Skip to main content

Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

Abstract

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem:

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

where \(a,b>0\) are constants, and the potential \(V(x)\) is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.

Introduction and main result

This paper is concerned with the following Kirchhoff-type problem:

$$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(a,b>0\) are constants, and the potential \(V(x)\) is indefinite in sign, f satisfies some conditions which will be stated later.

In recent years, more and more attention has been devoted to study the following Kirchhoff-type problem:

$$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx )\Delta u+V(x)u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}),\end{cases}\displaystyle \end{aligned}$$
(1.2)

where \(V:\mathbb{R}^{N}\rightarrow \mathbb{R}\) and \(a,b>0\) are constants. (1.2) is a nonlocal problem as the appearance of the term \(\int _{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\Delta u\), which implies that (1.2) is not a pointwise identity. This causes some mathematical difficulties which make the study of (1.2) particularly interesting. Problem (1.2) appears in an interesting physical context. Indeed, if we consider the case \(V(x) = 0\) and replace \(\mathbb{R}^{N}\) with a bounded domain \(\Omega \subset \mathbb{R}^{N}\) in (1.2), then we get the following Dirichlet problem of Kirchhoff type:

$$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\Omega } \vert \nabla u \vert ^{2}\,dx )\Delta u=f(x,u) & \text{in }\Omega , \\ u=0 & \text{on }\Omega ,\end{cases}\displaystyle \end{aligned}$$
(1.3)

which is a nonlocal problem due to the presence of the nonlocal term \(b\int _{\Omega }|\nabla u|^{2}\,dx\Delta u\) and is related to the stationary analogue of the equation

$$\begin{aligned} u_{tt}- \biggl(a+b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u=f(x,u). \end{aligned}$$
(1.4)

(1.4) was first proposed by Kirchhoff in [12] as a generalization of the classical D’Alembert wave equations, particularly taking into account the subsequent change in string length caused by oscillations. The readers can learn some early classical research of Kirchhoff equations from [4, 22]. For the results concerning the existence of sign-changing solutions for (1.3), we refer the reader to papers [20, 24, 34], which depend heavily on the nonlinear term with 4-superlinear growth at infinity in the sense that

$$\begin{aligned} \lim_{|t|\rightarrow \infty }\frac{F(x,t)}{t^{4}}=+\infty,\quad x \in \Omega, \end{aligned}$$

where \(F(x,t)=\int ^{t}_{0}f(x,s)\,ds\). And [19, 32] deal with the fact that the nonlinearity \(f(x,u)\) may not be 4-superlinear at infinity.

Motivated by the strong physical background, equations (1.2) and (1.3) have been extensively studied in recent years under variant assumptions on V and f. There are many papers involving the existence of nontrivial solutions of equation (1.2). In [21], Perera and Zhang obtained a nontrivial solution of (1.2) via Yang index and critical group. By using the local minimum methods and the fountain theorems, He and Zou [9] obtained infinitely many solutions. Later, Jin and Wu [11] proved the existence of infinitely many radial solutions by applying a fountain theorem. Using the Nehari manifold and fibering map methods, equation (1.2) was studied with concave and convex nonlinearities, the existence of multiple positive solutions was obtained by Chen et al. [6]. Moreover, the existence of infinitely many solutions to equation (1.2) has been derived by a variant version of fountain theorem in [18]. Subsequently, in [13] Li and Ye, using a monotone method and a global compactness lemma, showed the existence of a positive ground state solution for the corresponding limiting problem of equation (1.2). After that, Guo [8] generalized the result in [13] to a general nonlinearity. In [26] Tang and Cheng proposed a new approach to recover compactness for the \((PS)\)-sequence, and they proved that equation (1.3) possesses one ground state sign-changing solution, and its energy is strictly larger than twice that of the ground state solutions of Nehari type. In [25] Tang and Chen proved that equation (1.2) admits a ground state solution of Nehari–Pohozaev type and a least energy solution under some mild assumptions on V and f by using a new approach to recover compactness for the minimizing sequence.

Recently, Xiang et al. [31] considered the existence and multiplicity of solutions for a class of Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. By using the concentration compactness principle in fractional Sobolev spaces, they obtained m pairs of solutions, by using Krasnoselskii’s genus theory, the existence of infinitely many solutions were obtained. Later, Xiang et al. [30] developed the fractional Trudinger–Moser inequality in the singular case and used it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff-type problems with singular exponential nonlinearity. For further important and interesting results, one can refer to [3, 10, 14, 27, 29] and the references therein.

In all the above-mentioned studies, we notice that the potential \(V(x)\) was assumed to be equipped with some “compact” condition or positive definite. But in this paper the potential \(V(x)\) is indefinite in sign, the methods and arguments for the cases \(V(x)\geq 0\) are not applicable to the indefinite cases. So, this article is a complement to the indefinite Kirchhoff problems in the literature. Our main aim is to study the existence of nontrivial solutions for problem (1.1) by means of Morse theory and local linking, which are different from the literature mentioned above. Before stating our main results, we need to describe the eigenvalue of Schrödinger operator \(-a\Delta +V\). Consider the following increasing sequence \(\lambda _{1}\leq \lambda _{2}\leq \cdot \cdot \cdot \) of minimax values defined by

$$\begin{aligned} \lambda _{n}:=\inf_{S\in \Gamma _{n}}\sup_{u\in S\backslash \{ 0 \} } \frac{\int _{\mathbb{R}^{3}} (a \vert \nabla u \vert ^{2}+V(x)u^{2} )\,dx}{\int _{\mathbb{R}^{3}}u^{2}\,dx}, \end{aligned}$$

