k}_k}\) satisfying (3.1). From Lemma 3.3, it is a critical point of \(J_b\) in \({\mathcal {H}}^{\varvec{\rho }_k}_k\) and satisfying (2.3). Using the strong maximum principle, each component \((-1)^{i+1}\xi ^{\varvec{\rho }_k}_i>0\) in \(B^{\varvec{\rho }_k}_i\), for \(i=1,\ldots ,k+1\). The proof is complete. \(\square \)

Lemma 3.5

For any \(p\in (4,3+\alpha )\), \(q\in (4,6)\) and \(\varvec{\rho }_k=(\rho _1,\ldots ,\rho _k)\in \Gamma _k\)

  1. (i)

    For uniformly bounded \(\varvec{\rho }_k\), if \(\rho _i-\rho _{i-1}\rightarrow 0\) for some \(i\in \left\{ 1,\ldots ,k\right\} \), then \(d(\varvec{\rho }_k)\rightarrow +\infty \).

  2. (ii)

    If \(\rho _k\rightarrow \infty \), then \(d(\varvec{\rho }_k)\rightarrow +\infty \).

  3. (iii)

    d is continuous in \(\Gamma _k\). Therefore, there exists a \(\bar{\varvec{\rho }}_k\in \Gamma _k\) such that

    $$\begin{aligned}d(\bar{\varvec{\rho }}_k)=\inf _{\varvec{\rho }_k\in \Gamma _k}d(\varvec{\rho }_k).\end{aligned}$$

Proof

(i) By lemma 3.4, it is easy to see that for each \({\varvec{\rho }}_k\in \Gamma _k\), there exists a solution \(\varvec{\xi }^{\varvec{\rho }_k}=(\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho _k}}_{k+1})\in {{\mathcal {M}}^{\varvec{\rho }_k}_k}\) such that \(d({\varvec{\rho }_k})=J_b(\xi ^{\varvec{\rho }_k}_1,\ldots ,\xi ^{\varvec{\rho }_k}_{k+1})\). By Hardy–Littlewood–Sobolev inequality, Hölder inequality and embedding inequality, we have

$$\begin{aligned} \Vert \xi ^{\varvec{\rho }_k}_i\Vert ^2_i&=\int _{\mathbb {R}^3}\int _{B^{\varvec{\rho }_k}_i} \frac{|\sum \limits _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j(y)|^p|\xi ^{\varvec{\rho }_k}_i(x)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y+ \int _{B^{\varvec{\rho }_k}_i}|\xi ^{\varvec{\rho }_k}_i|^q{\text {d}}x\\&-\int _{\mathbb {R}^3}\sum _{j=1}^{k+1}| \nabla \xi ^{\varvec{\rho }_k}_j(x)|^2{\text {d}}x\int _{B^{\varvec{\rho }_k}_i}|\nabla \xi ^{\varvec{\rho }_k}_i|^2{\text {d}}x\\&\le c\left( |\sum _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j|^p_{L^{\frac{6p}{3+\alpha }}}|\xi ^{\varvec{\rho }_k}_i|^p_{L^{\frac{6p}{3+\alpha }}} +|\xi ^{\varvec{\rho }_k}_i|^q_{L^q}\right) \\&\le c\left( |\sum _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j|^p_{L^{\frac{6p}{3+\alpha }}}|\xi ^{\varvec{\rho }_k}_i|^p_{L^6}|B^{\varvec{\rho }_k}_i| ^{\frac{3+\alpha -p}{6}}+|\xi ^{\varvec{\rho }_k}_i|^q_{L^6}|B^{\varvec{\rho }_k}_i|^{\frac{6-q}{6}}\right) \\&\le c\left[ \left( \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\right) \Vert \xi ^{\varvec{\rho }_k}_i\Vert ^p_i|B^{\varvec{\rho }_k}_i|^{\frac{3+\alpha -p}{6}}+ \Vert \xi ^{\varvec{\rho }_k}_i\Vert ^q_i|B^{\varvec{\rho }_k}_i|^{\frac{6-q}{6}}\right] . \end{aligned}$$

