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A Kirchhoff-type problem involving concave-convex nonlinearities

Abstract

A Kirchhoff-type problem with concave-convex nonlinearities is studied. By constrained variational methods on a Nehari manifold, we prove that this problem has a sign-changing solution with least energy. Moreover, we show that the energy level of this sign-changing solution is strictly larger than the double energy level of the ground state solution.

Introduction

We study the following Kirchhoff-type equation with concave-convex nonlinearities:

$$ \textstyle\begin{cases} (a+\lambda \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}+\lambda b \int _{\mathbb{R}^{3}}u^{2} )(-\Delta u+b u) \\ \quad =Q(x) \vert u \vert ^{p-1} u+ \kappa G(x) \vert u \vert ^{q-1} u ,\quad x\in \mathbb{R}^{3}, \\ u\in H^{1}_{r}(\mathbb{R}^{3}),\end{cases} $$
(1.1)

where \(a>0\), \(b>0\), \(\lambda >0\), \(\kappa <0\), \(p\in (3,5)\), \(q\in (0,1)\), and \(Q,G\in C(\mathbb{R}^{3},\mathbb{R}^{+})\) satisfying the following conditions:

\((Q_{1})\):

There exists \(\beta \in [0,p-2)\) such that \(\limsup_{x\rightarrow +\infty }\frac{Q(x)}{|x|^{\beta }}<+\infty \);

\((G_{1})\):

\(G(x)\in L^{2}(\mathbb{R}^{3},\mathbb{R}^{+})\).

In recent years, the following elliptic problem has been investigated by many researchers [1, 3, 6, 9, 17, 20]:

$$ \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2} )\Delta u=f(x,u),\quad x\in \mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}), \end{cases} $$
(1.2)

where \(f\in C(\mathbb{R}^{3}\times \mathbb{R},\mathbb{R})\) and \(a>0\), \(b>0\). The term \(\int _{\mathbb{R}^{3}}|\nabla u|^{2}\) in (1.2) has an interesting physical application. Moreover, this problem is related to the stationary analogue of the following equation proposed by Kirchhoff [10]:

$$ u_{tt}- \biggl(a+b \int _{\Omega } \vert \nabla u \vert ^{2} \biggr) \Delta u=f(x,u). $$
(1.3)

Inspired by the variational framework given by Lions [12], problem (1.3) has been investigated by many researchers, and the reader is referred to [5, 7, 11, 13, 19, 22] and the references therein for more details.

Shuai [16] studied the ground state sign-changing solution of problem (1.2) by using Brouwer degree theory, where \(f(x,u)\) is replaced with \(f(u)\) with the following hypotheses:

\((f_{1}^{\prime })\)::

\(f(s)=o(|s|)\) as \(s\rightarrow 0\);

\((f_{2}^{\prime })\)::

For some constant \(p\in (4,2^{*})\), \(\lim_{s\rightarrow \infty }\frac{f(s)}{s^{p-1}}=0\), where \(2^{*}=+\infty \) for \(N=1,2\) and \(2^{*}=6\) for \(N=3\);

\((f_{3}^{\prime })\)::

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

\((f_{4}^{\prime })\)::

\(\frac{f(s)}{|s|^{3}}\) is an increasing function with respect to \(s\in \mathbb{R}\setminus \{0\}\).

Huang and Liu [8] obtained the ground state sign-changing solutions of problem (1.4) with accurately two nodal domains

$$ - \biggl(1+\lambda \int _{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}+V(x)u^{2}\bigr) \biggr)\bigl[\Delta u+V(x)u\bigr]= \vert u \vert ^{p-1}u, \quad x\in \mathbb{R}^{N}, $$
(1.4)

where \(p\in (3,5)\), \(\lambda >0\) and \(V\in C(\mathbb{R}^{N},\mathbb{R})\) is to ensure the establishment of compactness.

Deng et al. [4] showed the existence of radial sign-changing solutions \(u_{k}^{b}\) of problem (1.5)

$$ \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2} )\Delta u+V(x) u=f(x,u),\quad x\in \mathbb{R}^{3}, \\ u\in H^{1}_{r}(\mathbb{R}^{3}), \end{cases} $$
(1.5)

by constrained minimization on the Nehari manifold, where k is any positive integer. Ye [21] studied the existence of least energy sign-changing solutions for problem (1.5), where \(f(x,u)\) is replaced with \(f(u)\).

Shao and Mao [15] got at least one sign-changing solution of problem (1.6) with concave-convex nonlinearities

$$ \textstyle\begin{cases} - (a+b\int _{\Omega } \vert \nabla u \vert ^{2} )\Delta u=\mu g(x,u)+ f(x,u),\quad \mbox{in }\Omega , \\ u=0,\quad \mbox{on }\partial \Omega , \end{cases} $$
(1.6)

by using the method of invariant sets of descending flow.

Motivated by the aforementioned works, we prove the existence of sign-changing solutions with least energy for problem (1.1) with concave-convex nonlinearities and unbounded potential by constrained variational methods on a Nehari manifold.

Now we will give the main results by Theorems 1.1 and 1.2.

Theorem 1.1

Assume that \((Q_{1})\) and \((G_{1})\) hold, then, for \(a>0\), \(b>0\), \(\lambda >0\), and \(\kappa <0\), problem (1.1) has one least energy sign-changing solution with accurately two nodal domains.

Theorem 1.2

Assume that \((Q_{1})\) and \((G_{1})\) hold, then, for \(a>0\), \(b>0\), \(\lambda >0\), and \(\kappa <0\), problem (1.1) has one least energy solution. Moreover \(m_{\lambda }>2c_{\lambda }\), where \(m_{\lambda }\) and \(c_{\lambda }\) are defined by (2.3) and (2.5) respectively.

Remark 1.3

Comparing with Shuai [16], Huang and Liu [8], Deng et al. [4], and Ye [21], the difference is to consider Kirchhoff-type equation with concave and convex terms, where \(Q(x)\) is unbounded at infinity. Moreover, since \(H^{1}_{r}(\mathbb{R}^{3})\hookrightarrow L^{{q+1}}(\mathbb{R}^{3})\) is not compact for \({q}\in (0,1)\), this means that the appearance of concave and convex terms has greatly increased the difficulty of problem (1.1). Shao and Mao [15] got sign-changing solutions for Kirchhoff equation with concave and convex terms by using the method of invariant sets of descending flow. However, we want to obtain ground state sign-changing solutions of (1.1) by variational methods and constrained minimization on the sign-changing Nehari manifold. It should be addressed that our methods are different to those in [15].