where \(\Gamma _{n}\) denotes the family of n-dimensional subspaces of \(C_{0}^{\infty }(\mathbb{R}^{3})\), remember \(a\neq 0\). Let

$$\begin{aligned} \lambda _{\infty }:=\lim_{n\rightarrow \infty }\lambda _{n}, \end{aligned}$$

then \(\lambda _{\infty }\) is the bottom of the essential spectrum of \(-a\Delta +V\) if it is finite, and for every \(n\in \mathbb{N}\), the inequality \(\lambda _{n}<\lambda _{\infty }\) implies that \(\lambda _{n}\) is an eigenvalue of \(-a\Delta +V\) of finite multiplicity (see [23, Chapt. XIII] for details). Note that if V is bounded from below, then \(\lambda _{n}\) is well defined and is finite.

Set \(F(x,u):=\int ^{u}_{0}f(x,t)\,dt\). We assume that V and f satisfy the following conditions:

\(( V ) \):

\(V(x)\in C(\mathbb{R}^{3},\mathbb{R})\) bounded and there exists an integer \(k\geq 1\) such that \(\lambda _{k}<0<\lambda _{k+1}\).

\(( f_{1} ) \):

\(f\in C({\mathbb{R}^{3}}\times {\mathbb{R}},{\mathbb{R}})\), and there exist \(C>0\) and \(p\in (2,6)\) such that

$$\begin{aligned} \bigl\vert f(x,u) \bigr\vert \leq C \bigl(1+ \vert u \vert ^{p-1} \bigr)\quad \text{for all }(x,u)\in \mathbb{R}^{3} \times \mathbb{R}. \end{aligned}$$
\(( f_{2} ) \):

\(f(x,u)=o(u)\) as \(u\rightarrow 0\) uniformly in \(x\in \mathbb{R}^{3}\).

\(( f_{3} ) \):

There exists \(\mu >4\) such that \(0<\mu F(x,u)\leq f(x,u)u\) for all \((x,u)\in \mathbb{R}^{3}\times \mathbb{R}\) and \(u\neq 0\).

\(( f_{4} ) \):

For any \(r>0\), we have

$$\begin{aligned} \lim_{ \vert x \vert \rightarrow \infty }\sup_{0< \vert u \vert \leq r} \biggl\vert \frac{f(x,u)}{u} \biggr\vert =0. \end{aligned}$$

It is easy to see that condition \((f_{3})\) implies that

$$\begin{aligned} \lim_{|u|\rightarrow \infty }\frac{F(x,u)}{u^{4}}=+\infty. \end{aligned}$$

Concerning the existence of solutions for problem (1.1), we have the following result.

Theorem 1.1

Suppose that \((V)\) and \((f_{1})\)\((f_{4})\) hold. Then problem (1.1) has at least one nontrivial solution.

Remark 1.2

We should also mention two recent papers [15, 33] related to problem (1.1). In these two papers, the variational functional is coercive and bounded from below. When the nonlinearity is odd, infinitely many nontrivial solutions of (1.1) were obtained in both [15] and [33] by using critical point theory of even functional; while if the nonlinearity is not odd, two nontrivial solutions were obtained in [15] via Morse theory. In the current paper, the variational functional is neither bounded from above nor bounded from below, this is quite different from the situation in [15, 33].

Remark 1.3

To deal with problem (1.1), one encounters various difficulties. On the one hand, the Sobolev embedding \(H^{1}(\mathbb{R}^{3})\hookrightarrow L^{2}(\mathbb{R}^{3})\) is not compact. To overcome this, one can restrict the energy functional Φ to a subspace of \(H^{1}(\mathbb{R}^{3})\), which embeds compactly into \(L^{2}(\mathbb{R}^{3})\) with certain qualifications or consists of radially symmetric functions. In [7], Chen and Liu considered the standing waves of (1.1) with the nonlinearity f is 4-superlinear and the potential V satisfying assumption \((V)\) and

$$\begin{aligned} \mu \bigl(V^{-1}(-\infty,M] \bigr)< \infty \end{aligned}$$
(1.5)

for all \(M>0\), where μ denotes the Lebesgue measure in \(\mathbb{R}^{N}\), then the working space

$$\begin{aligned} X:= \biggl\{ u\in H^{1} \bigl(\mathbb{R}^{N} \bigr)\bigg| \int _{\mathbb{R}^{N}}V(x)u^{2}\,dx< \infty \biggr\} \end{aligned}$$

embeds compactly into \(L^{2}(\mathbb{R}^{N})\), which is crucial in verifying the Palais–Smale condition. However, our assumptions on V are much weaker.

On the other hand, under our assumptions the potential \(V(x)\) is indefinite in sign, then the quadratic part of the functional Φ (defined in (2.2)) possesses a nontrivial negative space, and the functional Φ does not satisfy the linking geometry any more, so that the linking theorem is not applicable. We will use the idea of local linking to overcome the difficulty. To our knowledge, there are a few results on this case.

The remainder of this paper is organized as follows. In Sect. 2, we give the variational framework for problem (1.1) and prove that Φ satisfies the \((PS)\) condition. In Sect. 3, we recall some concepts and results in infinite-dimensional Morse theory [5] and give the proof of Theorem 1.1.

Variational setting and Palais–Smale condition

In this section, we give the variational setting for problem (1.1) and establish the compactness conditions. By \(|\cdot |_{s} \) as follows, we denote the usual \(L^{s}\)-norm for \(s\geq 2\), and \(C,C_{i}\) stand for different positive constants.

