Nontrivial Solutions for 4-Superlinear Schrödinger–Kirchhoff Equations with Indefinite Potentials

Abstract

This paper is devoted to the 4-superlinear Schrödinger–Kirchhoff equation where , . The potential here is indefinite so that the Schrödinger operator possesses a finite-dimensional negative space. By using the Morse theory, we obtain nontrivial solutions for this problem.

1. Introduction and Main Results

In this work, we consider the Schrödinger–Kirchhoff type equation of the formwhere , are constants. This equation arises when we look for stationary solutions of the equationproposed by Kirchhoff [1] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings, where , , , and are positive constants. In [2], J.L. Lions introduced an abstract functional analysis framework to the equationwhere is a bounded domain. After that, (3) received much attention. For more details on the physical and mathematical background of this problem, we refer to [1–3] for references.

Problem (1) has been studied extensively by many researchers. Some interesting studies by variational methods can be found in, for example, [3–13] and references therein. Under suitable conditions, it is well known that weak solutions to (1) correspond to critical points of the energy functional ,where d. We emphasize that in the papers mentioned above, the authors only considered the case that the Schrödinger operator is positively definite. In this situation, a typical way to deal with (1) is to use the mountain pass theorem (cf. [14]). However, when the potential is negative somewhere, the zero function is no longer a local minimizer of . In this case, the functional would not enjoy the general linking geometry due to the nonlocal term d. Hence, the classical linking theorem (see, e.g., ([15], Theorem 2.12)) is also not applicable.

Such an indefinite situation was studied in [16]. To overcome these difficulties and the difficulty that the Sobolev embedding is not compact, it is assumed in [16] thatso that the related weighted Sobolev space is compactly embedded into . Then, via Morse theory, they obtained nontrivial critical points of .

In this paper, we will consider the case of more general such that the abovementioned compact embedding may not be true. We assume the potential satisfies.

(V) is bounded such that the quadratic formwhich is nondegenerate and the negative space of is finite-dimensional.

For the nonlinearity , we make the following assumptions:

(f₁) there exist and such that

(f₂) as uniformly in and

(f₃) for and ;

(f₄) for any , there holds

Remark 1. (i)(V) is a generic assumption on . For example, if such thatand does not happen to be a spectrum point of the Schrödinger operator , then such satisfies our assumption (V).(ii)Note that in (f₂), we have not required that the limit (8) holds uniformly(iii)In order to produce critical points of , eventually, we will encounter the compactness problem. For this issue, we assume assumption (f₄). It is easily see that let be continuous, and , thensatisfies (f₁)–(f₄).

Now, we are ready to state our main results.

Theorem 2. Assume that (V) and (f₁)–(f₄) are satisfied. Then, problem ((1)) has a nontrivial solution.

Theorem 3. Assume that (V) and (f₁)–(f₄) are satisfied. Moreover, ifis odd, then problem ((1)) has a sequence of solutionssuch that the energy.

It is known that if the quadratic form is indefinite, usually, it is more difficult to verify the boundedness of the (PS) sequence. In [16], this is done by taking advantage of the compact embedding mentioned before. Under our present setting, the related Sobolev embedding is not compact. We will illustrate a general technique to establish the boundedness of (PS) sequences. Moreover, it is also worth to point out that the weak limit of the bounded (PS) sequence is not obviously a critical point of . Since we cannot easily see that is weakly sequentially continuous in by direct calculations due to the existence of nonlocal term d. Indeed, in general, we do not know dd from in .

The paper is organized as follows. In the next section, we prove that the (PS) sequences of are bounded and satisfies the (PS) condition. In Section 3, we recall some concepts and results in infinite-dimensional Morse theory; then, we analyze the critical groups of at infinity; finally, we give the proof of Theorem 11. Having established the (PS) condition, the proof of Theorem 3 is quite similar to that of ([16], Theorem 3); therefore, we omit it here.

2. Palais-Smale Condition

Throughout of this paper, we always denote . In view of assumption (), we may choose an equivalent norm on such thatwhere and are positive and negative spaces of , respectively, and . Here and in what follows, denotes the orthogonal projection of on . Then, the functional can be rewritten as

By (f₁), is of class on with the derivative given byfor all . Then, solves (1) if and only if it is a critical point of .

