EXISTENCE OF NONTRIVIAL SOLUTIONS FOR SCHR¨ODINGER-KIRCHHOFF EQUATIONS WITH INDEFINITE POTENTIALS

. We consider a class of Schr¨odinger-Kirchhoﬀ equations in R 3 with a general nonlinearity g and coercive sign-changing potential V so that the Schr¨odinger operator − a ∆ + V is indeﬁnite. The nonlinearity considered here satisﬁes the Ambrosetti-Rabinowitz type condition g ( t ) t ≥ µG ( t ) > 0 with µ > 3. We obtain the existence of nontrivial solutions for this problem via Morse theory.


Introduction
In this article, we consider the Schrödinger-Kirchhoff type problem where a and b are positive constants, V ∈ C(R 3 ) is the potential and g ∈ C(R) is the nonlinearity.
The above problem is nonlocal as the term R 3 |∇u| 2 dx implies that (1.1) is not a pointwise identity.This feature causes some mathematical difficulties, which make the investigation of (1.1) particularly interesting.Problem (1.1) arises in an interesting physical context.Indeed, if we set V (x) = 0 and replace R 3 by a bounded domain Ω ⊂ R 3 , problem (1.1) reduces to the Dirichlet problem which is related to the stationary analogue of the equation proposed by Kirchhoff [10] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where ρ, P 0 , h, E and L are positive constants.Lions [15] introduced an abstract functional analysis framework for the equation Since then, problem (1.4) have received much attention.Problem (1.1) has been studied extensively in recent years (in bounded or unbounded domain), see, for example, [1,7,14,22,23,26,28] and references therein.We emphasize that in all these papers, the potential V is assumed to be nonnegative.In this case, the quadratic part of the variational functional Φ given in (2.1) is positively definite, the zero function u = 0 is a local minimizer of Φ and the mountain pass theorem [2] can be applied.However, when the potential V is negative somewhere so that the quadratic part of Φ is indefinite, the zero function u = 0 is no longer a local minimizer of Φ, the mountain pass theorem is not applicable anymore.For the semilinear problem (b = 0) with indefinite Schrödinger operator −∆ + V , one usually applies the linking theorem to obtain a solution, see e.g.[11,19].For problem (1.1), it seems hard to verify the linking geometry because of the nonlocal term R 3 |∇u| 2 dx, which prevents the functional Φ to be nonpositive on the negative space of the Schrödinger operator, see Remark 1.2 for detail.Hence, the classical linking theorem [25, Lemma 2.12] is also not applicable.
For this reason, there are very few results about (1.1) with indefinite potential.To the best of our knowledge, the first work on this situation is due to Chen and Liu [6].To overcome the above difficulty, a crucial observation in [6] is that Φ has a local linking at u = 0, thus for certain i ∈ N, the i-th critical group of Φ at u = 0 is nontrivial.Assuming that g is 3-superlinear with suitable technical conditions it was shown in [6] that all critical groups of Φ at infinity are trivial, then a nonzero critical point of Φ can be found via Morse theory.See also [13] for a recent result for 3-superlinear nonlinearity.
For other results on the indefinite problem (1.1), we mention [27] and [9], where the nonlinearity g is sublinear and subquadratic, respectively.In these two papers the variational functional is coercive.The three critical points theorem of Liu and Su [16] and the classical Clark theorem are applied to obtain multiple solutions.
Having the above results in mind, it is natural to ask what will happen when g is 2-superlinear (g satisfies (1.5) with 3 replaced by 2)?This is the motivation of the current paper.
The main difficulty under this assumption is that it is not know whether Palais-Smale (or even Cerami) sequences are bounded or not.Motivated by Liu and Mosconi [20] on the study of nonlinear Schrödinger-Poisson systems (see also [8]), we add a dummy variable and consider an augmented functional Φ : R × X → R, see (2.4).It turns out that Φ satisfies the (P S) condition (see Lemma 2.6), and if (s, ū) is a critical point of Φ, then ū is a critical point of Φ (see Lemma 2.5).Moreover, Φ has a local linking at zero, and all critical groups of Φ at infinity are trivial.Eventually, a nonzero critical point of Φ can be obtained by using Morse theory, which give rise to a nonzero critical point of Φ.
Without loss of generality, we assume that a = b = 1.Then the problem (1.1) can be rewritten as follows On the linear subspace where from now on all integrals are taken over R 3 except stated explicitly, we equip with the inner product and the corresponding norm • .Note that if the assumptions (A1) below holds, then X is a Hilbert space and by Bartsch and Wang [4] we have a compact embedding X → L s (R 3 ) for s ∈ [2,6).Now we present our assumptions on the potential V (x) and the nonlinearity g(u).
Consider the bilinear form then X = X + ⊕ X − ⊕ X 0 , where X + , X − and X 0 are positive, negative, and null eigenspaces of the Schrödinger operator.It is well known that there exist constants η ± > 0 such that We are now ready to state our main result.
Remark 1.2.As we have mentioned, because of the nonlocal term R 3 |∇u| 2 dx, our functional Φ (see (2.1)) does not satisfy the geometric assumption of the linking theorem.For R > r > 0 set provided R is large enough and r is small enough.However, because the integral |∇u| 2 2 in our functional, Φ may be very large, it is possible that Φ(u) > b for some u ∈ ∂M ∩ (X − ⊕ X 0 ).Therefore, the following geometric assumption of the linking theorem inf can not be satisfied.Thus, the linking theorem is not applicable for problem (1.6).

