Nontrivial solutions for periodic Schr\"odinger equations with sign-changing nonlinearities

Using a new infinite-dimensional linking theorem, we obtained nontrivial solutions for strongly indefinite periodic Schr\"odinger equations with sign-changing nonlinearities.


Introduction and statement of results
In this paper, we consider the following semilinear Schrödinger equation: where N ≥ 1. The potential V is a continuous function and 1-periodic in x j for j = 1, · · · , N . In this case, the spectrum of the operator −∆ + V is a purely continuous spectrum that is bounded below and consists of closed disjoint intervals ([16, Theorem XIII.100]). Thus, the complement R \ σ(L) consists of open intervals called spectral gaps. More precisely, for V , we assume (v). V ∈ C(R N ) is 1-periodic in x j for j = 1, · · · , N , 0 is in a spectral gap (−µ −1 , µ 1 ) of −∆ + V and −µ −1 and µ 1 lie in the essential spectrum of −∆ + V. Denote µ 0 := min{µ −1 , µ 1 }.
And for f, we assume (f 1 ). f ∈ C(R N × R) is 1-periodic in x j for j = 1, · · · , N . And there exist constants C > 0 and 2 < p < 2 * such that |f (x, t)| ≤ C(1 + |t| p−1 ), ∀(x, t) ∈ R N × R and A solution u of (1.1) is called nontrivial if u ≡ 0. Our main result is the following theorem: As an application of this theorem, we have the following corollary: Corollary 1.2. Suppose that 2 < p < q < 2 * , and (v) is satisfied. Then there exists λ 0 > 0 such that, for 0 < λ < λ 0 , −∆u + V u = λ|u| p−2 u − |u| q−2 u has a nontrivial solution. Under the assumption (v), the quadratic form R N (|∇u| 2 +V (x)u 2 )dx has infinite-dimensional negative and positive spaces. This case is called strongly indefinite. Semilinear periodic Schrödinger equations with strongly indefinite linear part have attracted much attention in recent years due to its numerous applications in mathematical physics. In [2], the authors used a dual variational method to obtain a nontrivial solution of (1.1) with f (x, t) = −W (x)|t| p−2 t, where W is a perturbed periodic function and 2 < p < 2 * . In [19], Troestler and Willem used critical point theory to obtain a nontrivial solution of (1.1) by assuming Then, Kryszewski and Szulkin [8] proved a new infinite-dimensional linking theorem. Using it, they generalized Troestler and Willem's result by assuming that f ∈ C(R N × R) and g = −f satisfies the Ambrosetti-Rabinowitz condition. Similar results were also obtained by Pankov and Pflüger in [13,14] by an approximation method and a variant Nehari method. Equation (1.1) with asymptotically linear nonlinearities and other super-linear nonlinearities has also been studied by many authors. One can see [5,6,7,10] for the asymptotically linear case and [1,3,4,7,9,12,17,22] for the super-linear case. Moreover, equation (1.1) with 0 belonging to the spectrum of −∆ + V was studied in [6,21,23]. Finally, we should mention that the methods of studying the strongly indefinite periodic Schrödinger can shed light on other strongly indefinite problems, such as, the Hamiltonian systems or the discrete nonlinear Schrödinger equations. One can consult [6,18] or [15,24].
All the existence results for equation (1.1) we mentioned above are obtained under the assumption that f does not change sign in R N × R, i.e., f ≥ 0 in R N × R or f ≤ 0 in R N × R. However, under our assumptions (f 1 ) − (f 3 ), f (x, t) can be negative in {(x, t) | |t| < ρ}. Together with (1.4), this implies that f (x, ·) may change sign in R. As we know, this situation has never been studied before. And it is the novelty of our main results Theorem 1.1 and Corollary 1.2.
The difficulties of equation (1.1) with sign-changing nonlinearity come from two aspects. The first is that the classical infinite-dimensional linking theorem (see [20,Theorem 6.10] or [8]) cannot be used to deal (1.1) in this case. To use this linking theorem, the functional corresponding to (1.1) must satisfy some upper semi-continuous assumption. However, when f (x, t) is sign-changing, the functional corresponding to (1.1) does not satisfy this assumption. The second is that the sign-changing nonlinearity brings more difficulty in the proof of boundedness of Palais-Smale sequence.
In this paper, a variant infinite-dimensional linking theorem (see Theorem 4.2) is given. This theorem replaced the upper semi-continuous assumption in the classical infinite-dimensional linking theorem ([20, Theorem 6.10]) with other assumptions. For the reader's convenience, we presented this theorem and its proof in the appendix. Using this theorem, a (C) c sequence (see Definition 4.1) of equation ( , we can prove that this sequence is bounded in H 1 (R N ) (see Lemma 3.2). Then, we obtained a nontrivial solution of (1.1) from the (C) c sequence through the concentrationcompactness principle.
Notation. B r (a) denotes the open ball of radius r and center a. For a Banach space E, we denote the dual space of E by E ′ , and denote strong and weak convergence in E by → and ⇀, respectively. For ϕ ∈ C 1 (E; R), we denote the Fréchet derivative of ϕ at u by ϕ ′ (u). The Gateaux derivative of ϕ is denoted by