Since \(\varvec{\rho }_k\) is uniformly bounded, then if \(\rho _i-\rho _{i-1}\rightarrow 0\) for some \(i\in \left\{ 1,\ldots ,k\right\} \), we have \(|B^{\varvec{\rho }_k}_i|\rightarrow 0\). From \(p\in (4,3+\alpha )\), \(q\in (4,6)\), we have \(\sum \limits _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\rightarrow \infty \). By the same arguments as in (3.2) or (3.3), we have

$$\begin{aligned}d(\varvec{\rho }_k)\ge \left( \frac{1}{2}-\frac{1}{2p}\right) \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^2_j\rightarrow \infty ,\end{aligned}$$

or

$$\begin{aligned}d(\varvec{\rho }_k)\ge \left( \frac{1}{2}-\frac{1}{q}\right) \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^2_j\rightarrow \infty .\end{aligned}$$

Then, \((i)\) holds.

\((ii)\) By the Strauss inequality [2], for every radial function \(u\in H_V\), we can find \(a_0>0\) such that \(u(x)\le \frac{a_0\Vert u\Vert _V}{|x|}\) for a.e. \(|x|>1\).This combined with Hardy–Littlewood–Sobolev inequality and embedding theorem, yields

$$\begin{aligned} \Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^2_{k+1}&=\int _{\mathbb {R}^3}\int _{B^{\varvec{\rho }_k}_{k+1}} \frac{|\sum \limits _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j(y)|^p|\xi ^{\varvec{\rho }_k}_{k+1}(x)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y+\int _{B^{\varvec{\rho }_k}_{k+1}}|\xi ^{\varvec{\rho }_k}_{k+1}|^q{\text {d}}x\\ -&\int _{\mathbb {R}^3}\sum _{j=1}^{k+1}|\nabla \xi ^{\varvec{\rho }_k}_j(x)|^2{\text {d}}x \int _{B^{\varvec{\rho }_k}_{k+1}}|\nabla \xi ^{\varvec{\rho }_k}_{k+1}(x)|^2{\text {d}}x\\&\le \int _{\mathbb {R}^3}\int _{B^{\varvec{\rho }_k}_{k+1}}\frac{|\sum \limits _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j(y)|^p |\xi ^{\varvec{\rho }_k}_{k+1}(x)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y+ \int _{B^{\varvec{\rho }_k}_{k+1}}|\xi ^{\varvec{\rho }_k}_{k+1}|^q{\text {d}}x\\&\le c\left( |\sum _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j|^p_{L^{\frac{6p}{3+\alpha }}}\right) \left( \int _{B^{\varvec{\rho }_k}_{k+1}} |\xi ^{\varvec{\rho }_k}_{k+1}|^{\frac{6p}{3+\alpha }}{\text {d}}x\right) ^{\frac{3+\alpha }{6}} +c\int _{B^{\varvec{\rho }_k}_{k+1}}|\xi ^{\varvec{\rho }_k}_{k+1}|^{q}{\text {d}}x\\&\le c\left( |\sum _{j=1}^{k+1}\xi ^{\varvec{\rho }_k}_j|^p_{L^{\frac{6p}{3+\alpha }}}\right) \left( \int _{B^{\varvec{\rho }_k}_{k+1}} |\xi ^{\varvec{\rho }_k}_{k+1}|^{\frac{6p}{3+\alpha }-2}|\xi ^{\varvec{\rho }_k}_{k+1}|^2{\text {d}}x\right) ^{\frac{3+\alpha }{6}}\\&\quad +c\int _{B^{\varvec{\rho }_k}_{k+1}}|\xi ^{\varvec{\rho }_k}_{k+1}|^{q-2}|\xi ^{\varvec{\rho }_k}_{k+1}|^2{\text {d}}x\\&\le c\left( \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\right) \Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{\left( \frac{6p}{3+\alpha }-2\right) \frac{3+\alpha }{6}}_{k+1} \left( \int _{B^{\varvec{\rho }_k}_{k+1}}|x|^{-\left( \frac{6p}{3+\alpha }-2\right) }|\xi ^{\varvec{\rho }_k}_{k+1}(x)|^2{\text {d}}x\right) ^{\frac{3+\alpha }{6}}\\&+c\Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{q-2}_{k+1}\left( \int _{B^{\varvec{\rho }_k}_{k+1}}|x|^{-(q-2)}| \xi ^{\varvec{\rho }_k}_{k+1}(x)|^2{\text {d}}x\right) \\&\le c\left( \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\right) |\rho _k|^{-\left( \frac{6p}{3+\alpha }-2\right) \frac{3+\alpha }{6}} \Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{\left( \frac{6p}{3+\alpha }-2\right) \frac{3+\alpha }{6}}_{k+1}\\&\quad \left( \int _{B^{\varvec{\rho }_k}_{k+1}} |\xi ^{\varvec{\rho }_k}_{k+1}(x)|^2{\text {d}}x\right) ^{\frac{3+\alpha }{6}}\\ +&c|{\rho }_k|^{-(q-2)}\Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{q-2}_{k+1}\left( \int _{B^{\varvec{\rho }_k}_{k+1}} |\xi ^{\varvec{\rho }_k}_{k+1}(x)|^2{\text {d}}x\right) \\&\le c\left( \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\right) |\rho _k|^{-\left( \frac{6p}{3+\alpha }-2\right) \frac{3+\alpha }{6}} \Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^p_{k+1}+c|\rho _k|^{-(q-2)}\Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{q}_{k+1}\\&=c\left( \sum _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\right) |\rho _k|^{-\left( p-\frac{3+\alpha }{3}\right) } \Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^p_{k+1}+c|\rho _k|^{-(q-2)}\Vert \xi ^{\varvec{\rho }_k}_{k+1}\Vert ^{q}_{k+1}, \end{aligned}$$

