Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing

Abstract

This paper deals with the following Kirchhoff–Schrödinger–Poisson system:

$$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$

where a, b are positive constants, $K(x)$, $V(x)$ are positive continuous functions vanishing at infinity, and $f(u)$ is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions.

1 Introduction

In this paper, we discuss the existence of a least energy nodal solution of the following Kirchhoff–Schrödinger–Poisson system:

$$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u) &\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$

(1.1)

where a, b are positive constants, $K(x)$, $V(x)$ are positive continuous functions vanishing at infinity, and $f(u)$ is a continuous function.

When $a=1$ and $b=0$, system (1.1) stems from the following Schrödinger–Poisson system:

$$ \textstyle\begin{cases} -\Delta u + V(x)u+\phi u=f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}. \end{cases} $$

(1.2)

Recently, many authors payed their attentions to finding nodal solutions to problems like (1.2), and indeed some interesting results were obtained; see, for example, [2–4, 7, 17, 18, 22, 23, 30, 36–38, 41, 50] and the references therein.

Note that system (1.1) is also related to the following Kirchhoff-type equations:

$$ - \biggl(a+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=f(u) \quad \text{in } \mathbb{R}^{3}. $$

(1.3)

Equation (1.3) has arisen the interest of many mathematicians. Especially, there are many papers on nodal solutions to problems like (1.3) [6, 9, 11–13, 15, 20, 21, 25, 26, 29, 31, 33, 34, 39, 40, 46–48].

Since there are both nonlocal operator and nonlocal nonlinear terms, the study of system (1.1) becomes more complicated. In recent years, some scholars began to show interest to problem like (1.1); see [8, 10, 14, 19, 24, 27, 43, 44, 49] and references therein. However, to the best of our knowledge, few papers considered nodal solutions to problem like (1.1). Via a gluing method, Deng and Yang [10] studied the nodal solutions for system (1.1) with $f(u)=|u|^{p-2}u$, $p\in (4,6)$. Wang, Li, and Hao [35]studied the existence and asymptotic behavior of a least energy nodal solution for system (1.1) by using the constraint variation methods.

Inspired by the works mentioned, especially by [10, 13, 15, 35, 45], in this paper, we find the nodal solutions to system (1.1) under some weaker assumptions on f. As in [1], we say that $(V,K)\in \mathcal{K}$ if continuous functions $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ satisfy the following conditions:

$(VK_{0})$:

$V(x),K(x)>0$ for all $x\in \mathbb{R}^{3}$ and $K\in L^{\infty }(\mathbb{R}^{3})$;

$(VK_{1})$:

If $\{A_{n}\}_{n}\subset \mathbb{R}^{3}$ is a sequence of Borel sets such that their Lebesgue measures $|A_{n}|\leq R$ for all $n\in \mathbb{N}$ and some $R>0$, then

$$ \lim_{r\rightarrow +\infty } \int _{A_{n}\cap B^{c}_{r}(0)}K(x)=0\quad \text{uniformly in } n\in \mathbb{N}. $$

Moreover, one of the following two conditions holds:

$(VK_{2})$:

$\frac{K}{V}\in L^{\infty }(\mathbb{R}^{3})$;

or

$(VK_{3})$:

there exists $p\in (2,6)$ so that

$$ \frac{K(x)}{V(x)^{\frac{6-p}{4}}}\rightarrow 0 \quad \text{as } \vert x \vert \rightarrow +\infty . $$

As for the function f, we assume that $f\in C(\mathbb{R},\mathbb{R})$ and satisfies the following conditions:

$(f_{1})$:

$\lim_{t\rightarrow 0}\frac{f(t)}{t^{3}}=0$ if $(VK_{2})$ holds, and $\lim_{t\rightarrow 0}\frac{f(t)}{|t|^{p-1}}=0$ for some $p\in (4,6)$ if $(VK_{3})$ holds;

$(f_{2})$:

$\lim_{|t|\rightarrow +\infty }\frac{f(t)}{t^{5}}=0$;

$(f_{3})$:

$\lim_{|t|\rightarrow \infty }\frac{F(t)}{t^{4}}=+\infty$, where $F(t)=\int ^{t}_{0}f(s)\,ds$;

$(f_{4})$:

$\frac{f(t)}{t^{3}}$ is nondecreasing for $t>0$, and $\frac{f(t)}{t^{3}}$ is nonincreasing for $t<0$.

Since the potential function $V(x)$ vanishes at infinity, as in [5], we consider following function space:

$$ W= \biggl\{ u\in D^{1,2} \bigl(\mathbb{R}^{3} \bigr): \int _{\mathbb{R}^{3}} \int _{ \mathbb{R}^{3}}\frac{u^{2}(x)u^{2}(y)}{ \vert x-y \vert }\,dx\,dy< \infty \biggr\} $$

with the norm

$$ \Vert u \Vert ^{2}_{W}= \int _{\mathbb{R}^{3}}a \vert \nabla u \vert ^{2}\,dx+ \biggl( \frac{1}{4\pi } \int _{\mathbb{R}^{3}} \int _{\mathbb{R}^{3}} \frac{u^{2}(x)u^{2}(y)}{ \vert x-y \vert }\,dx\,dy\biggr)^{\frac{1}{2}}. $$

Then it follows from [16] that W is a uniformly convex Banach space.

According to the Lax–Milgram theorem, for any $u\in W$, there exists unique $\phi _{u}\in D^{1,2}(\mathbb{R}^{3})$ such that $-\Delta \phi _{u}=u^{2}$. Furthermore,

$$ \phi _{u}(x)=\frac{1}{4\pi } \int _{\mathbb{R}^{3}} \frac{u^{2}(y)}{ \vert x-y \vert }\,dy. $$

(1.4)

By (1.4) we know that system (1.1) is a single equation on u:

$$\begin{aligned} - \biggl(a+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u+\phi _{u} u=K(x)f(u) \quad \text{in } \mathbb{R}^{3}. \end{aligned}$$

(1.5)

Since $(V, K)\in \mathcal{K}$, we know that the space

$$ E= \biggl\{ u\in D^{1,2} \bigl(\mathbb{R}^{3} \bigr): \int _{\mathbb{R}^{3}}V(x)u^{2}\,dx< + \infty \biggr\} $$

with the norm

$$ \Vert u \Vert ^{2}_{E}= \int _{\mathbb{R}^{3}}\bigl(a \vert \nabla u \vert ^{2}+V(x)u^{2}\bigr)\,dx $$

is compactly embedded into the weighted Lebesgue space $L^{q}_{k}(\mathbb{R}^{3})$ for some $q\in (2,6)$ (see Lemma 2.2), where

$$ L^{q}_{k} \bigl(\mathbb{R}^{3} \bigr)= \biggl\{ u: u\text{ is measurable on }\mathbb{R}^{3} \text{ and } \int _{\mathbb{R}^{3}}K(x) \vert u \vert ^{q}\,dx< + \infty \biggr\} $$

endowed with the norm

$$ \Vert u \Vert _{K}= \biggl( \int _{\mathbb{R}^{3}}K(x) \vert u \vert ^{q}\,dx \biggr)^{\frac{1}{q}}. $$

In this paper, we discuss our problem on the new space

$$ H:= \biggl\{ u\in D^{1,2} \bigl(\mathbb{R}^{3} \bigr): \int _{\mathbb{R}^{3}}V(x) \vert u \vert ^{2}\,dx< \infty , \int _{\mathbb{R}^{3}} \int _{\mathbb{R}^{3}} \frac{u^{2}(x)u^{2}(y)}{ \vert x-y \vert }\,dx\,dy< \infty \biggr\} $$

with the norm

$$ \Vert u \Vert ^{2}= \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr)\,dx+ \biggl( \frac{1}{4\pi } \int _{\mathbb{R}^{3}} \int _{\mathbb{R}^{3}} \frac{u^{2}(x)u^{2}(y)}{ \vert x-y \vert }\,dx\,dy \biggr)^{\frac{1}{2}}. $$

