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

where Ω ⊂R3 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 Φ : H1ðR3Þ→R,


Introduction and Main Results
In this work, we consider the Schrödinger-Kirchhoff type equation of the form where a > 0, b ≥ 0 are constants. This equation arises when we look for stationary solutions of the equation proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where ρ, h, P 0 , and L are positive constants. In [2], J.L. Lions introduced an abstract functional analysis framework to the equation where Ω ⊂ ℝ 3 is a bounded domain. After that, (3) received much attention. For more details on the physical and math-ematical background of this problem, we refer to [1][2][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][4][5][6][7][8][9][10][11][12][13] and references therein. Under suitable conditions, it is well known that weak solutions to (1) correspond to critical points of the energy functional Φ : where Fðx, tÞ = Ð t 0 f ðx, sÞds. We emphasize that in the papers mentioned above, the authors only considered the case that the Schrödinger operator −Δ + V 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 V is negative somewhere, the zero function u = 0 is no longer a local minimizer of Φ. In this case, the functional Φ would not enjoy the general linking geometry due to the nonlocal term Ð ℝ 3 j∇uj 2 dx. 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 H 1 ðℝ 3 Þ°L 2 ðℝ 3 Þ is not compact, it is assumed in [16] that so that the related weighted Sobolev space is compactly embedded into L 2 ðℝ 3 Þ. Then, via Morse theory, they obtained nontrivial critical points of Φ.
In this paper, we will consider the case of more general V such that the abovementioned compact embedding may not be true. We assume the potential V satisfies.
(V) V ∈ Cðℝ 3 Þ is bounded such that the quadratic form which is nondegenerate and the negative space of Q is finitedimensional.
and 0 does not happen to be a spectrum point of the Schrödinger operator −Δ + V, then such V satisfies our assumption (V).
(ii) Note that in (f 2 ), we have not required that the limit (8) holds uniformly (iii) In order to produce critical points of Φ, eventually, we will encounter the compactness problem. For this issue, we assume assumption (f 4 ). It is easily see that let a : ℝ 3 → ð0,∞Þ be continuous, lim jxj→∞ aðxÞ = 0 and satisfies (f 1 )-(f 4 ). Now, we are ready to state our main results.
It is known that if the quadratic form Q 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 H 1 ðℝ 3 Þ°L 2 ðℝ 3 Þ 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 Φ. Since we cannot easily see that Φ′ is weakly sequentially continuous in The paper is organized as follows. In the next section, we prove that the (PS) sequences of Φ are bounded and Φ 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 Φ 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.

Palais-Smale Condition
Throughout of this paper, we always denote E = H 1 ðℝ 3 Þ. In view of assumption (V), we may choose an equivalent norm k·k on E such that where E + and E − are positive and negative spaces of Q, respectively, and Here and in what follows, u ± denotes the orthogonal projection of u on E ± . Then, the functional Φ can be rewritten as By (f 1 ), Φ is of class C 1 on E with the derivative given by Then, fu n g is bounded in E.
Proof. Suppose by contradiction that ku n k → ∞. Let v n = ku n k −1 u n . Then, passing to a subsequence, there exists v ∈ E such that If By (f 3 ) we deduce that, for n large enough, contradicting ku n k → ∞. Now suppose that v ≠ 0. Then, the set Θ = fx ∈ ℝ 3 : vðx Þ ≠ 0g has positive Lebesgue measure. For x ∈ Θ, we have j u n ðxÞj → ∞, and hence (8) implies Then, the Fatou lemma yields ð On the other hand, for large n, where M > 0 is a constant and j·j s denotes the standard norm in L s ðℝ 3 Þ. This is also a contradiction.
In conclusion, we deduce that the (PS) sequence fu n g is bounded.
To get a convergent subsequence of the (PS) sequence, we need the following lemma.
Proof. Since u n ⇀ u in E, we have ∇u n ⇀ ∇u in L 2 ðℝ 3 Þ.
Define ψðuÞ = Ð ℝ 3 j∇uj 2 dx in E. It is easy to see that ψ is continuous and convex. Hence, ψ is weakly lower semicontinuous in E, so that Consequently, The lemma is proved.

Lemma 6. Φ satisfies the (PS) condition.
Proof. Let fu n g be a (PS) sequence. From Lemma 4, we know that fu n g is bounded in E. Up to a subsequence, we may assume that u n ⇀ u in E. Hence, Consequently,

Journal of Function Spaces
Because dim E − < ∞, we have u − n → u − and thus ku − n k → ku − k. Collecting all infinitesimal terms, we obtain We claim that lim Given ε > 0. For r ≥ 1, by (f 1 ) and Hölder's inequality, we have Since p < 6, we may fix r large enough such that, ð for all n. Moreover, it follows from (f 4 ) that there exists R > 0 such that for all n. Finally, from (f 1 ) and (f 2 ) we deduce that, for any ε > 0, there exists c ε > 0 such that Note that u + n → u + in L s ðB R ð0ÞÞ for every s ∈ 2, 2 * Þ. Consequently, for n large enough, Combining (31), (32), and (34), we obtain that ð for n large enough. Since dim E − < ∞, we obtain that (29) holds by the arbitrariness of ε. Now using Lemma 5, we deduce from (28) that lim Combining this with the weakly lower semicontinuous of the norm, we have That is ku + n k → ku + k. Reminding ku − n k → ku − k, we get ku n k → kuk. Thus, u n → u in E.

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 X be a Banach space, φ : X → ℝ be a C 1 functional, u be an isolated critical point of φ and φðuÞ = c. Then, is called the qth critical group of φ at u, where φ c ≔ φ −1 ð− ∞,c and H * stands for the singular homology with coefficients in ℤ.
If φ satisfies the (PS) condition and the critical values of φ are bounded from below by α, then following Bartsch and Li [18], we call the qth critical group of φ at infinity. It is well known that the homology on the right hand does not depend on the choice of α.
Proposition 8 (see [19]). Suppose φ ∈ C 1 ðX, ℝÞ has a local linking at 0 , i.e., X = Y ⊕ Z and Journal of Function Spaces For the proof of Theorem 11, we may assume the Φ has only finitely many critical points. Since Φ satisfies the (PS) condition, the critical group C * ðΦ,∞Þ of Φ at infinity makes sense. To study C * ðΦ,∞Þ, 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 H 1 ðℝ 3 Þ, which can not compactly embedded into L 2 ðℝ 3 Þ.

Lemma 9.
There exists A > 0 such that, if ΦðuÞ ≤ −A , then Proof. Otherwise, there exists a sequence fu n g ⊂ E such that Φðu n Þ ≤ −n but Consequently, Let v n = ku n k −1 u n and v ± n be the orthogonal projection of v n on E ± . Then, v − n → v − for some v − ∈ E − , because dim E − < ∞.
If v − ≠ 0, then for some v ∈ E \ f0g we have v n ⇀ v in E. Similar to (20), we obtain Hence, by (42), we get 0 ≤ Φ′ u n ð Þ, u n