which yields that \(\sum \limits _{j=1}^{k+1}\Vert \xi ^{\varvec{\rho }_k}_j\Vert ^p_j\rightarrow \infty \) as \(\rho _k\rightarrow \infty \), due to \(p\in (4,3+\alpha ), q\in (4,6), \alpha \in (1,3)\). So \(d(\varvec{\rho }_k)\rightarrow \infty \) and \((ii)\) follows.

\((iii)\) Take a sequence \(\left\{ \varvec{\rho }^n_k\right\} _{n=1}^\infty \) satisfying \(\varvec{\rho }^n_k\rightarrow \varvec{\bar{\rho }}_k\in \Gamma _k\). We will prove the conclusion by showing \(d(\varvec{\bar{\rho }}_k)\ge \limsup \nolimits _{n\rightarrow \infty }d(\varvec{\rho }^n_k)\), \(d(\varvec{\bar{\rho }}_k)\le \liminf \nolimits _{n\rightarrow \infty }d(\varvec{\rho }^n_k)\).

First, we prove that \(d(\varvec{\bar{\rho }}_k)\ge \limsup \limits _{n\rightarrow \infty }d(\varvec{\rho }^n_k)\). In order to emphasize that \(v^{\varvec{\rho }^n_k}_i\) is radial in \(B^{\varvec{\rho }^n_k}_i\), we will rewrite \(v^{\varvec{\rho }^n_k}_i(|x|){=}v^{\varvec{\rho }^n_k}_i(\rho )\). Define \(v^{\varvec{\rho }^n_k}_i{:}[\rho ^n_{i-1},\rho ^n_i] \rightarrow \mathbb {R}\) by

$$\begin{aligned} v^{\varvec{\rho }^n_k}_i(\rho )= \left\{ \begin{array}{ll} t^n_i\xi ^{\varvec{\bar{\rho }}_k}_i\left( \bar{\rho }_{i-1}+\frac{\bar{\rho }_i-\bar{\rho }_{i-1}}{\rho ^n_i-\rho ^n_{i-1}}(\rho -\rho ^n_{i-1})\right) ,\ \ \ &{} \mathrm{}\ \quad i=1,\ldots ,k,\\ t^n_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}\left( \frac{\bar{\rho }_k}{\rho ^n_k}\rho \right) , \ \ \ &{} \mathrm{}\ \quad i=k+1, \end{array} \right. \end{aligned}$$