Since $(W,\|\cdot \|)$ and $(E,\|\cdot \|)$ are Banach spaces, $(H,\|\cdot \|)$ is also a Banach space. Denote the usual norm in $L^{p}(\mathbb{R}^{3})$ by $|\cdot |_{p}$. So, it follows that $H\hookrightarrow E\hookrightarrow D^{1,2}(\mathbb{R}^{3}) \hookrightarrow L^{6}(\mathbb{R}^{3})$ is a continuous embedding. Let

$$ S_{6}=\inf_{u\in E, u\neq 0} \frac{\int _{\mathbb{R}^{3}}(a \vert \nabla u \vert ^{2}+Vu^{2})\,dx}{(\int _{\mathbb{R}^{3}}u^{6}\,dx)^{\frac{1}{3}}}. $$

(1.6)

The energy functional associated with system (1.1) is defined by

$$\begin{aligned} J(u) =&\frac{1}{2} \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr)\,dx+ \frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{1}{4} \int _{\mathbb{R}^{3}}\phi _{u}u^{2}\,dx \\ &{}- \int _{\mathbb{R}^{3}}K(x)F(u)\,dx,\quad \forall u\in H. \end{aligned}$$

(1.7)

Under assumptions $(f_{1})$–$(f_{4})$, $J\in C^{1}(H,\mathbb{R})$, and

$$\begin{aligned} \bigl\langle J'(u),v \bigr\rangle = {}& \int _{\mathbb{R}^{3}} \bigl(a\nabla u\nabla v+V(x)uv \bigr)\,dx+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \int _{\mathbb{R}^{3}}\nabla u \nabla v \,dx \\ &{}+ \int _{\mathbb{R}^{3}}\phi _{u}uv \,dx- \int _{\mathbb{R}^{3}}K(x)f(u)v \,dx, \quad \forall u,v\in H. \end{aligned}$$

(1.8)

The solution of system (1.1) is the critical point of the functional J. Especially, we call u a nodal solution to (1.1) if u is a solution of (1.1) with $u^{\pm }\neq 0$, where

$$ u^{+}(x)=\max \bigl\{ u(x),0 \bigr\} \quad \text{and}\quad u^{-}(x)=\min \bigl\{ u(x),0 \bigr\} . $$

The following two lemmas are useful for dealing with the compactness of (PS) sequence of the functional J; the details can be found in [16, 28].

Lemma 1.1

• (1) If$\{u_{n}\}\subset W$, then$u_{n}\rightarrow u$inWif and only if$u_{n}\rightarrow u$and$\phi _{u_{n}}\rightarrow \phi _{u}$in$D^{1,2}(\mathbb{R}^{3})$. Moreover, $u_{n}\rightharpoonup u$inWif and only if$u_{n}\rightharpoonup u$in$D^{1,2}(\mathbb{R}^{3})$and$\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{u_{n}^{2}(x)u_{n}^{2}(y)}{|x-y|}\,dx\,dy$is bounded, and then$\phi _{u_{n}}\rightharpoonup \phi _{u}$in$D^{1,2}(\mathbb{R}^{3})$.

• (2) $\phi _{tu}(x)=t^{2}\phi _{u}(x)\geq 0$, $\forall u\in D^{1,2}(\mathbb{R}^{3})$, $\forall t\in \mathbb{R}$.

For simplicity, we define $T:W^{4}\rightarrow \mathbb{R}$ by

$$ T(u,v,w,z)= \int _{\mathbb{R}^{3}} \int _{\mathbb{R}^{3}} \frac{u(x)v(x)w(y)z(y)}{4\pi \vert x-y \vert }\,dx\,dy. $$

Lemma 1.2

Suppose that$u_{n}\rightharpoonup u$, $v_{n}\rightharpoonup v$, $w_{n}\rightharpoonup w$, and$z\in W$. Then

$$ T(u_{n},v_{n},w_{n},z)\rightarrow T(u,v,w,z). $$

The following theorem is the main result of this paper.

Theorem 1.1

Let$(V,K)\in \mathcal{K}$. Under assumptions$(f_{1})$–$(f_{4})$, system (1.1) has at least one energy nodal solution. In addition, iffis an odd function, that is, $f(-t)=-f(t)$for$t\in \mathbb{R}$, then system (1.1) has infinitely many nontrivial solutions.

2 The variation framework and basic conclusions

Lemma 2.1

([1])

Let$(V,K)\in \mathcal{K}$. If$(VK_{2})$holds, thenEis continuously embedded in$L^{q}_{K}(\mathbb{R}^{3})$for every$q\in [2,6]$; if$(VK_{3})$holds, thenEis continuously embedded in$L^{p}_{K}(\mathbb{R}^{3})$.

Lemma 2.2

([1])

Let$(V,K)\in \mathcal{K}$. If$(VK_{2})$holds, thenEis compactly embedded in$L^{q}_{K}(\mathbb{R}^{3})$for every$q\in (2,6)$; if$(VK_{3})$holds, thenEis compactly embedded in$L^{p}_{K}(\mathbb{R}^{3})$.

Lemma 2.3

([1])

Suppose that$(V,K)\in \mathcal{K}$and$(f_{1})$–$(f_{2})$hold. If$\{v_{n}\}\subset E$and$v_{n}\rightharpoonup v$inE, then

$$ \int _{\mathbb{R}^{3}}K(x)F(v_{n})\,dx\rightarrow \int _{\mathbb{R}^{3}}K(x)F(v)\,dx,\qquad \int _{\mathbb{R}^{3}}K(x)f(v_{n})v_{n}\,dx \rightarrow \int _{ \mathbb{R}^{3}}K(x)f(v)v\,dx. $$

The following results are very important, because they allow us to overcome the non-differentiability of $\mathcal{N}:=\{u\in H\backslash \{0\}:\langle I_{b}'(u),u\rangle=0 \}$.

Lemma 2.4

Suppose that$(V,K)\in \mathcal{K}$and$(f_{1})$–$(f_{4})$hold. Then:

• (1) Define$h_{u}:\mathbb{R}_{+}\rightarrow \mathbb{R}$by$h_{u}(t)=J(tu)$, $u\in H\backslash \{0\}$. Then there exists a unique$t_{u}>0$such that$h'_{u}(t)>0$, $t\in (0,t_{u})$. However, $h'_{u}(t)<0$, $t\in (t_{u},\infty )$.

• (2) $t_{u}\geq \tau$, $u\in S$, whereτis independent of$u\in S$. Moreover, for any compact set$Q\subset S$, there is a constant$C_{Q}>0$such that$t_{u}\leq C_{Q}$, $u\in Q$.

• (3) The map$\hat{m}:H\backslash \{0\}\rightarrow \mathcal{N}$is continuous for$\hat{m}(u)=t_{u}u$, and$m:=\hat{m}|_{S}$is a homeomorphic mapping between sphereSand$\mathcal{N}$.