Palais-Smale condition
Now, we investigate the functional Φ.Under the assumptions (A1) and (A4) we can show that the functional Φ : X → R, is well defined and of class C 1 .The derivative of Φ is given by for u, v ∈ X.Consequently, critical points of Φ are weak solutions of problem (1.6).
Proposition 2.1 ([20, Lemma 3.1]).Assume (A3) holds and let Ṽ := V +m.Then for any t > 0 and x ∈ R 3 , we have For t > 0 and u ∈ X we define and define on the Hilbert space R × X (with natural norm (s, u) (2.7) Proof.By changing variables, we obtain Similarly, By the continuity of V , changing variables x = yt on any fixed ball B R and applying Proposition 2.1, we obtain Letting R → +∞ we deduce that From the above estimates and continuous embeddings, Together with (2.4), it is easy to see that Φ is well defined.Formula (2.5) directly follows from changing variables.Formula (2.6) can be computed in a standard way, while (2.7) is obtained by differentiating under the integral sign in (2.5).By (A4), we deduce that |3G(e s u) + g(e s u)e s u| ≤ C(e 2s |u| 2 + e ps |u| 6 ). (2.8) It follows from |V | ≤ Ṽ + m, (A3) and Proposition 2.1 that (2.9) Noting that, by the dominated convergence theorem, both Lemma 2.4 (Pohožaev identity).Suppose that (A3) and (A4) hold.Let u ∈ X be a weak solution to problem (1.6), then we have the following Pohožaev identity, Then, by (2.10), (2.11), and DΦ(u) = 0, we obtain In the following, we denote by D Φ the total differential of Φ with respect to both variables s and u.Lemma 2.5.Suppose that (A3) and (A4) hold, then Proof.The proof is similar to the proof of [20, Lemma 3.5], we include it here for the reader's convenience.(⇐) From the definition of D Φ, we have (2.12) Note that DΦ(ū) = 0 implies Therefore, it suffices to prove that ∂ s Φ(0, ū) = 0. From DΦ(ū) = 0, Lemma 2.4 gives d dt t=1 Φ(ū t ) = 0. (2.13) The map t → Φ(ū t ) is C 1 by Lemma 2.3 and (⇒) If D Φ(s, ū) = 0, combining (2.4), we immediately infer that Thus, we only have to prove that s = 0. Applying Lemma 2.4 to v := ūe s gives d By the chain rule and ∂ s Φ(s, ū) = 0 we have Lemma 2.6.Suppose that (A1)-(A5) hold.Then Φ satisfies the (P S) condition.
Proof.Let {(s n , u n )} be a (P S) sequence for Φ in R × X, that is Choose λ ∈ (3, µ), where µ > 3 is given by (A5).Then (2.14) The third integral is bounded from below through (A1)-(A3), and Hölder's inequality.Indeed, we set (2.15) As W λ is bounded on bounded sets, we let We write the integral over R 3 \ B R as the sum of the integrals over the intersections of R 3 \ B R with {V ≥ 0} and {V < 0}.Obviously, assumption (A1) implies that {V < 0} has finite measure.Because λ − 2 > 1, assumption (A2) implies that On the other hand, by (A3) and V ≥ −m we have For some possibly larger C λ , we have (2.17) Combining (2.15), (2.16), and (2.17), we obtain Condition (A5) implies that for some C > 0, G(t) ≥ C|t| µ .We deduce  To complete the proof of the boundedness of u n , let S > 0 be such that |s n | ≤ S. Since V ≥ −m on {V ≤ k}, we deduce where we used (A1) and (2.20) in the last inequality.From (2.21) we thus infer by (2.20), proving the boundedness of {u n } in X.Using the compact embedding X → L s for s ∈ [2,6), by standard argument as in [9, Lemma 3.2], we see that {u n } has a convergent subsequence.Hence {(s n , u n )} has a convergent subsequence.

Critical groups and proof of Theorem 1.1
Having established the (P S) condition for Φ, we are now ready to present the proof of Theorem 1.1.We start by recalling some concepts and results from infinitedimensional Morse theory (see e.g., Chang [5] and Mawhin and Willem [21,Chapter 8]).
Let X be a Banach space, ϕ : X → R be a C 1 functional, u be an isolated critical point of ϕ and ϕ(u) = c.Then is called the i-th critical group of ϕ at u, where ϕ c := ϕ −1 (−∞, c] and H * stands for the singular homology with coefficients in Z. If ϕ satisfies the (P S) condition and the critical values of ϕ are bounded from below by α, then following Bartsch and Li [3], we define the i-th critical group of ϕ at infinity by By the deformation lemma, it is well known that the homology on the right-hand side does not depend on the choice of α. ).Suppose ϕ ∈ C 1 (X, R) satisfies the (P S) condition and has a local linking at 0 with respect to the decomposition X = Y ⊕ Z, i.e., for some ε > 0, Since Φ satisfies the (P S) condition, the critical group C * ( Φ, ∞) of Φ at infinity makes sense.To study C * ( Φ, ∞) we need the following lemma.
for all t > 0, u ∈ X and τ ∈ R.
Choose λ ∈ (3, µ) and a < −M λ , where M λ ≥ 0 is given in Lemma 3.3.Then by Lemma 3.4, we have So there is t s,u > 0 such that Φ(s, u ts,u ) = a.A direct computation and Lemma 3.3 gives By the implicit function theorem, given (s, u) ∈ R × Ẋ, there is a unique solution t = T (s, u) for the equation Φ(s, u t ) = a, and the map is continuous.Using the continuous function T , it is standard (see [24,18]) to construct a deformation from R × Ẋ to the level set Φa = Φ−1 (−∞, a], and deduce Recalling that X + , X − and X 0 are the negative, positive and null eigenspaces of the bilinear form defined in (1.9).To study the critical group C * ( Φ, 0) of Φ at origin, we consider the decomposition ) dim X 0 > 0 and G(t) ≥ c|t| ν for some ν < 4, then the functional Φ has a local linking at 0 with respect to the decomposition R × X = X− ⊕ X+ .