where \((\xi ^{\varvec{\rho }^n_k}_1,\ldots ,\xi ^{\varvec{\rho }^n_k}_{k+1})\) and \((\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})\) are minimizers of \(J_b|_{{\mathcal {M}}^{\varvec{\rho }^n_k}_k}\) and \(J_b|_{{\mathcal {M}}^{\varvec{\bar{\rho }}_k}_k}\) respectively. \((t^n_1,\ldots ,t^n_{k+1})\) is the unique \((k+1)\) tuple of positive numbers such that \((v^{\varvec{\rho }^n_k}_1,\ldots ,v^{\varvec{\rho }^n_k}_{k+1})\in {\mathcal {M}}^{\varvec{\rho }^n_k}_k\). By the definition of \((\xi ^{\varvec{\rho }^n_k}_1,\ldots ,\xi ^{\varvec{\rho }^n_k}_{k+1})\), we know that

$$\begin{aligned} J_b(v^{\varvec{\rho }^n_k}_1,\ldots ,v^{\varvec{\rho }^n_k}_{k+1})\ge J_b(\xi ^{\varvec{\rho }^n_k}_1,\ldots ,\xi ^{\varvec{\rho }^n_k}_{k+1})=d(\varvec{\rho }^n_k). \end{aligned}$$
(3.5)

Since \(\varvec{\rho }^n_k\rightarrow \varvec{\bar{\rho }}_k\in \Gamma _k\), we can easily get the following equations,

$$\begin{aligned}&\int _{B^{\varvec{\rho }^n_k}_i}|v^{\varvec{\rho }^n_k}_i|^2=(t^n_i)^2\int _{B^{\varvec{\bar{\rho }}_k}_i}|\xi ^{\varvec{\bar{\rho }}_k}_i|^2{\text {d}}x+o(1)\\&\Vert v^{\varvec{\rho }^n_k}_i\Vert ^2_i=(t^n_i)^2\Vert \xi ^{\varvec{\bar{\rho }}_k}_i\Vert ^2_i+o(1)\\&\int _{B^{\varvec{\rho }^n_k}_i}|v^{\varvec{\rho }^n_k}_i|^q{\text {d}}x=(t^n_i)^q\int _{B^{\varvec{\bar{\rho }}_k}_i}|\xi ^{\varvec{\bar{\rho }}_k}_i|^q{\text {d}}x+o(1)\\&\int _{B^{\varvec{\rho }^n_k}_i}\int _{B^{\varvec{\rho }^n_k}_j}\frac{|v^{\varvec{\rho }^n_k}_i(x)|^p |v^{\varvec{\rho }^n_k}_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y\\&\quad = (t^n_i)^p(t^n_j)^p\int _{B^{\varvec{\bar{\rho }}_k}_i}\int _{B^{\varvec{\bar{\rho }}_k}_j}\frac{|\xi ^{\varvec{\bar{\rho }}_k}_i(x)|^p|\xi ^{\varvec{\bar{\rho }}_k}_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y+o(1), \end{aligned}$$

and

$$\begin{aligned}&\int _{B^{\varvec{\rho }^n_k}_i}|\nabla v^{\varvec{\rho }^n_k}_i(x)|^2{\text {d}}x\int _{B^{\varvec{\rho }^n_k}_j}|\nabla v^{\varvec{\rho }^n_k}_j(x)|^2{\text {d}}x\\&\quad =(t^n_i)^2(t^n_j)^2\int _{B^{\varvec{\bar{\rho }}_k}_i}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_i(x)|^2{\text {d}}x\int _{B^{\varvec{\bar{\rho }}_k}_j}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_j(x)|^2{\text {d}}x+o(1). \end{aligned}$$

Since \((v^{\varvec{\rho }^n_k}_1,\ldots ,v^{\varvec{\rho }^n_k}_{k+1})\in {\mathcal {M}}^{\varvec{\rho }^n_k}_k\) and \((\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})\in {\mathcal {M}}^{\varvec{\bar{\rho }}_k}_k\), there holds that