Let

$$\begin{aligned} H^{1} \bigl(\mathbb{R}^{3} \bigr):= \bigl\{ u\in L^{2} \bigl(\mathbb{R}^{3} \bigr)|\nabla u \in L^{2} \bigl(\mathbb{R}^{3} \bigr) \bigr\} , \end{aligned}$$

with the usual norm

$$\begin{aligned} \Vert u \Vert _{H}= \biggl( \int _{\mathbb{R}^{3}} \bigl( \vert \nabla u \vert ^{2}+u^{2} \bigr)\,dx \biggr)^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} E:= \biggl\{ u\in H^{1} \bigl(\mathbb{R}^{3} \bigr)\bigg| \int _{\mathbb{R}^{3}}V(x)u^{2}\,dx< \infty \biggr\} , \end{aligned}$$

be a linear subspace of \(H^{1}(\mathbb{R}^{3})\). Let \(E^{-}\) be the space spanned by the eigenfunctions with respect to \(\lambda _{1},\ldots,\lambda _{k}\) and \(E^{+}=(E^{-})^{\perp }\). From \((V)\), we deduce that \(E=E^{-}\oplus E^{+}\), where \(E^{-}\), \(E^{+}\) are the negative eigenspace and the positive eigenspace of the operator \(-a\Delta +V\). Moreover, \(k\leq \operatorname{dim}E^{-}<\infty \).

For any \(u,v\in E\), we define

$$\begin{aligned} (u,v):= \int _{\mathbb{R}^{3}} \bigl( a\nabla u^{+}\nabla v^{+}+V(x)u^{+}v^{+} \bigr)\,dx- \int _{\mathbb{R}^{3}} \bigl( a\nabla u^{-}\nabla v^{-}+V(x)u^{-}v^{-} \bigr)\,dx, \end{aligned}$$

where \(u=u^{-}+u^{+}\), \(v=v^{-}+v^{+}\), \(u^{+},v^{+}\in E^{+}\), \(u^{-},v^{-}\in E^{-}\). Then \((\cdot,\cdot )\) is an inner product in E. Hence, E is a Hilbert space with the norm \(\|u\|=(u,u)^{\frac{1}{2}}\), which is an equivalent norm on \(H^{1}(\mathbb{R}^{3})\). It is easy to see that

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

For any \(s\in [2,6]\), since the embedding \(E\hookrightarrow L^{s}(\mathbb{R}^{3})\) is continuous, then there exists a constant \(d_{s}>0\) such that

$$\begin{aligned} \vert u \vert _{s}\leq d_{s} \Vert u \Vert \quad \text{for all }u\in E. \end{aligned}$$
(2.1)

It is easy to see that, from \((f_{1})\) and \((f_{2})\), the functional \(\Phi: E\rightarrow \mathbb{R}\),

$$\begin{aligned} \Phi (u)=\frac{1}{2} \int _{\mathbb{R}^{3}}a \vert \nabla u \vert ^{2}\,dx+ \frac{1}{2} \int _{\mathbb{R}^{3}} V(x)u^{2}\,dx+\frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}- \int _{\mathbb{R}^{3}}F(x,u)\,dx, \end{aligned}$$
(2.2)

is of class \(C^{1}\) with derivative

$$\begin{aligned} \bigl\langle \Phi '(u),v \bigr\rangle ={}& \int _{\mathbb{R}^{3}} \bigl(a\nabla u \cdot \nabla v+V(x)uv \bigr)\,dx+b \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \int _{\mathbb{R}^{3}}\nabla u\cdot \nabla v\,dx \\ &{}- \int _{\mathbb{R}^{3}}f(x,u)v\,dx \end{aligned}$$
(2.3)

for all \(u,v\in E\). It can be proved that a weak solution of problem (1.1) is a critical point of the functional Φ.

We say that a functional \(I\in C^{1}(E,\mathbb{R})\) satisfies the \((PS)\) condition if any sequence \(\{u_{n}\}\subset E\) such that

$$\begin{aligned} \sup_{n} \bigl\vert I(u_{n}) \bigr\vert < \infty\quad \text{and} \quad I'(u_{n})\rightarrow 0\quad \text{in }E^{-1} \end{aligned}$$

(called a \((PS)\) sequence) has a convergent subsequence.

Lemma 2.1

Under assumptions \((V)\), \((f_{3})\), every \((PS)\) sequence of functional Φ is bounded in E.

Proof

Let \(\{u_{n}\}\) be a \((PS)\) sequence of functional Φ, that is,

$$\begin{aligned} \sup_{n} \bigl\vert \Phi (u_{n}) \bigr\vert < \infty\quad \text{and} \quad\Phi '(u_{n})\rightarrow 0 \quad \text{in }E^{-1}. \end{aligned}$$

If the conclusion is not true, we may assume \(\|u_{n}\|\rightarrow \infty \), as \(n\rightarrow \infty \). Let \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\|v_{n}\|=1\) up to a subsequence

$$\begin{aligned} v_{n}=v_{n}^{+}+v_{n}^{-} \rightharpoonup v=v^{+}+v^{-}\in E,\quad v_{n}^{ \pm },v^{\pm } \in E^{\pm }. \end{aligned}$$