The rest of the paper is organized as follows. In Sect. 2 we give some notations and the main lemmas related to the proof of our main results. Sections 3 and 4 give the proofs of Theorems 1.1 and 1.2, respectively.

Some notations and preliminary lemmas

Here are some notations to be used in this paper.

  • C denotes a positive constant;

  • \(H^{1}({\mathbb{R}}^{3})\) denotes the usual Sobolev space with the norm \(\|u\|^{2}=\int _{\mathbb{R}^{3}}(|\nabla u|^{2}+b |u|^{2})\);

  • \(|\cdot |\) denotes the usual norm \(L^{\bar{q}}(\mathbb{R}^{3})\) for \(\bar{q}\in [1,\infty )\);

  • \(H^{1}_{r}({\mathbb{R}}^{3}):=\{u:u\in H^{1}({\mathbb{R}}^{3}),u(x)=u(|x|) \}\);

  • \(u^{+}:=\max \{{u,0}\}\) and \(u^{-}:=\min \{{u,0}\}\).

Lemma 2.1

(see Berestycki and Lions [2])

Let \(N\geq 2\) and \(u\in H_{r}^{1}({\mathbb{R}}^{N})\), Then

$$ \bigl|u(r)\bigr|\leq C_{0}\|u\|r^{\frac{1-N}{2}} \quad \textit{for } r\geq 1 , $$

where \(C_{0}>0\) is only related to N.

Remark 2.2

For any \(u\in H_{r}^{1}({\mathbb{R}}^{3})\), by \((Q_{1})\), \((G_{1})\), and Lemma 2.1, we have

$$ 0\leq \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}\leq C_{1} \Vert u \Vert ^{p+1} $$

and

$$ \biggl\vert \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \biggr\vert \leq \int _{ \mathbb{R}^{3}} \bigl\vert G(x) \bigr\vert \vert u \vert ^{q+1}\leq \bigl\vert G(x) \bigr\vert _{2} \vert u \vert ^{q+1}_{2(q+1)}\leq C_{1} \Vert u \Vert ^{q+1}. $$

The energy functional \(J_{\lambda }\in C^{1}(H^{1}_{r}({\mathbb{R}}^{3}),\mathbb{R})\) is well defined by

$$ J_{\lambda }(u)=\frac{1}{2}a \Vert u \Vert ^{2} + \frac{1}{4}\lambda \Vert u \Vert ^{4}- \frac{1}{p+1} \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}- \frac{1}{q+1}\kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1}. $$
(2.1)

For each \(u,v\in H^{1}_{r}({\mathbb{R}}^{3})\),

$$ \bigl\langle J^{\prime }_{\lambda }(u),v\bigr\rangle =a(u,v)+\lambda \Vert u \Vert ^{2}(u,v)- \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p-1}uv-\kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q-1}uv. $$
(2.2)

In order to get a sign-changing solution \(u^{\pm }\neq 0\) of (1.1), the following functionals need to be established:

$$\begin{aligned}& J_{\lambda }(u)=J_{\lambda }\bigl(u^{+} \bigr)+J_{\lambda }\bigl(u^{-}\bigr)+ \frac{\lambda }{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \bigl\Vert u^{-} \bigr\Vert ^{2}, \\& \bigl\langle J_{\lambda }^{\prime }(u),u^{+} \bigr\rangle =\bigl\langle J_{\lambda }^{\prime }\bigl(u^{+} \bigr),u^{+}\bigr\rangle +\lambda \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}, \\& \bigl\langle J_{\lambda }^{\prime }(u),u^{-} \bigr\rangle =\bigl\langle J_{\lambda }^{\prime }\bigl(u^{-} \bigr),u^{-}\bigr\rangle +\lambda \bigl\Vert u^{+} \bigr\Vert ^{2} \bigl\Vert u^{-} \bigr\Vert ^{2}. \end{aligned}$$

Let us define

$$ \mathcal{M}_{\lambda }=\bigl\{ u\in H^{1}_{r} \bigl({\mathbb{R}}^{3}\bigr):u^{\pm }\neq 0, \bigl\langle J^{\prime }_{\lambda }(u),u^{+}\bigr\rangle =\bigl\langle J^{\prime }_{ \lambda }(u),u^{-}\bigr\rangle =0\bigr\} $$

and

$$ m_{\lambda }:=\inf \bigl\{ J_{\lambda }(u): u\in \mathcal{M}_{\lambda }\bigr\} . $$
(2.3)

In addition, we define

$$ \mathcal{N}_{\lambda }=\bigl\{ u\in H^{1}_{r} \bigl({\mathbb{R}}^{3}\bigr)\setminus \{0 \}:\bigl\langle J^{\prime }_{\lambda }(u),u\bigr\rangle =0\bigr\} $$
(2.4)

and

$$ c_{\lambda }:=\inf \bigl\{ J_{\lambda }(u):u\in \mathcal{N}_{\lambda } \bigr\} .$$
(2.5)

Lemma 2.3

Assume that \((Q_{1})\), \((G_{1})\), and \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\) hold, then

$$ \lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}}G(x) \vert u_{n} \vert ^{q+1}= \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1}. $$

In particular,

$$ \lim_{n\rightarrow \infty } \int _{\mathbb{R}^{3}}G(x) \bigl\vert u_{n}^{\pm } \bigr\vert ^{q+1}= \int _{\mathbb{R}^{3}}G(x) \bigl\vert u^{\pm } \bigr\vert ^{q+1}. $$

Proof

If \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\), then \(u_{n}\rightarrow u\) in \(L^{\bar{q}}(\mathbb{R}^{3})\) for \(\bar{q}\in (2,6)\). According to [18, Theorem A.4, p. 134], we can obtain that \(|u_{n}|^{q+1}\rightarrow |u|^{q+1}\) in \(L^{2}(\mathbb{R}^{3})\). By the Hölder inequality, we have

$$ \begin{aligned} & \biggl\vert \int _{\mathbb{R}^{3}}G(x) \vert u_{n} \vert ^{q+1}- \int _{ \mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \biggr\vert \\ &\quad \leq \int _{\mathbb{R}^{3}} \bigl\vert G(x) \bigr\vert \bigl\vert \vert u_{n} \vert ^{q+1}- \vert u \vert ^{q+1} \bigr\vert \\ &\quad \leq \bigl\vert G(x) \bigr\vert _{2} \bigl\vert |u_{n} \vert ^{q+1}- \vert u \vert ^{q+1}\bigr|_{2} \rightarrow 0. \end{aligned} $$

