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Theory and Modern Applications

A Kirchhoff-type problem involving concave-convex nonlinearities


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.

1 Introduction

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

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

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:


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


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

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

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

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]:

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

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

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

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)

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

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

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

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.

2 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

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


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

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

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

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

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

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

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


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

In addition, we define

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


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

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,

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


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,

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


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

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

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

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

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),

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

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

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


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

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 \(u^{\pm }\neq 0\).


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

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

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

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

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.


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

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

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 \(J_{\lambda }(u_{\lambda })=m_{\lambda }\). □

3 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 \(J^{\prime }_{\lambda }(u_{\lambda })=0\).


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

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

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

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


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

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. □

4 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 \(\inf_{\|u\|=\rho }J_{\lambda }(u)>J_{\lambda }(0)> J_{\lambda }(e)\).


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.


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

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


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


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

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


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

Lemma 4.4

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


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


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).

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