$$\begin{aligned}&\Vert \xi ^{\varvec{\bar{\rho }}_k}_i\Vert ^2_i+b\sum _{j=1}^{k+1}\int _{B^{\varvec{\bar{\rho }}_k}_i}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_i(x)|^2{\text {d}}x\int _{B^{\varvec{\bar{\rho }}_k}_j}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_j(x)|^2{\text {d}}x\\ -&\sum _{j=1}^{k+1}\int _{B^{\varvec{\bar{\rho }}_k}_i}\int _{B^{\varvec{\bar{\rho }}_k}_j}\frac{|\xi ^{\varvec{\bar{\rho }}_k}_i(x)|^p|\xi ^{\varvec{\bar{\rho }}_k}_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y-\int _{B^{\varvec{\bar{\rho }}_k}_i}|\xi ^{\varvec{\bar{\rho }}_k}_i|^q{\text {d}}x=0, \end{aligned}$$

and

$$\begin{aligned}&(t^n_i)^2\Vert \xi ^{\varvec{\bar{\rho }}_k}_i\Vert ^2_i+b(t^n_i)^2(t^n_j)^2\sum _{j=1}^{k+1}\int _{B^{\varvec{\bar{\rho }}_k}_i}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_i(x)|^2{\text {d}}x\int _{B^{\varvec{\bar{\rho }}_k}_j}|\nabla \xi ^{\varvec{\bar{\rho }}_k}_j(x)|^2{\text {d}}x\\&-(t^n_i)^p(t^n_j)^p\sum _{j=1}^{k+1}\int _{B^{\varvec{\bar{\rho }}_k}_i}\int _{B^{\varvec{\bar{\rho }}_k}_j}\frac{|\xi ^{\varvec{\bar{\rho }}_k}_i(x)|^p|\xi ^{\varvec{\bar{\rho }}_k}_j(y)|^p}{|x-y|^{3-\alpha }}{\text {d}}x{\text {d}}y -(t^n_i)^q\int _{B^{\varvec{\bar{\rho }}_k}_i}|\xi ^{\varvec{\bar{\rho }}_k}_i|^q{\text {d}}x=o(1). \end{aligned}$$

This combined with Lemma 3.1, we have \(\lim \limits _{n\rightarrow \infty }t^n_i=1\) for all i. Hence, from (3.5) we can see that

$$\begin{aligned} d(\varvec{\bar{\rho }}_k)=J_b(\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})=\limsup _{n\rightarrow \infty }J_b(v^{\varvec{\rho }^n_k}_1,\ldots ,v^{\varvec{\rho }^n_k}_{k+1}) \ge \limsup _{n\rightarrow \infty }d(\varvec{\rho }^n_k). \end{aligned}$$
(3.6)

Next, we prove that \(d(\varvec{\bar{\rho }}_k)\le \liminf \limits _{n\rightarrow \infty }d(\varvec{\rho }^n_k)\). By the same argument as former case, let \(u^{\varvec{\rho }^n_k}_i=[\bar{\rho }_{i-1},\bar{\rho }_i]\rightarrow \mathbb {R}\) be defined by

$$\begin{aligned} u^{\varvec{\rho }^n_k}_i(\rho )= \left\{ \begin{array}{ll} s^n_i\xi ^{\varvec{\rho }^n_k}_i\left( \rho ^n_{i-1}+\frac{\rho ^n_i-\rho ^n_{i-1}}{\bar{\rho }_i-\bar{\rho }_{i-1}}(\rho -\bar{\rho }_{i-1})\right) ,\ \ \ &{} \mathrm{if}\ \ i=1,\ldots ,k,\\ s^n_{k+1}\xi ^{\varvec{\rho }^n_k}_{k+1}\left( \frac{\rho ^n_k}{\bar{\rho }_k}\rho \right) , \ \ \ &{} \mathrm{if}\ \ i=k+1, \end{array} \right. \end{aligned}$$

where \((s^n_1,\ldots ,s^n_{k+1})\in (\mathbb {R}_+)^{k+1}\) such that \((u^{\varvec{\rho }^n_k}_1,\ldots ,u^{\varvec{\rho }^n_k}_{k+1})\in {\mathcal {M}}^{\varvec{\bar{\rho }}_k}_k\).