Thus, \(\lim_{n\rightarrow \infty }\int _{\mathbb{R}^{3}}G(x)|u_{n}|^{q+1}= \int _{\mathbb{R}^{3}}G(x)|u|^{q+1}\). Similarly, \(\lim_{n\rightarrow \infty }\int _{\mathbb{R}^{3}}G(x)|u_{n}^{\pm }|^{q+1}= \int _{\mathbb{R}^{3}}G(x)|u^{\pm }|^{q+1}\). □

Lemma 2.4

Under the assumptions of Theorem 1.1. If \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\), there exists a unique pair \((s_{u},t_{u})\in (0,+\infty )\times (0,+\infty )\) such that \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\). Moreover,

$$ J_{\lambda }\bigl(s_{u}u^{+}+t_{u}u^{-} \bigr)=\max_{s,t\geq 0}J_{\lambda }\bigl(su^{+}+tu^{-} \bigr). $$

Proof

Let \(u\in H^{1}(\mathbb{R}^{3})\) with \(u^{\pm }\neq 0\). Define

$$\begin{aligned}& g_{1}(s,t)=a s^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}+\lambda s^{4} \bigl\Vert u^{+} \bigr\Vert ^{4}+\lambda s^{2}t^{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \bigl\Vert u^{-} \bigr\Vert ^{2} \\& \hphantom{g_{1}(s,t)={}}{}-s^{p+1} \int _{\mathbb{R}_{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}- \kappa s^{q+1} \int _{\mathbb{R}_{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1}, \end{aligned}$$
(2.6)
$$\begin{aligned}& g_{2}(s,t)=a t^{2} \bigl\Vert u^{-} \bigr\Vert ^{2}+\lambda t^{4} \bigl\Vert u^{-} \bigr\Vert ^{4}+\lambda s^{2}t^{2} \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \\& \hphantom{g_{2}(s,t)={}}{}-t^{p+1} \int _{\mathbb{R}_{3}}Q(x) \bigl\vert u^{-} \bigr\vert ^{p+1}- \kappa t^{q+1} \int _{\mathbb{R}_{3}}G(x) \bigl\vert u^{-} \bigr\vert ^{q+1}. \end{aligned}$$
(2.7)

According to Remark 2.2, for \(\kappa <0\), we have \(g_{i}(s,s)>0\) as \(s>0\) small and \(g_{i}(t,t)<0\) as \(t>0\) large, where \(i=1,2\). Then there exists \(0<\mu <\nu \) such that

$$ g_{i}(\mu ,\mu )>0,\qquad g_{i}(\nu ,\nu )< 0. $$
(2.8)

By (2.6), (2.7), (2.8), we have that

$$\begin{aligned}& g_{1}(\mu ,t)>0,\qquad g_{1}(\nu ,t)< 0,\quad t\in [\mu ,\nu ], \\& g_{2}(s,\mu )>0,\qquad g_{2}(s,\nu )< 0,\quad s\in [\mu ,\nu ]. \end{aligned}$$

From Miranda’s theorem [14], there exists a pair \((s_{u},t_{u})\) such that

$$ g_{1}(s_{u},t_{u})=0,\qquad g_{2}(s_{u},t_{u})=0,\quad \mu < s_{u},t_{u}< \nu . $$

Thus, \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\).

Secondly, we prove the uniqueness. Let both \((s_{1},t_{1})\) and \((s_{2},t_{2})\) satisfy \(u_{i}=s_{i}u^{+}+t_{i}u^{-}\in \mathcal{M}_{\lambda }\) (\(i=1,2\)) and \(u_{1}=s_{1}u^{+}+t_{1}u^{-}= m s_{2}u^{+}+n t_{2}u^{-}=mu_{2}^{+}+nu^{-}_{2}\), where \(m=\frac{s_{1}}{s_{2}}\), \(n=\frac{t_{1}}{t_{2}}\). By (2.6) and (2.7),

$$\begin{aligned}& g_{1}^{u_{1}}(1,1)=g_{1}^{u_{2}}(m,n)=g_{1}^{u_{2}}(1,1)=0, \end{aligned}$$
(2.9)
$$\begin{aligned}& g_{2}^{u_{1}}(1,1)=g_{2}^{u_{2}}(m,n)=g_{2}^{u_{2}}(1,1)=0. \end{aligned}$$
(2.10)

We only need to prove that \(m=n=1\). Now, assume that \(0< m\leq n\). By (2.9) and (2.10),

$$ g_{1}^{u_{2}}(1,1)-\frac{g_{1}^{u_{2}}(m,n)}{m^{4}}=0 $$
(2.11)

and

$$ g_{2}^{u_{2}}(1,1)-\frac{g_{2}^{u_{2}}(m,n)}{n^{4}}=0.$$
(2.12)

If \(m<1\), then

$$\begin{aligned}& \biggl(1-\frac{1}{m^{2}}\biggr)a \bigl\Vert u_{2}^{+} \bigr\Vert ^{2}+\biggl(1- \frac{n^{2}}{m^{2}}\biggr)\lambda \bigl\Vert u_{2}^{-} \bigr\Vert ^{2} \bigl\Vert u_{2}^{+} \bigr\Vert ^{2} \\& \quad =\bigl(1-m^{p-3}\bigr) \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u_{2}^{+} \bigr\vert ^{p+1}+\bigl(1-m^{q-3}\bigr) \kappa \int _{\mathbb{R}^{3}}G(x) \bigl\vert u_{2}^{+} \bigr\vert ^{q+1}, \end{aligned}$$

this is impossible for \(\kappa <0\). Then \(m\geq 1\). Similarly, if \(n>1\), (2.12) is impossible. Then \(n\leq 1\). Thus \(m=n=1\).

At last, let

$$ \begin{aligned} H_{\lambda }(s,t)&=J_{\lambda } \bigl(su^{+}+tu^{-}\bigr) \\ &=\frac{a}{2}s^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}+ \frac{\lambda }{4}s^{4} \bigl\Vert u^{+} \bigr\Vert ^{4}-\frac{s^{p+1}}{p+1} \int _{ \mathbb{R}^{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}-\frac{s^{q+1}}{q+1}\kappa \int _{ \mathbb{R}_{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1} \\ &\quad {} +\frac{a}{2}t^{2} \bigl\Vert u^{-} \bigr\Vert ^{2}+\frac{\lambda }{4}t^{4} \bigl\Vert u^{-} \bigr\Vert ^{4}- \frac{t^{p+1}}{p+1} \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u^{-} \bigr\vert ^{p+1}- \frac{t^{q+1}}{q+1}\kappa \int _{\mathbb{R}_{3}}G(x) \bigl\vert u^{-} \bigr\vert ^{q+1} \\ &\quad {} +\frac{\lambda }{2}s^{2}t^{2} \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}. \end{aligned} $$