If \(v=0\), then \(v_{n}^{-}\rightarrow v^{-}=0\) due to \(\operatorname{dim}E^{-}<\infty \). Since

$$\begin{aligned} \Vert v_{n} \Vert ^{2}= \bigl\Vert v_{n}^{-} \bigr\Vert ^{2}+ \bigl\Vert v_{n}^{+} \bigr\Vert ^{2}=1, \end{aligned}$$

for n large enough, we obtain

$$\begin{aligned} \bigl\Vert v_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert v_{n}^{-} \bigr\Vert ^{2}\geq \frac{1}{2}. \end{aligned}$$

Now, using \((f_{3})\), for n large enough, we deduce

$$\begin{aligned} & 1+\sup_{n} \bigl\vert \Phi (u_{n}) \bigr\vert + \Vert u_{n} \Vert \\ &\quad \geq \Phi (u_{n})- \frac{1}{\mu } \bigl\langle \Phi '(u_{n}),u_{n} \bigr\rangle \\ &\quad = \biggl(\frac{1}{2}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u_{n} \vert ^{2}+V(x)u_{n}^{2} \bigr)\,dx+ \biggl( \frac{1}{4}- \frac{1}{\mu } \biggr)b \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \biggr)^{2} \\ &\qquad{} + \int _{\mathbb{R}^{3}} \biggl(\frac{1}{\mu }f(x,u_{n})u_{n}-F(x,u_{n}) \biggr)\,dx \\ &\quad \geq \biggl(\frac{1}{2}-\frac{1}{\mu } \biggr) \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u_{n} \vert ^{2}+V(x)u_{n}^{2} \bigr)\,dx \\ &\quad = \biggl(\frac{1}{2}-\frac{1}{\mu } \biggr) \Vert u_{n} \Vert ^{2} \bigl( \bigl\Vert v_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert v_{n}^{-} \bigr\Vert ^{2} \bigr) \\ &\quad \geq \biggl(\frac{1}{4}-\frac{1}{2\mu } \biggr) \Vert u_{n} \Vert ^{2} , \end{aligned}$$
(2.4)

contradicting \(\|u_{n}\|\rightarrow \infty \), thus \(v\neq 0\).

It follows from (2.4) that

$$\begin{aligned} 0\leftarrow \frac{1+\sup_{n} \vert \Phi (u_{n}) \vert + \Vert u_{n} \Vert }{ \Vert u_{n} \Vert ^{4}}\geq \biggl(\frac{1}{2}-\frac{1}{\mu } \biggr) \biggl( \frac{ \Vert u_{n}^{+} \Vert ^{2}}{ \Vert u_{n} \Vert ^{4}}- \frac{ \Vert u_{n}^{-} \Vert ^{2}}{ \Vert u_{n} \Vert ^{4}} \biggr) + \biggl(\frac{1}{4}- \frac{1}{\mu } \biggr) b\frac{ \vert \nabla u_{n} \vert ^{4}_{2}}{ \Vert u_{n} \Vert ^{4}}, \end{aligned}$$

which implies that

$$\begin{aligned} \lim_{n\rightarrow \infty } \frac{ \vert \nabla u_{n} \vert ^{4}_{2}}{ \Vert u_{n} \Vert ^{4}}=0. \end{aligned}$$

Define \(\Upsilon (u):=|\nabla u|_{2}\), it is easy to see that ϒ is continuous and convex, hence it is weak lower semi-continuous. Consequently,

$$\begin{aligned} \liminf_{n\rightarrow \infty } \vert \nabla v_{n} \vert _{2}\geq \vert \nabla v \vert _{2}, \end{aligned}$$

and then

$$\begin{aligned} 0=\liminf_{n\rightarrow \infty } \frac{ \vert \nabla u_{n} \vert ^{4}_{2}}{ \Vert u_{n} \Vert ^{4}}=\liminf _{n\rightarrow \infty } \vert \nabla v_{n} \vert ^{4}_{2}\geq \vert \nabla v \vert ^{4}_{2}>0. \end{aligned}$$

This is a contradiction, thus the proof is completed. □

Lemma 2.2

Under the assumptions of Theorem 1.1, Φ satisfies the \((PS)\) condition.

Proof

Assume that \(\{u_{n}\}\) is a \((PS)\) sequence of Φ. It follows from Lemma 2.1 that \(\{u_{n}\}\) is bounded in E, then up to a subsequence

$$\begin{aligned} u_{n}\rightharpoonup u,\quad \text{in }E,\qquad u_{n} \rightarrow u\quad \text{in }L^{s}_{\mathrm{loc}} \bigl( \mathbb{R}^{3} \bigr), 2\leq s< 6,\qquad u_{n}(x)\rightarrow u(x)\quad \text{a.e. in }\mathbb{R}^{3}. \end{aligned}$$

We have

$$\begin{aligned} \int _{\mathbb{R}^{3}} \bigl(a\nabla u_{n}\cdot \nabla u+V(x)u_{n}u \bigr)\,dx\rightarrow \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u_{n} \vert ^{2}+V(x)u^{2} \bigr)\,dx= \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2}. \end{aligned}$$

Consequently,

$$\begin{aligned} o(1)={}& \bigl\langle \Phi '(u_{n}),u_{n}-u \bigr\rangle \\ ={}& \int _{\mathbb{R}^{3}} \bigl(a\nabla u_{n}\cdot \nabla (u_{n}-u)+V(x)u_{n}(u_{n}-u) \bigr)\,dx \\ &{} +b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx- \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx \\ ={}& \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u_{n} \vert ^{2}+V(x)u_{n}^{2} \bigr)\,dx- \int _{ \mathbb{R}^{3}} \bigl(a\nabla u_{n}\cdot \nabla u+V(x)u_{n}u \bigr)\,dx \\ &{} +b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx- \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx \\ ={}& \bigl( \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert u_{n}^{-} \bigr\Vert ^{2} \bigr)- \bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2} \bigr) \\ &{} +b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx- \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx+o(1) \\ ={}& \bigl( \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert u^{+} \bigr\Vert ^{2} \bigr)- \bigl( \bigl\Vert u^{-}_{n} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2} \bigr) \\ &{} +b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx- \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx+o(1) . \end{aligned}$$
(2.5)