Proof

Firstly, suppose that $(VK_{2})$ holds. According to $(f_{1})$ and $(f_{2})$, for a given $\varepsilon >0$, there exists $C_{\varepsilon }>0$ such that

$$\begin{aligned} \bigl\vert f(s) \bigr\vert \leq \varepsilon \vert s \vert +C_{\varepsilon } \vert s \vert ^{5}, \quad s\in \mathbb{R}. \end{aligned}$$

(2.1)

Hence we have that

$$\begin{aligned} J(tu) &= \frac{1}{2} \Vert tu \Vert ^{2}_{E}+ \frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla (tu) \bigr\vert ^{2}\,dx \biggr)^{2}+\frac{t^{4}}{4} \int _{\mathbb{R}^{3}}\phi _{u}u^{2}\,dx- \int _{\mathbb{R}^{3}}K(x)F(tu)\,dx \\ &\geq \frac{t^{2}}{2} \Vert u \Vert ^{2}_{E}- \varepsilon \int _{\mathbb{R}^{3}}K(x)t^{2}u^{2}\,dx -C_{\varepsilon } \int _{\mathbb{R}^{3}}K(x)t^{6}u^{6}\,dx \\ &\geq \biggl(\frac{1}{2}-\varepsilon \bigl\Vert KV^{-1} \bigr\Vert _{\infty } \biggr)t^{2} \Vert u \Vert ^{2}_{E}-S^{-3}_{6}C_{ \varepsilon }t^{6} \Vert K \Vert _{\infty } \Vert u \Vert ^{6}_{E}. \end{aligned}$$

Choosing $\varepsilon <\frac{1}{2}\|KV^{-1}\|_{\infty }$, there exists $t_{0}>0$ sufficiently small such that

$$\begin{aligned} 0< h_{u}(t)=J(tu),\quad \forall t\in (0,t_{0}). \end{aligned}$$

(2.2)

If $(VK_{3})$ holds, then according to discussion in [1], there is a constant $C_{p}>0$ such that, for each $\varepsilon \in (0,C_{p})$, there is $R>0$ satisfying

$$\begin{aligned} \int _{B_{R}(0)^{c}}K(x) \vert u \vert ^{p}\,dx\leq \varepsilon \int _{B_{R}(0)^{c}} \bigl(V(x)u^{2}+u^{6} \bigr)\,dx,\quad \forall u\in H. \end{aligned}$$

(2.3)

On the other hand, by $(f_{1})$ and $(f_{2})$ we derive that

$$\begin{aligned} J(tu)\geq & \frac{t^{2}}{2} \Vert u \Vert ^{2}_{E}-C_{1}t^{p} \int _{\mathbb{R}^{3}}K(x) \vert u \vert ^{p} \,dx-C_{2}t^{6} \int _{\mathbb{R}^{3}}K(x)u^{6}\,dx. \end{aligned}$$

Thanks to (2.3), the Hölder inequality, and $(VK_{0})$, we get

$$\begin{aligned} J(tu)\geq {}&\frac{t^{2}}{2} \Vert u \Vert ^{2}_{E}-C_{1} \varepsilon t^{p} \int _{B_{R}(0)^{c}} \bigl(V(x)u^{2}+u^{6} \bigr)\,dx -C_{1}t^{p} \Vert K \Vert _{\frac{6}{6-p}(B_{R}(0))} \biggl( \int _{B_{R}(0)}u^{6}\,dx \biggr)^{ \frac{p}{6}} \\ &{}-C_{2} \Vert K \Vert _{\infty }t^{6} \int _{\mathbb{R}^{3}}u^{6}\,dx \\ \geq {}&\frac{t^{2}}{2} \Vert u \Vert ^{2}_{E}-C_{2}t^{6} \Vert K \Vert _{\infty }S^{-3}_{6} \Vert u \Vert ^{6}_{E}-C_{1} \bigl(\varepsilon \Vert u \Vert ^{2}_{E} \\ &{}+S^{-3}_{6} \varepsilon \Vert u \Vert ^{6}_{E}+S^{-\frac{P}{2}}_{6} \Vert K \Vert _{\frac{6}{6-P}(B_{R}(0))} \Vert u \Vert ^{P}_{E} \bigr)t^{p}. \end{aligned}$$

Since $p>2$, we get that (2.2) also holds.

From $(f_{1})$ and $(f_{4})$ we have that $F(t)\geq 0$, $t\in \mathbb{R}$. Then

$$\begin{aligned} J(tu) \leq &\frac{1}{2} \Vert tu \Vert ^{2}_{E}+ \frac{bt^{4}}{4} \biggl( \int _{ \mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{t^{4}}{4} \int _{\mathbb{R}^{3}} \phi _{u}u^{2}\,dx - \int _{A}K(x)F(tu)\,dx, \end{aligned}$$

where $A\subset \operatorname{supp} u$ is a measurable set with finite measure. By Lemma 1.1, $(f_{3})$, and Fatou’s lemma we have

$$\begin{aligned} \lim_{t\rightarrow \infty }\sup \frac{J(tu)}{ \Vert tu \Vert ^{4}} \leq& \lim_{t\rightarrow \infty }\sup \frac{1}{2 \Vert tu \Vert ^{2}} + \frac{b+1}{4} \\ &{}- \lim_{t\rightarrow \infty }\inf \biggl\{ \int _{A}K(x) \biggl[ \frac{F(tu)}{(tu)^{4}} \biggr] \biggl( \frac{u}{ \Vert u \Vert } \biggr)^{4}\,dx \biggr\} \\ \rightarrow &-\infty . \end{aligned}$$

(2.4)

So, choosing $R>0$ large enough, we have

$$\begin{aligned} h_{u}(R)=J(Ru)< 0. \end{aligned}$$

(2.5)

Thanks to (2.2) and (2.5), from the continuity of $h_{u}$ and $(f_{4})$ it follows that $t_{u}>0$ is the global maximum point of $h_{u}$ and $t_{u}u\in \mathcal{N}$.

We assert that $t_{u}$ is the unique critical point of $h_{u}$. Indeed, if there are $t_{1}>t_{2}>0$ such that $h_{u}'(t_{1})=h_{u}'(t_{2})$, then from $(f_{4})$ we have that

$$\begin{aligned} 0>\frac{1}{ \Vert t_{1}u \Vert ^{2}_{E}}-\frac{1}{ \Vert t_{2}u \Vert ^{2}_{E}} = \frac{1}{ \Vert u \Vert ^{4}_{E}} \int _{\mathbb{R}^{3}}K(x) \biggl[ \frac{f(t_{1}u)}{(t_{1}u)^{3}}- \frac{f(t_{2}u)}{(t_{2}u)^{3}} \biggr]u^{4}\,dx \geq 0, \end{aligned}$$

which is a contradiction. So $t_{u}$ is the unique critical point of $h_{u}$.

Now we prove (2).

For any $u\in S$, by (1) and (2.1) we have that

$$\begin{aligned} t_{u}^{2} \Vert u \Vert ^{2}_{E}&\leq t_{u}^{2} \Vert u \Vert ^{2}_{E}+b t_{u}^{2} \biggl( \int _{ \mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+t_{u}^{4} \int _{\mathbb{R}^{3}} \phi _{u}u^{2}\,dx \\ &= \int _{\mathbb{R}^{3}}K(x)f(t_{u}u)t_{u}u\,dx \\ &\leq \int _{\mathbb{R}^{3}}K(x) \bigl[\varepsilon \vert t_{u}u \vert ^{2}+C_{ \varepsilon } \vert t_{u}u \vert ^{6} \bigr]\,dx \\ &\leq \varepsilon t_{u}^{2} \bigl\Vert KV^{-1} \bigr\Vert _{\infty } \Vert u \Vert ^{2}_{E}+S^{-3}_{6}t_{u}^{6}C_{ \varepsilon } \Vert K \Vert _{\infty } \Vert u \Vert ^{6}_{E}. \end{aligned}$$

(2.6)

So, there exists $\tau >0$, independent on u, such that $t_{u}\geq \tau$.