Lemma 4. Letbe a (PS) sequence of, that is,Then, is bounded in .

Proof. Suppose by contradiction that . Let . Then, passing to a subsequence, there exists such that

If , then since . Noting that , we have

By (f₃) we deduce that, for large enough,contradicting .

Now suppose that . Then, the set has positive Lebesgue measure. For , we have , and hence (8) implies

Then, the Fatou lemma yields

On the other hand, for large ,where is a constant and denotes the standard norm in . This is also a contradiction.

In conclusion, we deduce that the (PS) sequence is bounded.

To get a convergent subsequence of the (PS) sequence, we need the following lemma.

Lemma 5. Letin. Then,

Proof. Since in , we have in . Hence,

Define d in . It is easy to see that is continuous and convex. Hence, is weakly lower semicontinuous in , so that

Consequently,

The lemma is proved.

Lemma 6. satisfies the (PS) condition.

Proof. Let be a (PS) sequence. From Lemma 4, we know that is bounded in . Up to a subsequence, we may assume that in . Hence,

Consequently,

Because , we have and thus . Collecting all infinitesimal terms, we obtain

We claim that

Given . For , by (f₁) and Hölder’s inequality, we have

Since , we may fix large enough such that,for all . Moreover, it follows from (f₄) that there exists such thatfor all . Finally, from (f₁) and (f₂) we deduce that, for any , there exists such that

Note that in for every . Consequently, for large enough,

Combining (31), (32), and (34), we obtain thatfor large enough. Since , we obtain that (29) holds by the arbitrariness of .

Now using Lemma 5, we deduce from (28) that

Combining this with the weakly lower semicontinuous of the norm, we have

That is . Reminding , we get . Thus, in .

3. Critical Groups and the proof of Theorem 11

Before giving the proof of Theorem 11, we recall some concepts and results of infinite dimensional Morse theory [17].

Let be a Banach space, be a functional, be an isolated critical point of and . Then,is called the th critical group of at , where and stands for the singular homology with coefficients in .

If satisfies the (PS) condition and the critical values of are bounded from below by , then following Bartsch and Li [18], we callthe th critical group of at infinity. It is well known that the homology on the right hand does not depend on the choice of .

Proposition 7 (see [18]). Ifsatisfies the (PS) condition andfor some, thenhas a nonzero critical point.

Proposition 8 (see [19]). Supposehas a local linking at, i.e.,andfor some , where . If , then .

For the proof of Theorem 11, we may assume the has only finitely many critical points. Since satisfies the (PS) condition, the critical group of at infinity makes sense. To study , we need the following lemma.

Before state it, we point out that the proof of the following lemma is quite different and more general ([16], Lemma 2.4), because in our case, the working Sobolev space is , which can not compactly embedded into .

Lemma 9. There existssuch that, if, then

Proof. Otherwise, there exists a sequence such that but

Consequently,

Let and be the orthogonal projection of on . Then, for some , because .

If , then for some we have in . Similar to (20), we obtain

Hence, by (42), we geta contradiction. Therefore, . Fromwe see that . Consequently, for large enough,violating (43). Hence, the desired result is proved.

Lemma 10. for all

Proof. Let , be the unit sphere in , and be the number given in Lemma 9. Without loss of generality, we may assume that

Using (8), it is clear that for any ,

So there is such that . Set , then a direct computation and Lemma 9 gives

By the implicit function theorem, is a continuous function on . Using the function , as in [20–22], we can construct a strong deformation retract ,and deduce

Now, we are ready to prove our main result.

Proof of Theorem 11. It follows from (V) and (f₂) that as ,

Hence, there exists such that is positive in and negative in , that is, has a local linking with respect to the decomposition . Therefore, Proposition 8 yieldswhere . By Lemma 10, . Applying Proposition 7, we obtain that has a nonzero critical point. The proof of Theorem 11 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11701251, no. 11671185, and no. 11771195) and the Natural Science Foundation of Shandong Province (no. ZR2017BA015 and no. ZR2019YQ04).