Since \(\operatorname{dim}E^{-}<\infty \), we have \(u_{n}^{-}\rightarrow u^{-}\), thus \(\|u_{n}^{-}\|\rightarrow \|u^{-}\|\). Collecting all infinitesimal terms, we obtain

$$\begin{aligned} \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert u^{+} \bigr\Vert ^{2}={}& \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx \\ &{}-b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx+o(1). \end{aligned}$$
(2.6)

With the assumption \((f_{4})\), it has been shown in [2, p.29] that

$$\begin{aligned} \overline{\lim } \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx\leq 0. \end{aligned}$$

Because \(\Upsilon (u)=|\nabla u|_{2}\) is weak lower semi-continuous, we have

$$\begin{aligned} \underline{\lim } \int _{\mathbb{R}^{3}}\nabla u_{n}\cdot \nabla (u_{n}-u)\,dx & =\underline{\lim } \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx - \int _{\mathbb{R}^{3}}\nabla u_{n}\cdot \nabla u\,dx \biggr) \\ & \geq \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx- \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \\ & =0. \end{aligned}$$

Now, from (2.6), we have

$$\begin{aligned} &\overline{\lim } \bigl( \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}- \bigl\Vert u^{+} \bigr\Vert ^{2} \bigr)\\ &\quad = \overline{\lim } \biggl( \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx-b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{\mathbb{R}^{3}} \nabla u_{n}\cdot \nabla (u_{n}-u)\,dx \biggr) \\ &\quad \leq \overline{\lim } \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx- \underline{\lim }b \int _{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx \int _{ \mathbb{R}^{3}}\nabla u_{n}\cdot \nabla (u_{n}-u)\,dx \\ & \quad\leq \overline{\lim } \int _{\mathbb{R}^{3}}f(x,u_{n}) (u_{n}-u)\,dx \\ & \quad\leq 0, \end{aligned}$$

which implies \(\overline{\lim }\|u_{n}^{+}\|^{2}\leq \|u^{+}\|^{2}\). From the weak lower semi-continuity of the norm, we have

$$\begin{aligned} \bigl\Vert u^{+} \bigr\Vert ^{2}\leq \underline{\lim } \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}\leq \overline{\lim } \bigl\Vert u_{n}^{+} \bigr\Vert ^{2}\leq \bigl\Vert u^{+} \bigr\Vert ^{2}, \end{aligned}$$

that is, \(\|u_{n}^{+}\|^{2}\rightarrow \|u^{+}\|^{2}\), combining \(\|u_{n}^{-}\|^{2}\rightarrow \|u^{-}\|^{2}\), we get \(\|u_{n}\|\rightarrow \|u\|\). Thus \(u_{n}\rightarrow u\) in E. The proof is completed. □

Critical groups and the proof of Theorem 1.1

In Sect. 2, we have established the \((PS)\) condition for Φ. Now, we recall some concepts and results in infinite-dimensional Morse theory [5], then analyze the critical groups of Φ at infinity, and give the proof of Theorem 1.1.

Let X be a Banach space, \(\varphi:X\rightarrow \mathbb{R}\) be a \(C^{1}\)-functional, u be an isolated critical point of φ and \(\varphi (u)=c\). Then

$$\begin{aligned} C_{q}(\varphi,u):=H_{q} \bigl(\varphi _{c}, \varphi _{c}\backslash {\{u\}} \bigr),\quad q=0,1,2, \ldots, \end{aligned}$$

is called the qth critical group of φ at u, where \(\varphi _{c}:=\varphi ^{-1}(-\infty,c]\) and \(H_{q}(\cdot,\cdot )\) stands for the qth singular relative homology group with integer coefficients.

If φ satisfies the \((PS)\) condition and the critical values of φ are bounded from below by α, then following Bartsch and Li [1], we call

$$\begin{aligned} C_{q}(\varphi,\infty ):=H_{q}(X,\varphi _{\alpha }), \quad q=0,1,2,\ldots, \end{aligned}$$

the qth critical group of φ at ∞. It is well known that the homology on the right does not depend on the choice of α.

Proposition 3.1

([1])

If \(\varphi \in C^{1}(X,\mathbb{R})\) satisfies the \((PS)\) condition and \(C_{l}(\varphi,0)\neq C_{l}(\varphi,\infty )\) for some \(l\in \mathbb{N}\), then φ has a nonzero critical point.

Proposition 3.2

([16])

Suppose \(\varphi \in C^{1}(X,\mathbb{R})\) has a local linking at 0, that is, \(X=Y\oplus Z\) and

$$\begin{aligned} &\varphi (u)\leq 0 \quad \textit{for }u\in Y\cap B_{\rho }, \\ &\varphi (u)>0 \quad \textit{for }u\in \bigl(Z\backslash {\{0\}} \bigr)\cap B_{\rho }, \end{aligned}$$

for some \(\rho >0\), where \(B_{\rho }:=\{u\in X\mid \|u\|\leq \rho \}\). If \(l=\operatorname{dim}Y<\infty \), then \(C_{l}(\varphi,0)\neq 0\).