On the other hand, let $Q\subset S$ be a compact set. Suppose that there are a sequence $\{u_{n}\}\subset Q$ such that $t_{n}:=t_{u_{n}}\rightarrow \infty$ and $u\subset Q$ such that $u_{n}\rightarrow u$ in H. Then, by (2.4),

$$\begin{aligned} J(t_{u}u_{n})\rightarrow -\infty . \end{aligned}$$

(2.7)

From $(f_{4})$ we have that $\varTheta (t):=\frac{1}{4}f(t)t-F(t)\geq 0$ is nondecreasing for $t\geq 0$ and nonincreasing for $t\leq 0$. Then, for each $u\in \mathcal{N}$,

$$\begin{aligned} J(u)&=J(u)-\frac{1}{4} \bigl\langle J'(u),u \bigr\rangle \\ &=\frac{1}{4} \Vert u \Vert ^{2}- \int _{\mathbb{R}^{3}}K(x)F(u)\,dx+\frac{1}{4} \int _{\mathbb{R}^{3}}K(x)f(u)u\,dx \\ &=\frac{1}{4} \Vert u \Vert ^{2}+ \frac{1}{4} \int _{\mathbb{R}^{3}}K(x) \bigl(f(u)u-4F(u) \bigr)\,dx \\ &\geq 0. \end{aligned}$$

(2.8)

Since $t_{n}u_{n}\in \mathcal{N}$, setting $u=t_{n}u_{n}$ in (2.8), we get a contradiction with (2.7). Thus (2) holds.

Next, we prove (3). We assert that m̂, m, $m^{-1}$ are well defined. Indeed, for each $u\in H\backslash \{0\}$, by (1) there exists a unique $\hat{m}(u)\in \mathcal{N}$. On the other hand, if $u\in \mathcal{N}$, then $u\neq 0$, and thus $m^{-1}(u)=\frac{u}{\|u\|}\in S$ and $m^{-1}(u)$ are well defined. Furthermore,

$$\begin{aligned}& m^{-1} \bigl(m(u) \bigr)=m^{-1}(t_{u}u)= \frac{t_{u}u}{t_{u} \Vert u \Vert }=u, \quad \forall u\in S, \\& m^{-1} \bigl(m^{-1}(u) \bigr)=m \biggl( \frac{u}{ \Vert u \Vert } \biggr)=t_{(\frac{u}{ \Vert u \Vert })} \frac{u}{ \Vert u \Vert }=u,\quad \forall u \in \mathcal{N}. \end{aligned}$$

So m is bijective, and $m^{-1}$ continuous.

We now prove that $\hat{m}:H\backslash \{0\}\rightarrow \mathcal{N}$ is continuous. Let $\{u_{n}\}\subset H\backslash \{0\}$ and $u\in H\backslash \{0\}$ be such that $u_{n}\rightarrow u$ in $H\subset E$. From (2) we have that there is $t_{0}>0$ such that $\|u_{n}\|t_{u_{n}}=t_{(\frac{u_{n}}{\|u_{n}\|})}\rightarrow t_{0}$. Hence $t_{u_{n}}\rightarrow \frac{t_{0}}{\|u\|}:=t_{*}$. Since $t_{u_{n}}u_{n}\in \mathcal{N}$, we have that

$$\begin{aligned} t_{u_{n}}^{2} \Vert u_{n} \Vert ^{2}_{E}+bt_{u_{n}}^{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \biggr)^{2}+t_{u_{n}}^{4} \int _{\mathbb{R}^{3}}\phi _{u_{n}}u_{n}^{2} \,dx= \int _{\mathbb{R}^{3}}K(x)f(t_{u_{n}}u_{n})t_{u_{n}}u_{n} \,dx. \end{aligned}$$

So, passing to the limit as $n\rightarrow \infty$, we get

$$\begin{aligned} t_{*}^{2} \Vert u \Vert ^{2}_{E}+bt_{*}^{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+ \int _{\mathbb{R}^{3}}\phi _{t_{*}u}({t_{*}u})^{2} \,dx= \int _{ \mathbb{R}^{3}}K(x)f(t_{*}u)t_{*}u\,dx. \end{aligned}$$

Hence $t_{*}u\in \mathcal{N}$ and $t_{u}=t_{*}$, that is, $\hat{m}(u_{n})\rightarrow \hat{m}(u)$, and m̂ and m are continuous. □

Define $\widehat{\psi }:E\rightarrow \mathbb{R}$ and $\psi :\mathcal{S}\rightarrow \mathbb{R}$ by

$$ \widehat{\psi }(u)=J \bigl(\widehat{m}(u) \bigr)\quad \text{and}\quad \psi := \widehat{\psi }|_{\mathcal{S}}. $$

By Lemma 2.4 we have following results; for details, see the book [32].

Proposition 2.1

Suppose$(V,K)\in \mathcal{K}$andfsatisfies$(f_{1})$–$(f_{4})$. Then:

• (a) $\hat{\psi }\in C^{1}(H\backslash \{0\},\mathbb{R})$, and$\hat{\psi }'(u)v=\frac{\|\hat{m}(u)\|}{\|u\|}J'(\hat{m}(u))v$, $v\in H$, $u\in H\backslash \{0\}$.

• (b) $\psi \in C^{1}(S,\mathbb{R})$, $\psi '(u)v=\|{m}(u)\|J'(m(u))v$, $v\in T_{u}S$.

• (c) If$\{u_{n}\}$is a$(PS)_{d}$sequence forψ, then$\{m(u_{n})\}$is a$(PS)_{d}$sequence forJ. If$\{u_{n}\}\subset \mathcal{N}$is a bounded$(PS)_{d}$sequence forJ, then$\{m^{-1}(u_{n})\}$is a$(PS)_{d}$sequence forψ.

• (d) uis a critical point ofψif and only if$m(u)$is a nontrivial point ofJ. Moreover, the corresponding critical values coincide, and$\inf_{S}\psi =\inf_{\mathcal{N}}J$.

Proposition 2.2

If$(f_{1})$–$(f_{4})$hold, then

$$\begin{aligned} d_{\infty }:=\inf_{u\in \mathcal{N}}J(u)=\inf _{u \in H\backslash \{0\}}\max_{t>0}J(tu) =\inf _{u\in S} \max_{t>0}J(tu)>0. \end{aligned}$$

(2.9)

3 Technical lemmas

In this section, we give some technical lemmas related to the existence of a least energy nodal solution.

Set $\mathcal{M}=\{u\in H, u^{\pm }\neq 0, \langle J'(u),u^{\pm }\rangle =0 \}$. For $u\in H$ with $u^{\pm }\neq 0$, let

$$ G_{u}(t,s)=J \bigl(tu^{+}+su^{-} \bigr), \quad s>0, t>0. $$

Lemma 3.1

Assume that$(V,K)\in \mathcal{K}$andfsatisfies$(f_{1})$–$(f_{4})$. If$u\in H$with$u^{\pm }\neq 0$, then:

• (1) $(t,s)$is a critical point of$G_{u}$with$t,s>0$if and only if$tu^{+}+su^{-}\in \mathcal{M}$.

• (2) $G_{u}$has a unique critical point$(t_{+},s_{-})$with$t_{+}=t_{+}(u)>0$and$s_{-}=s_{-}(u)>0$, which is the unique maximum global point of$G_{u}$.