Variational and linking structure for equation (1.1)
Under the assumptions (v), and (f 1 ), the functional is of class C 1 on X := H 1 (R N ). The derivative of J is and the critical points of J are weak solutions of (1.1). There is a standard variational setting for the quadratic form R N (|∇u| 2 +V (x)u 2 )dx. For the reader's convenience, we state it here. One can consult [7] or [6] for more details.
Assume that (v) holds and let S = −∆ + V be the self-adjoint operator acting on L 2 (R N ) with domain D(S) = H 2 (R N ). By virtue of (v), we have the orthogonal decomposition with equivalent norms. Therefore, X continuously embeds in L q (R N ) for all 2 ≤ q ≤ 2 * . In addition, we have the decomposition where Y = X ∩ L 1 , Z = X ∩ L 2 and Y, Z are orthogonal with respect to both (·, ·) L 2 and (·, ·). Let P : X → Y and Q : X → Z be orthogonal projections. Therefore, for every u ∈ X , there is a unique decomposition u = P u + Qu with (P u, Qu) = 0 and and Therefore, We denote Then by (2.1) and (2.3), Φ can be written as And the derivative of Φ is given by The following Lemma shows that Φ satisfies the linking condition (see (4.5)) of Theorem 4.2 in the appendix.
where N, M and ∂M are defined in (4.3) and (4.4) in the appendix.
Proof. We divide the proof into several steps.

Combining
Step 1-Step 3, we get the desired results of this Lemma.
It follows that ||v n || = O(||u n ||). Together with the fact that {u n } is a (C) c sequence for Φ = −J, this implies that Then by (2.2), (3.1), and the fact that u n ≥ v n ≥ 0 on ̟ + n , we get that where and v n ≥ u n /2 > 0 on ̟ + n , we get from (3.3) that

Appendix
In this section, we give a variant linking theorem which is a generalization of the classical infinitedimensional linking theorem of [20, Theorem 6.10] (see also [8]). Before state this theorem, we give some notations and definitions. Let X be a separable Hilbert space with inner product (·, ·) and norm ||·||, respectively. X ± are closed subspaces of X and X = X + ⊕ X − . Let {e − k } be the total orthonormal sequence in X − . Let be the orthogonal projections. We define on X. Then ||Qu|| ≤ |||u||| ≤ ||u||, ∀u ∈ X.
Moreover, if ||u n || is bounded and |||u n − u||| → 0, then {u n } weakly converges to u in X. The topology generated by ||| · ||| is denoted by τ , and all topological notations related to it will include this symbol.
Let R > r > 0 and u 0 ∈ X + with ||u 0 || = 1. Set Then, M is a submanifold of X − ⊕ R + u 0 with boundary The main result of this section is the following theorem:    and From the definition of ||| · |||, we deduce that if a sequence {u n } ⊂ E ∩ B R τ -converges to u ∈ X, i.e., |||u n − u||| → 0, then u n ⇀ u in X (see Remark 6.1 of [20]). By the weakly sequential continuity of H ′ , we get that for any ϕ ∈ X, H ′ (u n ), ϕ → H ′ (u), ϕ . This implies that H ′ is τ -sequentially continuous in E ∩ B R . By (4.12), the τ -sequential continuity of H ′ in E ∩ B R and the weakly lower semi-continuity of the norm || · ||, we get that there exists a τ -open neighborhood V u of u such that and ||(1 + ||u||)φ u || = 1 + ||u|| ≤ 2(1 + ||v||), ∀v ∈ V u . (4.14) Because B R is a bounded convex closed set in the Hilbert space X, B R is a τ -closed set. Therefore, X \ B R is a τ -open set.
The family  Since the τ -open covering M of V is local finite, each u ∈ V belongs to only finite many sets M i . Therefore, for every u ∈ V, the sum in (4.15) is only a finite sum. It follows that, for any u ∈ V, there exist a τ -open neighborhood U u ⊂ V of u and L u > 0 such that ξ(U u ) is contained in a finite-dimensional subspace of X and Moreover, by the definition of ξ, (4.13) and (4.14), we get that, for every u ∈ V, ||ξ(u)|| ≤ 1 + ||u|| and H ′ (u), ξ(u) ≥ 0 (4.17) and for every u ∈ E ∩ B R , Step2. Let θ be a smooth function satisfying 0 ≤ θ ≤ 1 in R and From (4.16), (4.18) and the definition of χ, we deduce that the mapping χ satisfies that This means that χ is locally Lipschitz continuous and τ -locally Lipschitz continuous, (c).