To prove Theorem 1.1 with Proposition 3.1, we need the following lemma to investigate the critical groups of Φ at infinity.

Lemma 3.3

Under the assumptions of Theorem 1.1, there exists \(A>0\) such that, if \(\Phi (u)\leq -A\), then

$$\begin{aligned} \bigl\langle \Phi '(u),u \bigr\rangle =\frac{d}{dt}\bigg|_{t=1} \Phi (tu)< 0. \end{aligned}$$

Proof

We argue by contradiction. Assume that there exists a sequence \(\{u_{n}\}\subset E\) such that \(\Phi (u_{n})\leq -n\), but

$$\begin{aligned} \bigl\langle \Phi '(u_{n}),u_{n} \bigr\rangle =\frac{d}{dt}\bigg|_{t=1}\Phi (tu_{n}) \geq 0. \end{aligned}$$
(3.1)

By \((f_{3})\), we have

$$\begin{aligned} \bigl\Vert u^{+}_{n} \bigr\Vert ^{2}- \bigl\Vert u^{-}_{n} \bigr\Vert ^{2} & \leq \bigl( \bigl\Vert u^{+}_{n} \bigr\Vert ^{2}- \bigl\Vert u^{-}_{n} \bigr\Vert ^{2} \bigr)+ \int _{\mathbb{R}} \bigl[f(x,u_{n})u_{n}-4F(x,u_{n}) \bigr]\,dx \\ & =4\Phi (u_{n})- \bigl\langle \Phi '(u_{n}),u_{n} \bigr\rangle \\ & \leq -4n. \end{aligned}$$
(3.2)

This implies \(\|u_{n}\|\rightarrow \infty \). Let \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\), and \(v^{\pm }_{n}\) be the orthogonal projection of \(v_{n}\) on \(E^{\pm }\). Then \(v^{-}_{n}\rightarrow v^{-}\) for some \(v^{-}\in E^{-}\) since \(\operatorname{dim}E^{-}<\infty \).

If \(v^{-}\neq 0\), then for some \(v\in E\backslash \{0\}\), we have \(v_{n}\rightharpoonup v\) in E, and the set \(\Theta =\{x\in \mathbb{R}^{3}:v\neq 0\}\) has positive Lebesgue measure. For \(x\in \Theta \), we have \(|u_{n}(x)|\rightarrow \infty \), from \((f_{3})\), that

$$\begin{aligned} \frac{F(x,u_{n}(x))}{u_{n}^{4}(x)}v_{n}^{4}(x)\rightarrow +\infty. \end{aligned}$$

Then, by Fatou’s lemma and \((f_{3})\), we have

$$\begin{aligned} \int _{\mathbb{R}^{3}}\frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert ^{4}}\,dx & \geq \int _{\mathbb{R}^{3}}\frac{\mu F(x,u_{n})}{ \Vert u_{n} \Vert ^{4}}\,dx \\ & \geq \mu \int _{v\neq 0}\frac{F(x,u_{n})}{u_{n}^{4}(x)}v_{n}^{4}(x) \\ & \rightarrow +\infty . \end{aligned}$$

Hence, using (3.1), we have

$$\begin{aligned} 0 & \leq \frac{\langle \Phi '(u_{n}),u_{n}\rangle }{ \Vert u_{n} \Vert ^{4}} \\ & = \frac{ \Vert u^{+}_{n} \Vert ^{2}- \Vert u^{-}_{n} \Vert ^{2} }{ \Vert u_{n} \Vert ^{4}}+b \frac{ \vert \nabla u_{n} \vert _{2}^{4}}{ \Vert u_{n} \Vert ^{4}}- \int _{\mathbb{R}^{3}} \frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert ^{4}}\,dx \\ & \leq \frac{1}{ \Vert u_{n} \Vert ^{2}}+C_{1}b- \int _{\mathbb{R}^{3}} \frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert ^{4}}\,dx \\ & \leq 1+C_{1}b- \int _{\mathbb{R}^{3}} \frac{f(x,u_{n})u_{n}}{ \Vert u_{n} \Vert ^{4}}\,dx \\ & \rightarrow -\infty , \end{aligned}$$

a contradiction. Therefore \(v^{-}=0\), from

$$\begin{aligned} \Vert v_{n} \Vert ^{2}= \bigl\Vert v^{+}_{n} \bigr\Vert ^{2}+ \bigl\Vert v^{-}_{n} \bigr\Vert ^{2}=1, \end{aligned}$$

we see that \(\|v^{+}_{n}\|\rightarrow 1\). Consequently, for n large enough, we have

$$\begin{aligned} \bigl\Vert u^{+}_{n} \bigr\Vert = \Vert u_{n} \Vert \bigl\Vert v^{+}_{n} \bigr\Vert \geq \Vert u_{n} \Vert \bigl\Vert v^{-}_{n} \bigr\Vert = \bigl\Vert u^{-}_{n} \bigr\Vert , \end{aligned}$$

a contradiction to (3.2). Thus the conclusion of the lemma is true. □

Lemma 3.4

Under the assumptions of Theorem 1.1, \(C_{i}(\Phi,\infty )\cong 0\) for all \(i\in \mathbb{N}\).