• (3) The maps$\alpha _{+}(r)=\frac{\partial G_{u}}{\partial t}(r,s_{-})$and$\alpha _{-}(r)=\frac{\partial G_{u}}{\partial s}(t_{+},s)$are such that$\alpha _{+}(r)>0$for$r\in (0,t_{+})$and$\alpha _{+}(r)<0$for$r\in (t_{+},\infty )$, and$\alpha _{-}(r)>0$for$r\in (0,s_{-})$and$\alpha _{-}(r)<0$for$r\in (s_{-},\infty )$.

Proof

By direct calculation we have

$$\begin{aligned} \nabla G_{u}(t,s)&= \biggl(\frac{\partial G_{u}}{\partial t}(t,s), \frac{\partial G_{u}}{\partial t}(t,s) \biggr) \\ &= \bigl( \bigl\langle J' \bigl(tu^{+}+su^{-} \bigr),u^{+} \bigr\rangle , \bigl\langle J' \bigl(tu^{+}+su^{-} \bigr),u^{-} \bigr\rangle \bigr) \\ &= \biggl(\frac{1}{t} \bigl\langle J' \bigl(tu^{+}+su^{-} \bigr),tu^{+} \bigr\rangle ,\frac{1}{s} \bigl\langle J' \bigl(tu^{+}+su^{-} \bigr),su^{-} \bigr\rangle \biggr), \end{aligned}$$

from which (1) directly follows.

Now we prove (2). Firstly, we prove the existence of a critical point for $G_{u}$. Fixing $u\in H$ with $u^{\pm }\neq 0$ and $t_{0}\geq 0$, we define $g_{1}:[0,\infty )\rightarrow \mathbb{R}$ by $g_{1}(t)=G_{u}(t,s_{0})$. Similarly as in the proof (1) of Lemma 2.3, we get that $g_{1}$ has a positive maximum point. Suppose $t_{1}>t_{2}>0$ are all critical points of $g_{1}(t)$, that is, $g'_{1}(t_{1})=g'_{1}(t_{2})=0$. Then

$$\begin{aligned} &t_{i}^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}_{E}+bt_{i}^{4} \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx \biggr)^{2}+ \int _{\mathbb{R}^{3}}\phi _{t_{i}u^{+}+s_{0}u^{-}} \bigl({t_{i}u^{+}} \bigr)^{2}\,dx \\ &\qquad {} +bs^{2}_{0}t_{i}^{2} \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx \\ &\quad = \int _{\mathbb{R}^{3}}K(x)f \bigl(t_{i}u^{+} \bigr)t_{i}u^{+}\,dx,\quad i=1,2. \end{aligned}$$

From these equalities we have that

$$\begin{aligned} & \biggl(\frac{1}{t_{1}^{2}}-\frac{1}{t_{2}^{2}} \biggr) \biggl( \bigl\Vert u^{+} \bigr\Vert ^{2}_{E}+bs^{2}_{0} \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx+s_{0}^{2} \int _{\mathbb{R}^{3}}\phi _{u^{+}} \bigl(u^{-} \bigr)^{2}\,dx \biggr) \\ &\quad = \int _{\mathbb{R}^{3}}K(x) \biggl[\frac{f(t_{1}u^{+})}{(t_{1}u^{+})^{3}}- \frac{f(t_{2}u^{+})}{(t_{2}u^{+})^{3}} \biggr] \bigl(u^{+} \bigr)^{4} \,dx, \end{aligned}$$

(3.1)

a contradiction thanks to $(f_{4})$ and $0< t_{2}< t_{1}$. So there is a unique $t_{0}=t_{0}(u,s_{0})>0$ such that

$$ g'_{1}(t)>0,\quad t\in (0,t_{0});\qquad g'_{1}(t_{0})=0;\qquad g'_{1}(t)< 0,\quad t\in (t_{0}, \infty ). $$

Then we define the map $\varphi _{1}:[0,\infty )\rightarrow [0,\infty )$ by $\varphi _{1}(s)=t(u,s)$, where $t(u,s)$ has properties similar to $t_{0}=t_{0}(u,s_{0})$ mentioned before.

By the definition of $\varphi _{1}$ we get

$$\begin{aligned} g'_{1} \bigl(\varphi _{1}(s) \bigr)= \frac{\partial G_{u}}{\partial t} \bigl(\varphi _{1}(s),s \bigr)=0,\quad \forall s\geq 0. \end{aligned}$$

Furthermore, $\varphi _{1}$ has the following properties:

$(P_{1})$:

$\varphi _{1}$ is continuous;

$(P_{2})$:

$\varphi _{1}(0)>0$;

$(P_{3})$:

$\varphi _{1}(s)\leq s$ for s large.

Similarly, we can define a map $\varphi _{2}(t)$ satisfying properties $(P_{1})$–$(P_{3})$.

Fix $s_{1}>0$ and set

$$\begin{aligned} t_{n}=\varphi _{1}(s_{n}) \quad \text{and}\quad s_{n+1}=\varphi _{2}(t_{n}). \end{aligned}$$

(3.2)

Then by the definitions of $\varphi _{1}$ and $\varphi _{2}$ we have that

$$\begin{aligned} \frac{\partial G_{u}}{\partial t}(t_{n},s_{n})= \frac{\partial G_{u}}{\partial s}(t_{n},s_{n+1})=0,\quad \forall n \in \mathbb{N}. \end{aligned}$$

(3.3)

We claim that $\{t_{n}\}$ and $\{s_{n}\}$ are bounded. Suppose, by contradiction, there is a subsequence $t_{n}\rightarrow \infty$. By $(P_{3})$ there is $C_{1}>0$ such that when $t_{n}>C_{1}$ for some n, we have $s_{n+1}=\varphi _{2}(t_{n})\leq t_{n}$.

In the same way, there exists $C_{2}>0$ such that $s\geq C_{2}$ implies $\varphi _{1}(s)\leq s$. So, if $s_{n+1}\geq C_{2}$, then we get

$$\begin{aligned} t_{n+1}=\varphi _{1}(s_{n+1}) \leq s_{n+1}\leq t_{n}. \end{aligned}$$

(3.4)

On the other hand, by $(P_{1})$, if $s_{n+1}\leq C_{2}$, then we derive that

$$\begin{aligned} t_{n+1}=\varphi _{1}(s_{n+1}) \leq C_{3}:=\max_{s\in [0,C_{2}]} \varphi _{1}(s). \end{aligned}$$

(3.5)

From (3.4) and (3.5) we deduce that $\{t_{n}\}$ is bounded. By applying $(P_{3})$ again we can prove that $\{s_{n}\}$ is also bounded.

Therefore, in subsequence sense, there are $t_{+}\geq 0$ and $s_{-}\geq 0$ such that $t_{n}\rightarrow t_{+}$ and $s_{n}\rightarrow s_{-}$. Therefore from (3.2) and the continuity of $\varphi _{i}$, $i=1,2$, we have

$$\begin{aligned} t_{+}=\varphi _{1}(s_{-}) \quad \text{and}\quad s_{-}=\varphi _{2}(t_{+}). \end{aligned}$$

(3.6)

Furthermore, since $\varphi _{i}>0$, we get $t_{+}>0$ and $s_{-}>0$. Hence by (3.3) we obtain that $(t_{+},s_{-})$ is a critical point for $G_{u}$.

Now we prove the uniqueness of $(t_{+},s_{-})$. By standard arguments here we only prove the uniqueness in the case $w\in \mathcal{M}$.