By the same arguments, we can deduce \(s^n_i\rightarrow 1\) as \(n\rightarrow \infty \) for all \(i={1,\ldots ,k+1}\). Thus

$$\begin{aligned} d(\varvec{\bar{\rho }}_k)&=J_b(\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})\\&\le \liminf _{n\rightarrow \infty }J_b(u^{\varvec{\rho }^n_k}_1,\ldots ,u^{\varvec{\rho }^n_k}_{k+1})=\liminf _{n\rightarrow \infty } J_b(\xi ^{\varvec{\rho }^n_k}_1,\ldots ,\xi ^{\varvec{\rho }^n_k}_{k+1}) =\liminf _{n\rightarrow \infty }d(\rho ^n_k). \end{aligned}$$

This combined with (3.6) yields that d is continuous in \(\Gamma _k\). Furthermore, this combined with \((i)\), \((ii)\), we know that there is a \(\varvec{\bar{\rho }}_k\in \Gamma _k\) such that \(d(\varvec{\bar{\rho }}_k)=\inf \limits _{\varvec{\rho }_k\in \Gamma _k}d(\varvec{\rho }_k)\). Hence, \((iii)\) holds. \(\square \)

Proof of Theorem 1.1

By Lemmas 3.4 and  3.5, there exist \(\varvec{\bar{\rho }}_k\in \Gamma _k\) and \((\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})\in {\mathcal {M}}^{\varvec{\bar{\rho }}_k}_k\) with \((-1)^{i+1}\xi ^{\varvec{\rho }_k}_i>0\) in \(B^{\varvec{\rho }_k}_i\) such that

$$\begin{aligned} J_b(\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})=d(\varvec{\bar{\rho }}_k)=\inf _{\varvec{\rho }_k\in \Gamma _k}d(\varvec{\rho }_k). \end{aligned}$$

This implies that

$$\begin{aligned} c_k=d(\varvec{\bar{\rho }}_k)=I_b\left( \sum _{i=1}^{k+1}\xi ^{\varvec{\bar{\rho }}_k}_i\right) . \end{aligned}$$

\(\square \)

We claim that \(u^b_k=\sum \limits _{i=1}^{k+1}\xi ^{\varvec{\bar{\rho }}_k}_i\) is a solution of (1.3). Suppose by contradiction that the claim does not hold, that is, \(u^b_k\) is not a weak solution of (1.3). Then by the density argument, there is a radial function \(\phi \in \mathbb {C}^\infty _0(\mathbb {R}^3)\) such that

$$\begin{aligned} I^\prime _b\left( \sum _{i=1}^{k+1}\xi ^{\varvec{\bar{\rho }}_k}_i\right) \phi =-2. \end{aligned}$$
(3.7)

For \(\varvec{s}=(s_1,\ldots ,s_{k+1})\) and \(\mathbf{1} =(1,\ldots ,1)\in {\mathbb {R}^{k+1}}\), we define function \(g:= \mathbb {R}^{k+1}\times \mathbb {R}\rightarrow H^1(\mathbb {R}^3)\) by

$$\begin{aligned} g(\varvec{s},\epsilon ):=\sum \limits _{i=1}^{k+1}s^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i+\epsilon \phi . \end{aligned}$$

Since \(\sum \limits _{i=1}^{k+1}\xi ^{\varvec{\bar{\rho }}_k}_i\) is continuous and has k nodes, we know that there exists a neighborhood \(B_\tau (\mathbf{1} ):=\left\{ \varvec{s}\in \mathbb {R}^{k+1}:|\varvec{s}-\mathbf{1} |<\tau \right\} \) such that \(g(\varvec{s},\tau )\) also changes signs exactly k times and

$$\begin{aligned} J^\prime _b(g(\varvec{s},\epsilon ))\phi <-1 \quad \forall (\varvec{s},\epsilon )\in B_\tau (\mathbf{1} )\times [0,\tau ] \end{aligned}$$
(3.8)

for all \((\varvec{s},\epsilon )\in B_\tau (\mathbf{1} )\times [0,\tau ]\).

