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Existence of solutions for a class of Kirchhoff-type equations with indefinite potential
Boundary Value Problems volume 2021, Article number: 74 (2021)
Abstract
In this paper, we consider the existence of solutions of the following Kirchhoff-type problem:
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.
1 Introduction and main result
This paper is concerned with the following Kirchhoff-type problem:
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:
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:
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
(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
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
where \(\Gamma _{n}\) denotes the family of n-dimensional subspaces of \(C_{0}^{\infty }(\mathbb{R}^{3})\), remember \(a\neq 0\). Let
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
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
for all \(M>0\), where μ denotes the Lebesgue measure in \(\mathbb{R}^{N}\), then the working space
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.
2 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
with the usual norm
and
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
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
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
It is easy to see that, from \((f_{1})\) and \((f_{2})\), the functional \(\Phi: E\rightarrow \mathbb{R}\),
is of class \(C^{1}\) with derivative
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
(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,
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
If \(v=0\), then \(v_{n}^{-}\rightarrow v^{-}=0\) due to \(\operatorname{dim}E^{-}<\infty \). Since
for n large enough, we obtain
Now, using \((f_{3})\), for n large enough, we deduce
contradicting \(\|u_{n}\|\rightarrow \infty \), thus \(v\neq 0\).
It follows from (2.4) that
which implies that
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,
and then
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
We have
Consequently,
Since \(\operatorname{dim}E^{-}<\infty \), we have \(u_{n}^{-}\rightarrow u^{-}\), thus \(\|u_{n}^{-}\|\rightarrow \|u^{-}\|\). Collecting all infinitesimal terms, we obtain
With the assumption \((f_{4})\), it has been shown in [2, p.29] that
Because \(\Upsilon (u)=|\nabla u|_{2}\) is weak lower semi-continuous, we have
Now, from (2.6), we have
which implies \(\overline{\lim }\|u_{n}^{+}\|^{2}\leq \|u^{+}\|^{2}\). From the weak lower semi-continuity of the norm, we have
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. □
3 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
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
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
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
Proof
We argue by contradiction. Assume that there exists a sequence \(\{u_{n}\}\subset E\) such that \(\Phi (u_{n})\leq -n\), but
By \((f_{3})\), we have
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
Then, by Fatou’s lemma and \((f_{3})\), we have
Hence, using (3.1), we have
a contradiction. Therefore \(v^{-}=0\), from
we see that \(\|v^{+}_{n}\|\rightarrow 1\). Consequently, for n large enough, we have
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
Then, for \(v\in S\) and \((f_{3})\), we deduce that
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
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
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
If \(u\in E^{-}\), by (2.1) and the equivalence of norms on finite dimensional space \(E^{-}\), we have
If \(u\in E^{+}\), we have
So, there exists \(0<\rho <1\) small enough such that
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
where \(k=\operatorname{dim}E^{-}<\infty \). By Lemma 3.4, we have
Applying Proposition 3.1, we see that Φ has a nontrivial critical point. The proof of Theorem 1.1 is completed. □
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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).
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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
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DOI: https://doi.org/10.1186/s13661-021-01550-5