Proof

Let \(B=\{v\in E| \|v\|\leq 1\}\) be the unit ball in E, \(S=\partial B\) be the unit sphere. Let \(A>0\) be the number given in Lemma 3.3. Without loss of generality, we may assume that

$$\begin{aligned} \inf_{\|u\|\leq 2}\Phi (u)>-A. \end{aligned}$$

Then, for \(v\in S\) and \((f_{3})\), we deduce that

$$\begin{aligned} \Phi (sv) & =\frac{s^{2}}{2} \bigl( \bigl\Vert v^{+} \bigr\Vert ^{2}- \bigl\Vert v^{-} \bigr\Vert ^{2} \bigr)+ \frac{bs^{4}}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla v \vert ^{2}\,dx \biggr)^{2}- \int _{\mathbb{R}^{3}}F(x,sv)\,dx \\ & =s^{4} \biggl\{ \frac{ \Vert v^{+} \Vert ^{2}- \Vert v^{-} \Vert ^{2}}{2s^{2}}+ \frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla v \vert ^{2}\,dx \biggr)^{2}- \int _{\mathbb{R}^{3}}\frac{F(x,sv)}{s^{4}}\,dx \biggr\} \\ & \rightarrow -\infty,\quad \text{as }s\rightarrow +\infty. \end{aligned}$$

So there is \(s_{v}>0\) such that \(\Phi (s_{v}v)=-A\).

Set \(w=s_{v}v\), a direct computation and Lemma 3.3 give

$$\begin{aligned} \bigl\langle \Phi '(s_{v}v),v \bigr\rangle = \frac{d}{ds}\bigg|_{s=s_{v}}\Phi (sv)= \frac{1}{s_{v}} \frac{d}{dt}\bigg|_{t=1}\Phi (tw)< 0. \end{aligned}$$

By the implicit function theorem, \(T:v\rightarrow s_{v}\) is a continuous function on S. Using the function T and a standard argument (see, e.g., [17, 28]), we can construct a deformation from \(X\backslash B\) to \(\Phi _{-A}=\Phi ^{-1}(-\infty,-A]\), and deduce via the homotopic invariance of singular homology

$$\begin{aligned} C_{i}(\Phi,\infty )=H_{i}(X,\Phi _{-A})\cong H_{i}(X,X\backslash B)=0,\quad \text{for all }i\in \mathbb{N}. \end{aligned}$$

The proof is completed. □

Lemma 3.5

Under assumptions \((V)\), \((f_{1})\), and \((f_{2})\), the functional Φ has a local linking at 0 with respect to \(E=E^{-}\oplus E^{+}\).

Proof

By \((f_{1})\) and \((f_{2})\), there exists a constant \(C>0\) such that

$$\begin{aligned} \bigl\vert F(x,u) \bigr\vert \leq \varepsilon u^{2}+Cu^{p}. \end{aligned}$$

If \(u\in E^{-}\), by (2.1) and the equivalence of norms on finite dimensional space \(E^{-}\), we have

$$\begin{aligned} \Phi (u) & =\frac{1}{2} \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr)\,dx+\frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}- \int _{\mathbb{R}^{3}}F(x,u)\,dx \\ & =-\frac{1}{2} \Vert u \Vert ^{2}+\frac{b}{4} \vert \nabla u \vert ^{4}_{2}- \int _{ \mathbb{R}^{3}}F(x,u)\,dx \\ & \leq -\frac{1}{2} \Vert u \Vert ^{2}+C_{2} \frac{b}{4} \Vert u \Vert ^{4}+\varepsilon \vert u \vert ^{2}_{2}+C \vert u \vert ^{p}_{p} \\ & \leq \biggl(-\frac{1}{2}+\varepsilon d_{2}^{2} \biggr) \Vert u \Vert ^{2}+C_{2} \frac{b}{4} \Vert u \Vert ^{4}+Cd^{p}_{p} \Vert u \Vert ^{p}. \end{aligned}$$

If \(u\in E^{+}\), we have

$$\begin{aligned} \Phi (u) & =\frac{1}{2} \int _{\mathbb{R}^{3}} \bigl(a \vert \nabla u \vert ^{2}+V(x)u^{2} \bigr)\,dx+\frac{b}{4} \biggl( \int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}- \int _{\mathbb{R}^{3}}F(x,u)\,dx \\ & =\frac{1}{2} \Vert u \Vert ^{2}+\frac{b}{4} \vert \nabla u \vert ^{4}_{2}- \int _{ \mathbb{R}^{3}}F(x,u)\,dx \\ & \geq \frac{1}{2} \Vert u \Vert ^{2}-\varepsilon \vert u \vert ^{2}_{2}-C \vert u \vert ^{p}_{p} \\ & \geq \biggl(\frac{1}{2}-\varepsilon d_{2}^{2} \biggr) \Vert u \Vert ^{2}-Cd^{p}_{p} \Vert u \Vert ^{p}. \end{aligned}$$

So, there exists \(0<\rho <1\) small enough such that

$$\begin{aligned} & \Phi (u)\leq 0\quad \text{for }u\in E^{-}\text{ with }\Vert u \Vert \leq \rho, \\ & \Phi (u)> 0\quad \text{for }u\in E^{+}\text{ with }\Vert u \Vert \leq \rho. \end{aligned}$$

The proof is completed. □

We are now ready to prove Theorem 1.1.

Proof

It follows from Lemma 3.5 that Φ has a local linking at 0 with respect to \(E=E^{-}\oplus E^{+}\). Therefore Proposition 3.2 yields

$$\begin{aligned} C_{k}(\Phi,0)\neq 0, \end{aligned}$$

where \(k=\operatorname{dim}E^{-}<\infty \). By Lemma 3.4, we have

$$\begin{aligned} C_{k}(\Phi,\infty )= 0. \end{aligned}$$

Applying Proposition 3.1, we see that Φ has a nontrivial critical point. The proof of Theorem 1.1 is completed. □

Availability of data and materials

Not applicable.