Then, for \(\kappa <0\), we have \(H_{\lambda }(s,t)>0\) as \(|(s,t)|\rightarrow 0\), \(H_{\lambda }(s,t)<0\) as \(|(s,t)|\rightarrow \infty \), and \(H_{\lambda }\) cannot achieve the maximum point on \(\partial {\mathbb{R}^{+}}^{2}\). Without loss of generality, we only prove that \((0,t_{0})\) is not a maximum point of \(H_{\lambda }\). For \(s>0\) small enough,

$$ \begin{aligned} \frac{\partial H_{\lambda }}{\partial s}(s,t_{0})={}&a s \bigl\Vert u^{+} \bigr\Vert ^{2}+ \lambda s^{3} \bigl\Vert u^{+} \bigr\Vert ^{4}+\lambda st_{0}^{2} \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \\ &{}-s^{p} \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}-{s^{q}}\kappa \int _{ \mathbb{R}_{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1}>0, \end{aligned} $$

this implies that \(H_{\lambda }(s,t_{0})\) is an increasing function with respect to s, where \(s>0\) is small enough, then \((0,t_{0})\) is not a maximum point of \(H_{\lambda }\). Thus, there exists \((s_{u},t_{u})\in {\mathbb{R}^{+}}^{2}\) such that

$$ J_{\lambda }\bigl(s_{u}u^{+}+t_{u}u^{-} \bigr)=\max_{s,t\geq 0}J_{\lambda }\bigl(su^{+}+tu^{-} \bigr). $$

 □

Lemma 2.5

Under the assumptions of Theorem 1.1. If \(\langle J^{\prime }_{\lambda }(u),u^{\pm }\rangle \leq 0\), there exists \((s_{u},t_{u})\in (0,1]\times (0,1]\) such that \(s_{u}u^{+}+t_{u}u^{-}\in \mathcal{M}_{\lambda }\) for \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\).

Proof

Let \(u\in H^{1}_{r}({\mathbb{R}}^{3})\) with \(u^{\pm }\neq 0\), by Lemma 2.4, there exists a pair \((s_{u},t_{u})\) such that

$$\begin{aligned}& s_{u}^{2}a \bigl\Vert u^{+} \bigr\Vert ^{2}+s_{u}^{4} \lambda \bigl\Vert u^{+} \bigr\Vert ^{4}+s_{u}^{2}t_{u}^{2} \lambda \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \\& \quad{} -s_{u}^{p+1} \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}-s_{u}^{q+1} \kappa \int _{\mathbb{R}^{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1}=0. \end{aligned}$$
(2.13)

Since \(\langle J^{\prime }_{\lambda }(u),u^{\pm }\rangle \leq 0\), we have that

$$ a \bigl\Vert u^{+} \bigr\Vert ^{2}+ \lambda \bigl\Vert u^{+} \bigr\Vert ^{4}+\lambda \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}- \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}-\kappa \int _{ \mathbb{R}^{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1}\leq 0.$$
(2.14)

Now, assume that \(0< t_{u}\leq s_{u}\). If \(s_{u}>1\), by (2.13) and (2.14),

$$\begin{aligned}& \biggl(1-\frac{1}{s_{u}^{2}}\biggr)a \bigl\Vert u^{+} \bigr\Vert ^{2}+\biggl(1-\frac{t_{u}^{2}}{s_{u}^{2}} \biggr) \lambda \bigl\Vert u^{-} \bigr\Vert ^{2} \bigl\Vert u^{+} \bigr\Vert ^{2} \\& \quad \leq \bigl(1-s_{u}^{p-3}\bigr) \int _{ \mathbb{R}^{3}}Q(x) \bigl\vert u^{+} \bigr\vert ^{p+1}+\bigl(1-s_{u}^{q-3}\bigr)\kappa \int _{ \mathbb{R}^{3}}G(x) \bigl\vert u^{+} \bigr\vert ^{q+1}, \end{aligned}$$

which is contradictory for \(\kappa <0\). Then \(s_{u}\leq 1\). From \(0< t_{u}\leq s_{u}\), we obtain that \(0< t_{u}\leq s_{u}\leq 1\). □

Lemma 2.6

Under the assumptions of Theorem 1.1, \(m_{\lambda }>0\) can be achieved.

Proof

For all \(u\in \mathcal{M}_{\lambda }\), by the Sobolev embedding theorem, we have

$$ a \Vert u \Vert ^{2}\leq a \Vert u \Vert ^{2}+ \lambda \Vert u \Vert ^{4}= \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}+ \kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1}\leq C_{1} \Vert u \Vert ^{p+1}. $$

Then there exists \(C\geq C_{1}\) such that \(\|u\|\geq (\frac{a}{C} )^{\frac{1}{p-1}}>0\). Since

$$ \begin{aligned}[b] J_{\lambda }(u)&=J_{\lambda }(u)- \frac{1}{4}\bigl\langle J_{ \lambda }^{\prime }(u),u\bigr\rangle \\ &=\frac{a}{2} \Vert u \Vert ^{2}+\frac{\lambda }{4} \Vert u \Vert ^{4}-\frac{1}{p+1} \int _{ \mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}- \frac{1}{q+1}\kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \\ & -\frac{a}{4} \Vert u \Vert ^{2}- \frac{\lambda }{4} \Vert u \Vert ^{4}+\frac{1}{4} \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}+ \frac{1}{4}\kappa \int _{ \mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \\ &=\frac{a}{4} \Vert u \Vert ^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1} \biggr) \int _{ \mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}- \biggl( \frac{1}{q+1}-\frac{1}{4} \biggr) \kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \\ &\geq \frac{a}{8} \Vert u \Vert ^{2} \end{aligned} $$
(2.15)

for \(\kappa <0\). Then

$$ m_{\lambda }=\inf_{u\in {\mathcal{M}_{\lambda }}} J_{\lambda }(u)>0. $$

Let \(\{u_{n}\}\subset \mathcal{M}_{\lambda }\) and \(J_{\lambda }(u_{n})\rightarrow m_{\lambda }\). By Remark 2.2, we have

$$ 1+m_{\lambda } \geq J_{\lambda }(u_{n})- \frac{1}{p+1}\bigl\langle J_{\lambda }^{ \prime }(u_{n}),u_{n} \bigr\rangle \geq \frac{a}{8} \Vert u_{n} \Vert ^{2}. $$