For any $w\in \mathcal{M}$, we obtain

$$\begin{aligned} \nabla G_{w}(1,1)&= \biggl(\frac{\partial G_{w}}{\partial t}(1,1), \frac{\partial G_{w}}{\partial s}(1,1) \biggr) \\ &= \bigl( \bigl\langle J' \bigl(w^{+}+w^{-} \bigr),w^{+} \bigr\rangle , \bigl\langle J' \bigl(w^{+}+w^{-} \bigr),w^{-} \bigr\rangle \bigr) \\ &=(0,0). \end{aligned}$$

Obviously, $(1,1)$ is a critical point of $G_{w}$. Now we prove that $(1,1)$ is a unique critical point of $G_{w}$ with positive coordinates. Assume that $(t_{0},s_{0})$ are critical points of $G_{w}$ with $0< t_{0}\leq s_{0}$. Then

$$\begin{aligned} &s_{0}^{2} \bigl\Vert w^{-} \bigr\Vert ^{2}_{E}+bs_{0}^{4} \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+ \int _{\mathbb{R}^{3}}\phi _{t_{0}w^{+}+s_{0}w^{-}} \bigl({s_{0}w^{-}} \bigr)^{2}\,dx \\ &\qquad {}+bs^{2}_{0}t_{0}^{2} \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \\ &\quad = \int _{\mathbb{R}^{3}}K(x)f \bigl(s_{0}w^{-} \bigr)s_{0}w^{-}\,dx, \end{aligned}$$

(3.7)

$$\begin{aligned} &t_{0}^{2} \bigl\Vert w^{+} \bigr\Vert ^{2}_{E}+bt_{0}^{4} \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx \biggr)^{2}+ \int _{\mathbb{R}^{3}}\phi _{t_{0}w^{+}+s_{0}w^{-}} \bigl({t_{0}w^{+}} \bigr)^{2}\,dx \\ &\qquad {} +bs^{2}_{0}t_{0}^{2} \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \\ &\quad = \int _{\mathbb{R}^{3}}K(x)f \bigl(t_{0}w^{+} \bigr)t_{0}w^{+}\,dx. \end{aligned}$$

(3.8)

From (3.7) and $0< t_{0}\leq s_{0}$ we derive that

$$\begin{aligned} &\frac{ \Vert w^{-} \Vert ^{2}_{E}}{s_{0}^{2}}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+b \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{3}}\phi _{w} \bigl(w^{-} \bigr)^{2}\,dx \\ &\quad \geq\frac{ \Vert w^{-} \Vert ^{2}_{E}}{s_{0}^{2}}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+\frac{bt_{0}^{2}}{s_{0}^{2}} \int _{ \mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \\ &\qquad {}+\frac{t_{0}^{2}}{s_{0}^{2}} \int _{\mathbb{R}^{3}}\phi _{w^{+}} \bigl(w^{-} \bigr)^{2}\,dx+ \int _{\mathbb{R}^{3}}\phi _{w^{-}} \bigl(w^{-} \bigr)^{2}\,dx \\ &\quad = \int _{\operatorname{supp}(w^{-})}K(x)\frac{f(s_{0}w^{-})}{(s_{0}w^{-})^{3}} \bigl(w^{-} \bigr)^{4}\,dx. \end{aligned}$$

(3.9)

On the other hand, since $w\in \mathcal{M}$, we obtain

$$\begin{aligned} & \bigl\Vert w^{-} \bigr\Vert ^{2}_{E}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+b \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla w^{-} \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{3}}\phi _{w} \bigl({w^{-}} \bigr)^{2}\,dx \\ &\quad = \int _{\mathbb{R}^{3}}K(x)f \bigl(w^{-} \bigr)w^{-}\,dx. \end{aligned}$$

(3.10)

From (3.9) with (3.10) we get

$$\begin{aligned} \biggl(\frac{1}{s_{0}^{2}}-1 \biggr) \bigl\Vert w^{-} \bigr\Vert ^{2}_{E}+\geq \int _{\operatorname{supp}(w^{-})}K(x) \biggl( \frac{f(s_{0}w^{-})}{(s_{0}w^{-})^{3}}- \frac{f(w^{-})}{(w^{-})^{3}} \biggr) \bigl(w^{-} \bigr)^{4} \,dx. \end{aligned}$$

(3.11)

Hence, according to $(f_{4})$, we obtain that $0< t_{0}\leq s_{0}<1$.

Similarly, thanks to (3.8) and $(f_{4})$, we can prove that $t_{0}\geq 1$. Consequently, $t_{0}=s_{0}=1$, that is, $(1,1)$ is a unique critical point of $G_{w}$ with positive coordinates.

In the following, we prove that the map $G_{u}$ has a maximum global point $(s_{0},t_{0})\in (0,\infty )\times (0,\infty )$. Indeed, from $(f_{3})$ and $F\geq 0$ it follows that

$$\begin{aligned} \lim_{|(t,s)|\rightarrow \infty }G_{u}(t,s)=-\infty . \end{aligned}$$

(3.12)

Since $G_{u}$ is a continuous function, from (3.12) we conclude that $G_{u}$ has a global maximum at some point $(s_{0},t_{0})\in \mathbb{R}^{+}\times \mathbb{R}^{+}$, which is a critical point of $G_{u}$. Furthermore, for any $u\in H$ with $u^{\pm }\neq 0$, since $J(tu^{+})+J(su^{-})\leq J(tu^{+}+su^{-})$, $t,s\geq 0$, we get

$$ G_{u}(t,0)+G_{u}(0,s)\leq G_{u}(t,s), \quad t,s \geq 0. $$

So we derive that

$$\begin{aligned} \max_{t\geq 0}G_{u}(t,0)\leq \max _{t,s>0}G_{u}(t,s),\qquad \max_{s\geq 0}G_{u}(0,s) \leq \max_{t,s>0}G_{u}(t,s), \end{aligned}$$

which implies that $(s_{0},t_{0})\in (0,\infty )\times (0,\infty )$.

Finally, we prove (3). From (1) of Lemma 2.3 we have $\frac{\partial G_{u}}{\partial t}(r,s_{-})>0$ for $r\in (0,t_{+})$ and $\frac{\partial G_{u}}{\partial t}(t_{+},s_{-})=0$ and $\frac{\partial G_{u}}{\partial t}(r,s_{-})<0$ for $r\in (t_{+},\infty )$. Thus $\alpha _{+}$ and $\alpha _{-}$ have the same behavior. The proof is complete. □

Now we define

$$\begin{aligned} c_{\infty }=\inf_{u\in \mathcal{M}}J(u). \end{aligned}$$

Since $\mathcal{M}\subset \mathcal{N}$, we have $c_{\infty }\geq d_{\infty }$. On the other hand, for any $v \in H$ with $v^{\pm }\neq 0$, according to (1) of Lemma 2.4, there are $t^{+},s^{-}>0$ such that $t^{+}v^{+},s^{-}v^{-}\in \mathcal{N}$. So, by Lemma 3.1 we derive that

$$ J(v)\geq J \bigl(t^{+}v^{+}+s^{-}v^{-} \bigr)\geq J \bigl(t^{+}v^{+} \bigr)+J \bigl(s^{-}v^{-} \bigr) \geq 2d_{\infty },\quad v \in \mathcal{M}, $$

which implies

$$\begin{aligned} 2d_{\infty }\leq c_{\infty }. \end{aligned}$$

(3.13)

Lemma 3.2

The infimum$c_{\infty }$is achieved.