References

  1. Bartsch, T., Li, S.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419–441 (1997)

    MathSciNet  Article  Google Scholar 

  2. Bartsch, T., Liu, Z., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 2, 25–42 (2004)

    MATH  Google Scholar 

  3. Bensedik, A., Bouchekif, M.: On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity. Math. Comput. Model. 49, 1089–1096 (2009)

    MathSciNet  Article  Google Scholar 

  4. Bernstein, S.: Sur une classe d’équations fonctionelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. 4, 17–26 (1940)

    MATH  Google Scholar 

  5. Chang, K.-C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 6. Birkhäuser, Boston (1993)

    Book  Google Scholar 

  6. Chen, C.Y., Kuo, Y.C., Wu, T.F.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011)

    MathSciNet  Article  Google Scholar 

  7. Chen, S., Liu, S.: Standing waves for 4-superlinear Schrödinger-Kirchhoff equations. Math. Methods Appl. Sci. 38, 2185–2193 (2014)

    Article  Google Scholar 

  8. Guo, Z.: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2015)

    MathSciNet  Article  Google Scholar 

  9. He, X.M., Zou, W.M.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)

    MathSciNet  Article  Google Scholar 

  10. He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^{3}\). J. Differ. Equ. 2, 1813–1834 (2012)

    Article  Google Scholar 

  11. Jin, J., Wu, X.: Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 369, 564–574 (2010)

    MathSciNet  Article  Google Scholar 

  12. Kirchhoff, G., Hensel, K.: Vorlesungen Über Mathematische Physik:bd. Vorlesungen Über Mechanik. Teubner, Leipzig (1883)

    Google Scholar 

  13. Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^{3}\). J. Differ. Equ. 257, 566–600 (2014)

    Article  Google Scholar 

  14. Li, Y., Li, F., Shi, J.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285–2294 (2012)

    MathSciNet  Article  Google Scholar 

  15. Liu, H., Chen, H.: Multiple solutions for an indefinite Kirchhoff-type equation with sign-changing potential. Electron. J. Differ. Equ. 2015, 274 (2015)

    MathSciNet  Article  Google Scholar 

  16. Liu, J.Q.: The Morse index of a saddle point. Syst. Sci. Math. Sci. 2, 32–39 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Liu, S.: Existence of solutions to a superlinear p-Laplacian equation. Electron. J. Differ. Equ. 2011, 66 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Liu, W., He, X.: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473–487 (2012)

    MathSciNet  Article  Google Scholar 

  19. Lu, S.S.: Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains. J. Math. Anal. Appl. 432, 965–982 (2015)

    MathSciNet  Article  Google Scholar 

  20. Mao, A.M., Zhang, Z.T.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)

    MathSciNet  Article  Google Scholar 

  21. Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)

    MathSciNet  Article  Google Scholar 

  22. Pohožaev, S.I.: A certain class of quasilinear hyperbolic equations. Mat. Sb. (N. S.) 96, 152–166, 168 (1975). Russian

  23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  24. Shuai, W.: Sign-changing solutions for a class of Kirchhoff type problem in bounded domains. J. Differ. Equ. 259, 1256–1274 (2015)

    MathSciNet  Article  Google Scholar 

  25. Tang, X.H., Chen, S.T.: Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110 (2017)

    MathSciNet  Article  Google Scholar 

  26. Tang, X.H., Cheng, B.T.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261, 2384–2402 (2016)

    MathSciNet  Article  Google Scholar 

  27. Wang, J., Tian, L., Xu, J., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)

    MathSciNet  Article  Google Scholar 

  28. Wang, Z.Q.: On a superlinear elliptic equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, 43–57 (1991)

    MathSciNet  Article  Google Scholar 

  29. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 12, 1278–1287 (2011)

    MathSciNet  Article  Google Scholar 

  30. Xiang, M.Q., Radulescu, V.D., Zhang, B.L.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. 84(1), 915–954 (2021)

    MathSciNet  Article  Google Scholar 

  31. Xiang, M.Q., Zhang, B.L., Radulescu, V.D.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. Adv. Nonlinear Anal. 9, 690–709 (2020)

    MathSciNet  Article  Google Scholar 

  32. Yao, X.Z., Mu, C.L.: Infinitely many sign-changing solutions for Kirchhoff-type equations with power nonlinearity. Electron. J. Differ. Equ. 59, 1 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Zhang, Q., Xu, B.: Infinitely many solutions for Schrödinger–Kirchhoff-type equations involving indefinite potential. Electron. J. Qual. Theory Differ. Equ. 58, 1 (2017)

    Article  Google Scholar 

  34. Zhang, Z.T., Perera, K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referees and the editor for their valuable comments and suggestions.

Funding

This research was supported by the National Natural Science Foundation of China (No. 12161019) and Doctoral research project of Guizhou Normal University (No. GZNUD[2017]27, GZNUD[2019]14).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally and significantly in writing this paper. All authors wrote, read, and approved the final manuscript.

Corresponding author

Correspondence to Jian Zhou.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhou, J., Wu, Y. Existence of solutions for a class of Kirchhoff-type equations with indefinite potential. Bound Value Probl 2021, 74 (2021). https://doi.org/10.1186/s13661-021-01550-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-021-01550-5

Keywords

  • Kirchhoff-type equation
  • Variational methods
  • Palais–Smale condition
  • Local linking
  • Morse theory