This shows that \(\{u_{n}\}\) is bounded in \(H^{1}_{r}(\mathbb{R}^{3})\). Then there exists \(u_{\lambda }\in H^{1}_{r}(\mathbb{R}^{3})\) such that \(u_{n}^{\pm }\rightharpoonup u_{\lambda }^{\pm }\) in \(H^{1}_{r}(\mathbb{R}^{3})\), \(u_{n}^{\pm }\rightarrow u_{\lambda }^{\pm }\) in \(L^{q}(\mathbb{R}^{3})\) for \(q\in (2,6)\) and \(u_{n}^{\pm }(x) \rightarrow u_{\lambda }^{\pm }(x)\) a.e. on \(\mathbb{R}^{3}\). Since \(\{u_{n}\}\subset \mathcal{M}_{\lambda }\), we have

$$ 0< C\leq a \bigl\Vert u_{n}^{\pm } \bigr\Vert ^{2}+\lambda \bigl\Vert u_{n}^{\pm } \bigr\Vert ^{4}+\lambda \bigl\Vert u_{n}^{+} \bigr\Vert ^{2} \bigl\Vert u_{n}^{-} \bigr\Vert ^{2}= \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u_{n}^{\pm } \bigr\vert ^{p+1}+ \kappa \int _{\mathbb{R}^{3}}G(x) \bigl\vert u_{n}^{\pm } \bigr\vert ^{q+1}. $$

By Fatou’s lemma and Lemma 2.3,

$$ a \bigl\Vert u_{\lambda }^{\pm } \bigr\Vert ^{2}+\lambda \bigl\Vert u_{\lambda }^{\pm } \bigr\Vert ^{4}+\lambda \bigl\Vert u_{ \lambda }^{+} \bigr\Vert ^{2} \bigl\Vert u_{\lambda }^{-} \bigr\Vert ^{2}\leq \int _{\mathbb{R}^{3}}Q(x) \bigl\vert u_{ \lambda }^{\pm } \bigr\vert ^{p+1}+\kappa \int _{\mathbb{R}^{3}}G(x) \bigl\vert u_{\lambda }^{\pm } \bigr\vert ^{q+1}, $$

this implies that

$$ \bigl\langle J^{\prime }_{\lambda }(u_{\lambda }),u_{\lambda }^{\pm } \bigr\rangle \leq 0. $$

By Lemmas 2.4 and 2.5, there exists \((s_{{u}_{\lambda }},t_{{u}_{\lambda }})\in (0,1]\times (0,1]\) such that \(\widetilde{u}_{\lambda }=s_{{u}_{\lambda }}u^{+}_{\lambda }+t_{{u}_{ \lambda }}u^{-}_{\lambda }\in \mathcal{M}_{\lambda }\). Then

$$\begin{aligned} m_{\lambda } \leq& J_{\lambda }(\widetilde{u}_{\lambda })- \frac{1}{p+1}\bigl\langle J^{\prime }_{\lambda }( \widetilde{u}_{\lambda }), \widetilde{u}_{\lambda }\bigr\rangle \\ =& \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr)a \Vert \widetilde{u}_{\lambda } \Vert ^{2}+ \biggl( \frac{1}{4}-\frac{1}{p+1} \biggr)\lambda \Vert \widetilde{u}_{\lambda } \Vert ^{4}- \biggl( \frac{1}{q+1}-\frac{1}{p+1} \biggr)\kappa \int _{\mathbb{R}^{3}}G(x) \vert \widetilde{u}_{\lambda } \vert ^{q+1} \\ \leq& \frac{p-1}{2(p+1)}a \Vert {u}_{\lambda } \Vert ^{2}+ \frac{p-3}{4(p+1)}\lambda \Vert {u}_{\lambda } \Vert ^{4}-\frac{p-q}{(q+1)(p+1)} \kappa \int _{\mathbb{R}^{3}}G(x) \vert {u}_{\lambda } \vert ^{q+1} \\ \leq& \liminf_{n} \biggl\{ \frac{p-1}{2(p+1)}a \Vert {u}_{n} \Vert ^{2}+ \frac{p-3}{4(p+1)}\lambda \Vert {u}_{n} \Vert ^{4}-\frac{p-q}{(q+1)(p+1)} \kappa \int _{\mathbb{R}^{3}}G(x) \vert {u}_{n} \vert ^{q+1} \biggr\} \\ =&\liminf_{n} \biggl(J_{\lambda }(u_{n})- \frac{1}{p+1} \bigl\langle J^{\prime }_{\lambda }({u}_{n}),{u}_{n} \bigr\rangle \biggr) \\ =&m_{\lambda }, \end{aligned}$$

this implies that \(s_{u_{\lambda }}=t_{u_{\lambda }}=1\). Thus, \(\widetilde{u}_{\lambda }=u_{\lambda }\) and \(J_{\lambda }(u_{\lambda })=m_{\lambda }\). □

Sign-changing solutions

Lemma 3.1

Under the assumptions of Theorem 1.1. If \(u_{\lambda }\in \mathcal{M}_{\lambda }\) and \(J_{\lambda }(u_{\lambda })=m_{\lambda }\), then \(J^{\prime }_{\lambda }(u_{\lambda })=0\).

Proof

Suppose that \(J_{\lambda }^{\prime }(u_{\lambda })\neq 0\), then there are σ, \(\delta >0\) such that

$$ \bigl\Vert J_{\lambda }^{\prime }(u) \bigr\Vert \geq \sigma ,\quad \forall \Vert u-u_{ \lambda } \Vert \leq 3\delta . $$

Let \(D=(0.5,1.5)\times (0.5,1.5)\). By Lemma 2.4, we obtain that

$$ \iota :=\max_{(s,t)\in \partial D}J_{\lambda } \bigl(su_{\lambda }^{+}+tu_{ \lambda }^{-} \bigr)< m_{\lambda }. $$
(3.1)

For \(\varepsilon :=\min \{(m_{\lambda }-\iota )/2,\sigma \delta /8\}\) and \(S:=B(u_{\lambda },\delta )\), Willem [18, Lemma 2.3] produce a deformation η such that

  1. (i)

    \(\eta (1,u)=u\) if \(u\notin J_{\lambda }^{-1}([m_{\lambda }-2\varepsilon ,m_{\lambda }+2 \varepsilon ])\cap S_{2\delta }\);

  2. (ii)

    \(\eta (1,J_{\lambda }^{m_{\lambda }+\varepsilon }\cap S)\subset J_{ \lambda }^{m_{\lambda }-\varepsilon }\);

  3. (iii)

    \(J_{\lambda }(\eta (1,u))\leq J_{\lambda }(u)\) for all \(u\in H_{r}^{1}(\mathbb{R}^{3})\).