Proof

Let $\{u_{n}\}$ be a sequence in $\mathcal{M}$ such that

$$\begin{aligned} J(u_{n})\rightarrow c_{\infty }. \end{aligned}$$

(3.14)

Firstly, we prove that $\{u_{n}\}$ is bounded in H. By contradiction there is a subsequence, still denoted $\{u_{n}\}$, such that

$$\begin{aligned} \Vert u_{n} \Vert \rightarrow \infty . \end{aligned}$$

(3.15)

Set $v_{n}:=\frac{u_{n}}{\|u_{n}\|}$, $n\in \mathbb{N}$. Thus there is $v\in H$ such that

$$\begin{aligned} v_{n}\rightharpoonup v \quad \text{as } n \rightarrow \infty . \end{aligned}$$

(3.16)

From Lemma 2.2, up to a subsequence, we deduce that

$$\begin{aligned} v_{n}(x)\rightarrow v(x)\quad \text{a.e. in } \mathbb{R}^{3}. \end{aligned}$$

(3.17)

Thanks to Lemma 3.1 and $\{u_{n}\}\subset \mathcal{M}$, we have that $t_{+}(v_{n})=s_{-}(v_{n})=\|u_{n}\|$. Thus, using Lemma 3.1 again, we obtain

$$\begin{aligned} J \bigl( \Vert u_{n} \Vert v_{n} \bigr)\geq {}& J(tv_{n}) \\ = {}&\frac{t^{2}}{2} \Vert v_{n} \Vert ^{2}_{E}+ \frac{bt^{4}}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla v_{n} \vert ^{2}\,dx \biggr)^{2}+ \frac{t^{4}}{4} \int _{\mathbb{R}^{3}}\phi _{v_{n}}(v_{n})^{2} \,dx \\ &{}- \int _{\mathbb{R}^{3}}K(x)F(tv_{n})\,dx,\quad \forall t>0, n \in \mathbb{N}. \end{aligned}$$

(3.18)

Suppose that $v=0$. Then by (3.16) and Lemma 2.3 we have that

$$\begin{aligned} \int _{\mathbb{R}^{3}}K(x)F(tv_{n})\,dx\rightarrow 0, \quad \forall t>0. \end{aligned}$$

(3.19)

Passing to the limit as $n\rightarrow \infty$ in (3.18), it follows from (3.19) and $\|v_{n}\|=1$ that there exists $\alpha _{0}>0$ such that

$$\begin{aligned} c_{\infty }\geq \frac{t^{2}}{2}\lim_{n\rightarrow \infty } \biggl[ \Vert v_{n} \Vert ^{2}_{E}+ \frac{b}{2} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla v_{n} \vert ^{2}\,dx \biggr)^{2}+ \frac{1}{2} \int _{\mathbb{R}^{3}}\phi _{v_{n}}(v_{n})^{2} \,dx \biggr]:= \frac{t^{2}}{2}\alpha _{0}, \quad \forall t\geq 1, \end{aligned}$$

which is a contradiction. Hence we obtain $v\neq 0$.

On the other hand, we obtain

$$\begin{aligned} \frac{J(u_{n})}{ \Vert u_{n} \Vert ^{4}}={}&\frac{ \Vert u_{n} \Vert ^{2}_{E}}{2 \Vert u_{n} \Vert ^{4}}+ \frac{b}{4 \Vert u_{n} \Vert ^{4}} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \biggr)^{2} \\ &{}+ \frac{1}{4 \Vert u_{n} \Vert ^{4}} \int _{\mathbb{R}^{3}}\phi _{u_{n}}u_{n}^{2} \,dx- \int _{\mathbb{R}^{3}}K(x)\frac{F(u_{n})}{ \Vert u_{n} \Vert ^{4}}\,dx \\ \leq {}&\frac{1}{2 \Vert u_{n} \Vert ^{2}}+\frac{b+1}{4}- \int _{\mathbb{R}^{3}}K(x) \frac{F(v_{n} \Vert u_{n} \Vert )}{(v_{n} \Vert u_{n} \Vert )^{4}}v_{n}^{4} \,dx. \end{aligned}$$

(3.20)

Thanks to $v\neq 0$, from (3.15), (3.16), $(f_{3})$, and Fatou’s lemma we obtain

$$\begin{aligned} \int _{\mathbb{R}^{3}}K(x) \frac{F(v_{n} \Vert u_{n} \Vert )}{(v_{n} \Vert u_{n} \Vert )^{4}}v_{n}^{4} \,dx \rightarrow \infty . \end{aligned}$$

(3.21)

Thus by (3.15), (3.16), and (3.21), passing to the limit as $n\rightarrow \infty$ in (3.20), we get a contradiction. Therefore $\{u_{n}\}$ is bounded in H. So there is $u\in H$ such that

$$\begin{aligned} u_{n}\rightharpoonup u\quad \text{in } H, \qquad u_{n}\rightharpoonup u\quad \text{a.e. in } \mathbb{R}^{3}. \end{aligned}$$

(3.22)

Next, we prove that $u^{\pm }\neq 0$.

Firstly, we claim that there exists $\tau >0$ such that

$$\begin{aligned} \bigl\Vert u^{\pm } \bigr\Vert \geq \tau ,\quad \forall u\in \mathcal{M}. \end{aligned}$$

(3.23)

In fact, since $u\in \mathcal{M}$, we have that

$$\begin{aligned} \bigl\Vert u^{\pm } \bigr\Vert ^{2}_{E}\leq{}& \bigl\Vert u^{\pm } \bigr\Vert ^{2}_{E}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{\pm } \bigr\vert ^{2}\,dx \biggr)^{2} \\ &{}+b \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx \cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{3}} \phi _{u} \bigl({u^{\pm }} \bigr)^{2}\,dx \\ ={}& \int _{\mathbb{R}^{3}}K(x)f \bigl(u^{\pm } \bigr)u^{\pm }\,dx. \end{aligned}$$

(3.24)

So, similarly as in proof of (2.6), we can prove that there is $\tau >0$ such that

$$ \bigl\Vert u^{\pm } \bigr\Vert \geq \bigl\Vert u^{\pm } \bigr\Vert _{E}\geq \tau . $$

Since $\{u_{n}\}\subset \mathcal{M}$, we have

$$\begin{aligned} \int _{\mathbb{R}^{3}}K(x)f \bigl(u_{n}^{\pm } \bigr)u_{n}^{\pm }\,dx\geq \bigl\Vert u_{n}^{ \pm } \bigr\Vert ^{2}_{E} \geq \tau ^{2}. \end{aligned}$$

(3.25)

Combining Lemma 2.3 with (3.25), we have

$$\begin{aligned} \tau ^{2}\leq \int _{\mathbb{R}^{3}}K(x)f \bigl(u^{\pm } \bigr)u^{\pm }\,dx, \end{aligned}$$

which implies that $u^{\pm }\neq 0$.

So, according to Lemma 3.1, there exist $t_{+},s_{-}>0$ such that

$$\begin{aligned} t_{+}u^{+}+s_{-}u^{-} \in \mathcal{M}. \end{aligned}$$

(3.26)

We claim that $0< t_{+}$, $s_{-}\leq 1$. In fact, according to (3.22), Lemma 2.3, and Fatou’s lemma, we obtain

$$\begin{aligned}& \lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}}\phi _{u_{n}} \bigl(u_{n}^{ \pm } \bigr)^{2}\,dx \geq \int _{\mathbb{R}^{3}}\phi _{u} \bigl(u^{\pm } \bigr)^{2}\,dx, \end{aligned}$$

(3.27)

$$\begin{aligned}& \int _{\mathbb{R}^{3}}K(x)f \bigl(u_{n}^{\pm } \bigr)u_{n}^{\pm }\,dx\rightarrow \int _{\mathbb{R}^{3}}K(x)f \bigl(u^{\pm } \bigr)u^{\pm }\,dx. \end{aligned}$$

(3.28)