Let \(\eta \in {\mathbb {C}}^\infty ({\mathbb {R}}^3)\), \(0\le \eta \le 1\) with \(\eta (\varvec{s})=1\) if \(s\in \overline{B_{\frac{\tau }{4}}(1,\ldots ,1)} \) and \(\eta (\varvec{s})=0\) if \(\varvec{s}\not \in B_{\frac{\tau }{2}}(1,\ldots ,1)\). We define another continuous function \({\bar{g}}:\mathbb {R}^{k+1}\rightarrow \mathbb {H}_V\) by

$$\begin{aligned} {\bar{g}}(\varvec{s})=\sum _{i=1}^{k+1}s^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i+\tau \eta (\varvec{s})\phi . \end{aligned}$$

Obviously, for any \(\varvec{s}\in B_\tau (\mathbf{1} )\), \({\bar{g}}(\varvec{s})\) also changes signs exactly k times and has k nodes \(0<\rho _1(\varvec{s})<\cdots<\rho _{k}(\varvec{s})<\infty \). Moreover,

$$\begin{aligned} J^\prime _b(g(\varvec{s},\epsilon ))\phi <-1 \quad \forall (\varvec{s},\epsilon )\in B_\tau (\mathbf{1} )\times [0,\tau ]. \end{aligned}$$

Next, we will prove that there exists \(\varvec{{\bar{s}}}\in B_{\frac{\tau }{2}}(\mathbf{1} )\) such that \({\bar{g}}(\varvec{{\bar{s}}})\in {\mathcal {N}}_k\) changing sign k times. Denote

$$\begin{aligned} G(\mathbf{s} )=J_b\left( s^\frac{1}{p}_1\xi _1^{\varvec{\bar{\rho }}_k},\ldots ,s^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}\right) . \end{aligned}$$

For any \(\varvec{s}\in \partial B_{\frac{\tau }{2}}(\mathbf{1} )\), we have

$$\begin{aligned} \nabla G(\mathbf{s} )(\mathbf{1} -\varvec{s}) =&\nabla J_b\left( s^{\frac{1}{p}}_1\xi _1^{\varvec{\bar{\rho }}_k},\ldots ,s^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}\right) (\mathbf{1} -\varvec{s})\\ =&\sum \limits _{i=1}^{k+1}\frac{1}{p}s^{\frac{1}{p}-1}_i\left<\partial _{u_i}J_b\left( s^{\frac{1}{p}}_1\xi _1^{\varvec{\bar{\rho }}_k},\ldots ,s^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}\right) ,\xi ^{\varvec{\bar{\rho }}_k}_i\right>(1-s_i)\\ =&\sum _{i=1}^{k+1}\frac{1}{ps_i}(1-s_i)\left<\partial _{u_i}J_b(s^\frac{1}{p}_1\xi _1^{\varvec{\bar{\rho }}_k},\ldots ,s^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}),s^\frac{1}{p}_i\xi _i^{\varvec{\bar{\rho }}_k}\right>. \end{aligned}$$

By Lemma 3.1 and Corollary 3.2, we obtain \(G(\mathbf{s} )\) is strictly concave function in \(\mathbb {R}_+^{k+1}\) and attains its unique global maximum point at \(\varvec{1}\). By the Taylor expansion at \(\varvec{s}\ne \varvec{1}\) in \([0,\infty )^{k+1}\), and the strictly concavity, we have

$$\begin{aligned} 0<\varphi _b(\mathbf{1} )-\varphi _b(\mathbf{s} )-D^2\varphi _b(\varvec{s})(\mathbf{1} -\mathbf{s} )^2=\nabla G(\mathbf{s} )(\mathbf{1} -\mathbf{s} ), \end{aligned}$$

that is

$$\begin{aligned} \nabla G(\mathbf{s} )(\mathbf{1} -\mathbf{s} )>0. \end{aligned}$$

Set \({\widetilde{G}}_i(\mathbf{s} )=\frac{1}{p}s^{-1}_i\left<\partial _{u_j}J_b(s^{\frac{1}{p}}_1\xi _1^{\varvec{\bar{\rho }}_k},\ldots ,s^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}),s^{\frac{1}{p}}_i\xi _i^{\varvec{\bar{\rho }}_k}\right>\) and \({\widetilde{G}}(\mathbf{s} )=({\widetilde{G}}_1(\mathbf{s} ), \ldots ,{\widetilde{G}}_{k+1}(\mathbf{s} ))\). Define a map \(F(\theta ,\mathbf{s} ):[0,1]\times {\bar{B}}_{\frac{\tau }{2}}(\mathbf{1} )\rightarrow \mathbb {R}^{k+1}\) by

$$\begin{aligned} F(\theta ,\mathbf{s} )=\theta {\widetilde{G}}(\mathbf{s} )+(1-\theta )(\mathbf{1} -\mathbf{s} ). \end{aligned}$$