At first, we show that

$$ \max_{(s,t)\in \bar{D}}J_{\lambda }\bigl(\eta \bigl(1,su_{\lambda }^{+}+tu^{-}_{ \lambda } \bigr)\bigr)< m_{\lambda }. $$

For all \((s,t)\in \bar{D}\), by Lemma 2.4, we obtain \(J_{\lambda }(su_{\lambda }^{+}+tu_{\lambda }^{-})\leq m_{\lambda }< m_{ \lambda }+\varepsilon \), that is, \(su_{\lambda }^{+}+tu_{\lambda }^{-}\in J_{\lambda }^{m_{\lambda }+ \varepsilon }\). Therefore, \(J_{\lambda }(\eta (1,su_{\lambda }^{+}+tu_{\lambda }^{-}))\leq m_{ \lambda }-\varepsilon \).

Next, we prove that

$$ \eta \bigl(1,su_{\lambda }^{+}+tu_{\lambda }^{-} \bigr)\cap \mathcal{M}_{\lambda }\neq \emptyset ,\quad \forall (s,t)\in \bar{D}. $$

Define \(h(s,t)=\eta (1,su_{\lambda }^{+}+tu_{\lambda }^{-})\) and \(\psi :[0,1]\times \bar{D}\rightarrow \mathbb{R}^{2}\), for any \(\vartheta \in [0,1]\), we have

$$\begin{aligned} \psi \bigl(\vartheta ,(s,t)\bigr) =&\bigl(\bigl\langle J_{\lambda }^{\prime } \bigl(\eta \bigl(\vartheta ,su_{ \lambda }^{+}+tu_{\lambda }^{-} \bigr)\bigr),\bigl(\eta \bigl(\vartheta ,su_{\lambda }^{+}+tu_{ \lambda }^{-} \bigr)\bigr)^{+}\bigr\rangle , \\ &\bigl\langle J_{\lambda }^{\prime } \bigl(\eta \bigl( \vartheta ,su_{\lambda }^{+}+tu_{\lambda }^{-} \bigr)\bigr),\bigl(\eta \bigl(\vartheta ,su_{ \lambda }^{+}+tu_{\lambda }^{-} \bigr)\bigr)^{-}\bigr\rangle \bigr). \end{aligned}$$

Let

$$\begin{aligned}& \psi _{0}=\psi _{0}(1,\cdot )=\bigl\langle J_{\lambda }^{\prime }\bigl(su_{\lambda }^{+}+tu_{ \lambda }^{-} \bigr)su_{\lambda }^{+},J_{\lambda }^{\prime } \bigl(su_{\lambda }^{+}+tu_{ \lambda }^{-} \bigr)tu_{\lambda }^{-}\bigr\rangle , \\& \psi _{1}=\psi _{1}(1,\cdot )=\bigl\langle J_{\lambda }^{\prime }\bigl(h(s,t)\bigr)h^{+}(s,t),J_{ \lambda }^{\prime } \bigl(h(s,t)\bigr)h^{-}(s,t)\bigr\rangle . \end{aligned}$$

By a simple calculation, \(\operatorname{deg}(\psi _{0},D,0)=1\). According to (3.1), we obtain that \(u_{\lambda }=h\) on ∂D and from homotopy invariance that

$$ \deg (\psi _{1},D,0)=\deg (\psi _{0},D,0)=1. $$

Then there exists a pair \((s_{0},t_{0})\in D\) such that \(\psi _{1}(s_{0},t_{0})=0\) and \(\eta (1,s_{0}u_{\lambda }^{+}+t_{0}u^{-}_{\lambda })=h(s_{0},t_{0}) \in \mathcal{M}_{\lambda }\), which contradicts (3.1). Therefore, \(u_{\lambda }\) is a critical point of \(J_{\lambda }\), and so a sign-changing solution of (1.1). □

Proof of Theorem 1.1

Firstly, by the preceding lemmas, there exists \(u_{\lambda }\in \mathcal{M}_{\lambda }\) such that \(J_{\lambda }(u_{\lambda })=m_{\lambda }\) and \(J_{\lambda }^{\prime }(u_{\lambda })=0\). Thus, problem (1.1) has one least energy sign-changing solution \(u_{\lambda }\).

Secondly, we prove that \(u_{\lambda }\) has only two nodal domains. Assume that \(u_{\lambda }=u_{1}+u_{2}+u_{3}\) with

$$\begin{aligned}& u_{i} \not \equiv 0, \qquad u_{1}\geq 0, \qquad u_{2}\leq 0, \\& \operatorname{supp} (u_{i})\cap \operatorname{supp} (u_{j})=\emptyset , \quad i\neq j , i,j=1,2,3 . \end{aligned}$$

Setting \(w=u_{1}+u_{2}\) with \(w^{+}=u_{1}\) and \(w^{-}=u_{2}\), i.e., \(w^{\pm }\neq 0\). Since \(J_{\lambda }^{\prime }(u_{\lambda })=0\), we get

$$\begin{aligned}& \bigl\langle J_{\lambda }^{\prime }(w),w^{+}\bigr\rangle =\bigl\langle J_{\lambda }^{ \prime }(u_{1}+u_{2}),u_{1} \bigr\rangle \leq \bigl\langle J_{\lambda }^{\prime }(u_{\lambda }),u_{1} \bigr\rangle =0, \\& \bigl\langle J_{\lambda }^{\prime }(w),w^{-}\bigr\rangle =\bigl\langle J_{\lambda }^{ \prime }(u_{1}+u_{2}),u_{2} \bigr\rangle \leq \bigl\langle J_{\lambda }^{\prime }(u_{\lambda }),u_{2} \bigr\rangle =0. \end{aligned}$$

By Lemma 2.5, there exists \((s_{w},t_{w})\in (0,1]\times (0,1]\) such that

$$ s_{w}w^{+}+t_{w}w^{-}=s_{w}u_{1}+t_{w}u_{2} \in \mathcal{M}_{\lambda },\quad m_{\lambda }\leq J_{\lambda }(s_{w}u_{1}+t_{w}u_{2}). $$