By the weak semicontinuity of norm in $D^{1,2}(\mathbb{R}^{3})$ and E we have

$$\begin{aligned}& \begin{aligned} \int _{\mathbb{R}^{3}} \bigl(a\nabla uu^{\pm }+V(x)uu^{\pm } \bigr)\,dx&= \bigl\Vert u^{ \pm } \bigr\Vert ^{2}_{E} \\ &\leq \lim_{n\rightarrow \infty } \bigl\Vert u_{n}^{\pm } \bigr\Vert ^{2}_{E}=\lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}} \bigl(a\nabla u_{n}u_{n}^{\pm }+V(x)u_{n}u_{n}^{ \pm } \bigr)\,dx, \end{aligned} \\& \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \int _{\mathbb{R}^{3}}\nabla u \nabla u^{\pm }\,dx\leq \lim _{n\rightarrow \infty } \biggl( \int _{ \mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}}\nabla u_{n} \nabla u_{n}^{\pm }\,dx \biggr). \end{aligned}$$

From (3.22), (3.27), and (3.28) we obtain

$$\begin{aligned} \bigl\langle J'(u),u^{\pm } \bigr\rangle \leq \lim_{n\rightarrow \infty }\inf \bigl\langle J'(u_{n}),u_{n}^{\pm } \bigr\rangle =0. \end{aligned}$$

(3.29)

According to (3.29), we have that

$$\begin{aligned} & \bigl\Vert u^{-} \bigr\Vert ^{2}_{E}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+b \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{3}}\phi _{u} \bigl(u^{-} \bigr)^{2}\,dx \\ &\quad \leq \int _{\mathbb{R}^{3}}K(x)\frac{f(u^{-})}{(u^{-})^{3}} \bigl(u^{-} \bigr)^{4}\,dx. \end{aligned}$$

(3.30)

On the other hand, suppose that $0< t_{+}\leq s_{-}$. Then from (3.26) we have that

$$\begin{aligned} &\frac{ \Vert u^{-} \Vert ^{2}_{E}}{s_{-}^{2}}+b \biggl( \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx \biggr)^{2}+b \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{+} \bigr\vert ^{2}\,dx\cdot \int _{\mathbb{R}^{3}} \bigl\vert \nabla u^{-} \bigr\vert ^{2}\,dx+ \int _{\mathbb{R}^{3}}\phi _{u} \bigl(u^{-} \bigr)^{2}\,dx \\ &\quad \geq \int _{\mathbb{R}^{3}}K(x) \frac{f(s_{-}u^{-})}{(s_{-}u^{-})^{3}} \bigl(u^{-} \bigr)^{4}\,dx. \end{aligned}$$

(3.31)

Combining (3.30) with (3.31), we derive that

$$\begin{aligned} \biggl(\frac{1}{s_{-}^{2}}-1 \biggr) \bigl\Vert u^{-} \bigr\Vert ^{2}_{E}\geq \int _{\mathbb{R}^{3}}K(x) \biggl[ \frac{f(s_{-}u^{-})}{(s_{-}u^{-})^{3}}- \frac{f(u^{-})}{(u^{-})^{3}} \biggr] \bigl(u^{-} \bigr)^{4} \,dx. \end{aligned}$$

By $(f_{4})$ we conclude that $0< s_{-}\leq 1$, that is, $0< t_{+}$, $s_{-}\leq 1$.

Now we prove that

$$\begin{aligned} J \bigl(t_{+}u^{+}+s_{-}u^{-} \bigr)=c_{\infty }. \end{aligned}$$

Let $u_{\pm }:=t_{+}u^{+}+s_{-}u^{-}$. By the definition of $c_{\infty }$ and Fatou’s lemma we have

$$\begin{aligned} c_{\infty }&\leq J(u_{\pm }) \\ &=J(u_{\pm })-\frac{1}{4} \bigl\langle J'(u_{\pm }),u_{\pm } \bigr\rangle \\ &=\frac{1}{4} \Vert u_{\pm } \Vert ^{2}_{E}+ \frac{1}{4} \int _{\mathbb{R}^{3}}K(x) \bigl(f(u_{ \pm })u_{\pm }-4F(u_{\pm }) \bigr)\,dx \\ &\leq\frac{1}{4} \Vert u \Vert ^{2}_{E}+ \frac{1}{4} \int _{\mathbb{R}^{3}}K(x) \bigl(f(u)u-4F(u) \bigr)\,dx \\ &\leq\lim_{n\rightarrow \infty }\inf \biggl[\frac{1}{4} \Vert u_{n} \Vert ^{2}_{E}+ \frac{1}{4} \int _{\mathbb{R}^{3}}K(x) \bigl(f(u_{n})u_{n}-4F(u_{n}) \bigr)\,dx \biggr] \\ &=\lim_{n\rightarrow \infty }\inf \biggl[J(u_{n})- \frac{1}{4} \bigl\langle J'(u_{n}),u_{n} \bigr\rangle \biggr] \\ &=c_{\infty }. \end{aligned}$$

So $t^{+}=s^{-}=1$, and $c_{\infty }$ is achieved by $u:=u^{+}+u^{-}\in \mathcal{M}$. □

4 Proof of main results

Proof

According to Lemma 3.2, we just prove that the minimizer u for $c_{\infty }$ is indeed a nodal solution for system (1.1). By contradiction we suppose that $J'(u)\neq 0$. By continuity there are $\delta ,\mu >0$ such that

$$\begin{aligned} \bigl\Vert J'(v) \bigr\Vert _{H^{-1}}>\mu \quad \text{if } \Vert v-u \Vert < 3\delta . \end{aligned}$$

(4.1)

Choose $\rho \in (0,\min \{\frac{1}{2},\frac{\delta }{\sqrt{2}\|u\|}\})$ and let $D=(1-\rho ,1+\rho )\times (1-\rho ,1+\rho )$, $g(t,s)=tu^{+}+su^{-}$, $(t,s)\in D$. By Lemma 3.1 we get

$$\begin{aligned} \beta :=\max_{(t,s)\in \partial D}J \bigl(g(t,s) \bigr)< c_{\infty }. \end{aligned}$$

(4.2)

Let $\varepsilon \in (0,\min \{\frac{c_{\infty }-\beta }{4}, \frac{\mu \delta }{8}\})$. By Lemma 2.3 in [42] there is a deformation $\eta \in C([0,1]\times H,H)$ such that

• (a) $\eta (t,v)=v$ if $v\notin J^{-1}([c_{\infty }-2\varepsilon ,c_{\infty }+2\varepsilon ])$;

• (b) $J(\eta (1,v))\leq c_{\infty }-\varepsilon$ for each $v\in H$ such that $\|v-u\|\leq \delta$, and $J(v)\leq c_{\infty }+\varepsilon$;

• (c) $J(\eta (1,v))\leq J(v)$ for $v\in H$;

• (d) $\|\eta (t,v)-u\|\leq \delta$ for $v\in H$ and $t\in [0,1]$.

It follows from (4.2) and (b) that

$$\begin{aligned} \max_{(t,s)\in \bar{D}}J \bigl(\eta \bigl(1,g(t,s) \bigr) \bigr)< c_{\infty }. \end{aligned}$$

Then by arguments similar to those in [13, 15], using Brouwer degree theory, we can prove that

$$ \eta \bigl(1,g(D) \bigr)\cap \mathcal{M}\neq \emptyset , $$

a contradiction. Thus $u=u^{+}+u^{-}$ is a critical point of J, which is a least energy nodal solution of system (1.1).

Furthermore, if f is odd, then functional ψ is even. According to (2.9) and (3.13), we conclude that ψ is bounded from below in S. By Lemmas 2.2 and 2.3 we can prove that ψ satisfies the Palais–Smale condition on S. Hence by Lemma 2.4, Proposition 2.1, and [32] the functional J has infinitely many critical points. □