Obviously, \(F(0,\mathbf{s} )=\mathbf{1} -\mathbf{s} \), \(F(1,\mathbf{s} )={\widetilde{G}}(\mathbf{s} )\). Thus, \(\mathbf{1} -\mathbf{s} \) and \({\widetilde{G}}(\mathbf{s} )\) are homotopy. Moreover, for any \(\theta \in [0,1]\) and \(\mathbf{s} \in B_{\frac{\tau }{2}}(\mathbf{1} )\), we obtain \(F(\theta ,\mathbf{s} )\cdot (\mathbf{1} -\mathbf{s} )>0\). Thus, \(F(\theta ,\mathbf{s} )\ne \mathbf{0} \). By the Brouwer degree theory, we have

$$\begin{aligned} \deg ({\widetilde{G}},B_{\frac{\tau }{2}}(\mathbf{1} ),0)=\deg (1-id,B_{\frac{\tau }{2}}(\mathbf{1} ),0)=(-1)^{k+1}\ne 0. \end{aligned}$$

Therefore, there exists some \(\varvec{{\bar{s}}}\in B_{\frac{\tau }{2}}(\mathbf{1} )\) such that \({\bar{g}}(\varvec{{\bar{s}}})\in {\mathcal {N}}_k\) . The claim follows.

According to the claim, we have

$$\begin{aligned} J_b({\bar{g}}(\varvec{{\bar{s}}}))\ge c_k. \end{aligned}$$
(3.9)

On the other hand, by the mean value theorem and (3.8), we have

$$\begin{aligned} J_b({\bar{g}}(\varvec{{\bar{s}}}))&=I_b\left( \sum _{i=1}^{k+1}{\bar{s}}^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i\right) +\int _0^1\left<I^\prime _b\left( \sum _{i=1}^{k+1}s^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i+\theta \tau \eta (\varvec{{\bar{s}}})\phi \right) ,\tau \eta (\varvec{{\bar{s}}})\phi \right>d\theta \\&\le I_b\left( \sum _{i=1}^{k+1}{\bar{s}}^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i)-\tau \eta (\varvec{{\bar{s}}}\right) . \end{aligned}$$

If \(\varvec{s}\in B_{\frac{\tau }{2}}(\mathbf{1} )\) for each i, then \(\eta (\varvec{{\bar{s}}})>0\), by Corollary 3.2

$$\begin{aligned} J_b({\bar{g}}(\varvec{{\bar{s}}})) <I_b\left( \sum _{i=1}^{k+1}{\bar{s}}^{\frac{1}{p}}_i\xi ^{\varvec{\bar{\rho }}_k}_i\right)&=J_b({\bar{s}}^{\frac{1}{p}}_1\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,{\bar{s}}^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}) \\&\le J_b(\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})=d(\varvec{\bar{\rho }}_k)=c_k, \end{aligned}$$

and if \(\varvec{s}\not \in B_{\frac{\tau }{2}}(\mathbf{1} )\), then \(\eta (\varvec{{\bar{s}}})=0\), by corollary 3.2

$$\begin{aligned} J_b({\bar{g}}(\varvec{{\bar{s}}}))&=I_b\left( \sum _{j=1}^{k+1}{\bar{s}}^{\frac{1}{p}}_j\xi ^{\varvec{\bar{\rho }}_k}_j\right) =J_b\left( {\bar{s}}^{\frac{1}{p}}_1\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,{\bar{s}}^{\frac{1}{p}}_{k+1}\xi ^{\varvec{\bar{\rho }}_k}_{k+1}\right) \\&< J_b(\xi ^{\varvec{\bar{\rho }}_k}_1,\ldots ,\xi ^{\varvec{\bar{\rho }}_k}_{k+1})=d(\varvec{\bar{\rho }}_k)=c_k, \end{aligned}$$

which also contradicts to (3.9). Therefore, the function \(u^b_k\) is a solution of (1.3), such that \(J_b(u^b_k)=c_k\). The proof of Theorem 1.1 is complete.