Note that \(\langle J_{\lambda }^{\prime }(u_{\lambda }),u_{\lambda }\rangle =0\) and \(\langle J_{\lambda }^{\prime }(s_{w}u_{1}+t_{w}u_{2}),s_{w}u_{1}+t_{w}u_{2} \rangle =0\), we have

$$\begin{aligned} m_{\lambda } =&J_{\lambda }(u_{\lambda })- \frac{1}{p+1} \bigl\langle J_{\lambda }^{\prime }(u_{\lambda }),u_{\lambda } \bigr\rangle \\ =& \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr)a \Vert u_{\lambda } \Vert ^{2}+ \biggl(\frac{1}{4}- \frac{1}{p+1} \biggr)\lambda \bigl( \Vert u_{\lambda } \Vert ^{2}\bigr)^{2} \\ &{}- \biggl(\frac{1}{q+1}- \frac{1}{p+1} \biggr)\kappa \int _{\mathbb{R}^{3}}G(x) \vert u_{\lambda } \vert ^{q+1} \\ >& \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr)a \bigl( \Vert u_{1} \Vert ^{2}+ \Vert u_{2} \Vert ^{2} \bigr) \\ &{}+ \biggl(\frac{1}{4}-\frac{1}{p+1} \biggr) \lambda \bigl( \Vert u_{1} \Vert ^{4}+2 \Vert u_{1} \Vert ^{2} \Vert u_{2} \Vert ^{2}+ \Vert u_{2} \Vert ^{4} \bigr) \\ &{} - \biggl(\frac{1}{q+1}-\frac{1}{p+1} \biggr)\kappa \int _{\mathbb{R}^{3}}G(x) \bigl( \vert u_{1} \vert ^{q+1}+ \vert u_{2} \vert ^{q+1} \bigr) \\ \geq &\biggl(\frac{1}{2}-\frac{1}{p+1} \biggr)a \bigl( \Vert s_{w}u_{1} \Vert ^{2}+ \Vert t_{w}u_{2} \Vert ^{2} \bigr) \\ &{}+ \biggl( \frac{1}{4}-\frac{1}{p+1} \biggr)\lambda \bigl( \Vert s_{w}u_{1} \Vert ^{4}+2 \Vert s_{w}u_{1} \Vert ^{2} \Vert t_{w}u_{2} \Vert ^{2}+ \Vert t_{w}u_{2} \Vert ^{4} \bigr) \\ &{}- \biggl(\frac{1}{q+1}-\frac{1}{p+1} \biggr)\kappa \int _{\mathbb{R}^{3}}G(x) \bigl( \vert s_{w}u_{1} \vert ^{q+1}+ \vert t_{w}u_{2} \vert ^{q+1} \bigr) \\ =& \biggl(\frac{1}{2}-\frac{1}{p+1} \biggr)a \Vert s_{w}u_{1}+t_{w}u_{2} \Vert ^{2}+ \biggl(\frac{1}{4}-\frac{1}{p+1} \biggr) \lambda \bigl( \Vert s_{w}u_{1}+t_{w}u_{2} \Vert ^{2} \bigr)^{2} \\ &{} - \biggl(\frac{1}{q+1}-\frac{1}{p+1} \biggr)\kappa \int _{\mathbb{R}^{3}}G(x) \vert s_{w}u_{1}+t_{w}u_{2} \vert ^{q+1} \\ =&J_{\lambda }(s_{w}u_{1}+t_{w}u_{2})- \frac{1}{p+1}\bigl\langle J_{ \lambda }^{\prime }(s_{w}u_{1}+t_{w}u_{2}),s_{w}u_{1}+t_{w}u_{2} \bigr\rangle \\ =&J_{\lambda }(s_{w}u_{1}+t_{w}u_{2}) \\ \geq& m_{\lambda }, \end{aligned}$$

which is a contradiction. □

Ground state solutions

Lemma 4.1

(Mountain pass theorem [18])

Let X be a Banach space, \(I \in C^{1}(X, \mathbb{R})\), \(e \in X\), and \(\rho > 0\) such that \(\|e\|> \rho \) and

$$ \inf_{\|u\|=\rho } I(u) > I(0) \geq I(e). $$

If I satisfies the \((PS)_{c}\) condition with

$$\begin{aligned}& c := \inf_{\gamma \in \Gamma }\max_{t\in [0,1]}I\bigl(\gamma (t)\bigr), \\& \Gamma := \bigl\{ \gamma \in C\bigl([0, 1], X\bigr) : \gamma (0) = 0, \gamma (1) = e \bigr\} , \end{aligned}$$

then c is a critical value of I.

Lemma 4.2

Under the assumptions of Theorem 1.2, there exist \(e\in H^{1}_{r}(\mathbb{R}^{3})\) and \(\rho > 0\) such that \(\|e\|>\rho \) and \(\inf_{\|u\|=\rho }J_{\lambda }(u)>J_{\lambda }(0)> J_{\lambda }(e)\).

Proof

For all \(u\in H^{1}_{r}(\mathbb{R}^{3})\), by Remark 2.2,

$$ \begin{aligned} J_{\lambda }(u)&=\frac{a}{2} \Vert u \Vert ^{2}+\frac{\lambda }{4} \Vert u \Vert ^{4}- \frac{1}{p+1} \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}- \frac{\kappa }{q+1} \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1} \\ &\geq \frac{a}{2} \Vert u \Vert ^{2}+ \frac{\lambda }{4} \Vert u \Vert ^{4}- \frac{C_{1}}{p+1} \Vert u \Vert ^{p}, \end{aligned} $$

then there exists \(\rho >0\) such that

$$ b:=\inf_{\|u\|=\rho }J_{\lambda }(u)>0=J_{\lambda }(0).$$

Let \(t\geq 0\), we have

$$ J_{\lambda }(tu)=\frac{t^{2}}{2}a \Vert u \Vert ^{2}+\frac{t^{4}}{4}\lambda \Vert u \Vert ^{4}- \frac{t^{p+1}}{p+1} \int _{\mathbb{R}^{3}}Q(x) \vert u \vert ^{p+1}- \frac{t^{q+1}}{q+1}\kappa \int _{\mathbb{R}^{3}}G(x) \vert u \vert ^{q+1}, $$

then there exists \(e:=tu\) such that \(\|e\|>\rho \) and \(J_{\lambda }(e)<0\). □

Lemma 4.3

Under the assumptions of Theorem 1.2. \(J_{\lambda }\) satisfies the \((P S)_{c}\) condition.

Proof

Let \(\{u_{n}\}\subset H_{r}^{1}(\mathbb{R}^{3})\) and \(J_{\lambda }(u_{n})\rightarrow c\), \(J_{\lambda }(u_{n})\rightarrow 0\) as \(n\rightarrow \infty \). By (2.15) in Lemma 2.6 above, it is easy to see that \(\{u_{n}\}\) is bounded in \(H_{r}^{1}(\mathbb{R}^{3})\). Going if necessary to a subsequence, \(u_{n}\rightharpoonup u\) in \(H^{1}_{r}(\mathbb{R}^{3})\), \(u_{n}\rightarrow u\) in \(L^{s}(\mathbb{R}^{3})\) for \(s\in (2,6)\), and \(u_{n}(x)\rightarrow u(x)\) a.e. on \(\mathbb{R}^{3}\), then by \((G_{1})\) we have

$$\begin{aligned}& \biggl\vert \int _{\mathbb{R}^{3}}G(x) \vert u_{n} \vert ^{q}(u_{n}-u) \biggr\vert \\& \quad \leq \int _{\mathbb{R}^{3}} \bigl\vert G(x) \bigr\vert \bigl| \vert u_{n} \vert ^{q}|u_{n}-u \vert \bigr\vert \\& \quad \leq \biggl( \int _{\mathbb{R}^{3}} \bigl\vert G(x) \bigr\vert ^{2} \biggr)^{ \frac{1}{2}} \biggl( \int _{\mathbb{R}^{3}} \vert u_{n} \vert ^{2q} \vert u_{n}-u \vert ^{2} \biggr)^{\frac{1}{2}} \\& \quad \leq \bigl\vert G(x) \bigr\vert _{2} \biggl( \int _{\mathbb{R}^{3}} \vert u_{n} \vert ^{2q+2} \biggr)^{\frac{q}{2q+2}} \biggl( \int _{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{2q+2} \biggr)^{\frac{1}{2q+2}} \\& \quad \leq C \bigl\vert G(x) \bigr\vert _{2} \Vert u_{n} \Vert ^{q} \vert u_{n}-u \vert _{2q+2}\rightarrow 0. \end{aligned}$$

Since

$$\begin{aligned}& \bigl\langle J^{\prime }_{\lambda }(u_{n})-J^{\prime }_{\lambda }(u),u_{n}-u \bigr\rangle \rightarrow 0, \\& \int _{\mathbb{R}^{3}}Q(x) \bigl(\vert u_{n} \vert ^{p}-\vert u \vert ^{p}\bigr)(u_{n}-u)\rightarrow 0 \end{aligned}$$

and

$$ \begin{aligned} &\bigl(a+\lambda \Vert u_{n} \Vert ^{2}\bigr) \Vert u_{n}-u \Vert ^{2} \\ &\quad = \bigl\langle J^{\prime }_{\lambda }(u_{n})-J^{\prime }_{\lambda }(u),u_{n}-u \bigr\rangle +\lambda \bigl( \Vert u \Vert ^{2}- \Vert u_{n} \Vert ^{2} \bigr) \langle u,u_{n}-u \rangle \\ &\quad\quad {}+ \int _{\mathbb{R}^{3}}Q(x) \bigl(\vert u_{n} \vert ^{p}-\vert u \vert ^{p}\bigr)(u_{n}-u)+ \int _{\mathbb{R}^{3}}G(x) \bigl(\vert u_{n} \vert ^{p}-\vert u \vert ^{p}\bigr)(u_{n}-u). \end{aligned} $$

Thus, \(u_{n}\rightarrow u\) in \(H^{1}_{r}(\mathbb{R}^{3})\). □

Set

$$ c_{1}=\inf_{u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}}\max_{t \geq 0}J_{\lambda }(tu). $$

Lemma 4.4

Under the assumptions of Theorem 1.2, we have \({c}=c_{\lambda }=c_{1}\).

Proof

Similar to the proof of Lemma 2.4, for all \(u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}\), there exists unique \(t_{u}u\in \mathcal{N}\) such that \(J_{\lambda }(t_{u}u)=\max_{t\geq 0}J_{\lambda }(tu)\), this implies that \(c_{\lambda }\leq c_{1}\).

For each \(\gamma \in \Gamma \), it follows from the property of \(\mathcal{N}\) that \(\gamma (t)\) crosses \(\mathcal{N}\) as t varying over \([0,1]\). Since \(\gamma (0)=0\), \(J_{\lambda }(\gamma (1))<0\), then

$$ \max_{t\in [0,1]}J_{\lambda } \bigl(\gamma (t)\bigr)\geq \inf_{u\in \mathcal{N}}J_{\lambda }(u)=c_{\lambda }. $$

Therefore \(c\geq c_{\lambda }\). On the other hand, for \(u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}\), we have that \(J_{\lambda }(tu)<0\) for t large enough, and then

$$ \max_{t\geq 0}J_{\lambda }(tu) \geq \max_{t\in [0,1]}J_{\lambda }(tu) \geq \inf _{\gamma \in \Gamma }\max_{t\in [0,1]}J_{\lambda }\bigl( \gamma (t)\bigr)=c. $$

Therefore \(c_{1}\geq c\). □

Proof of Theorem 1.2

According to Lemmas 4.1, 4.2, 4.3, and 4.4, we obtain that problem (1.1) has one least energy solution.

Now we prove \(m_{\lambda }>2c_{\lambda }\). By the proof of Theorem 1.1, there exists \(u_{\lambda }\in \mathcal{M}_{\lambda }\) such that \(J_{\lambda }(u_{\lambda })=m_{\lambda }\). By Lemmas 2.4 and 4.4, we have

$$ \begin{aligned} m_{\lambda }&=J_{\lambda }(u_{\lambda }) \\ &\geq J_{\lambda }\bigl(su_{\lambda }^{+}+tu_{\lambda }^{-} \bigr) \\ &= J_{\lambda }\bigl(su_{\lambda }^{+} \bigr)+J_{\lambda }\bigl(tu_{\lambda }^{-}\bigr)+ \frac{s^{2}t^{2}}{2}\lambda \bigl\Vert u_{\lambda }^{+} \bigr\Vert ^{2} \bigl\Vert u_{\lambda }^{-} \bigr\Vert ^{2} \\ &> J_{\lambda }\bigl(su_{\lambda }^{+} \bigr)+J_{\lambda }\bigl(tu_{\lambda }^{-}\bigr) \\ &\geq 2c_{\lambda }. \end{aligned} $$

 □

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Acknowledgements

The authors would like to thank the referees for their useful suggestions which have significantly improved the paper.

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This work was supported financially by the National Natural Science Foundation of China (11871302).

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Gao, Y., Liu, L., Luan, S. et al. A Kirchhoff-type problem involving concave-convex nonlinearities. Adv Differ Equ 2021, 171 (2021). https://doi.org/10.1186/s13662-021-03331-x

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MSC

  • 35J20
  • 35J65
  • 35A15
  • 35J60

Keywords

  • Concave-convex nonlinearities
  • Kirchhoff-type problem
  • Nehari manifolds
  • Ground state sign-changing solutions
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