On nontrivial solutions of nonlinear Schrödinger equations with sign-changing potential

In this paper, we consider the superlinear Schrödinger equation with bounded potential well. The potential here is allowed to be sign-changing. Without assuming the Ambrosetti–Rabinowitz-type condition, we prove the existence of a nontrivial solution and multiplicity results.


Introduction and main results
This paper is concerned with the existence and multiplicity of nontrivial solutions for the superlinear Schrödinger equation of the form (1.1) With the aid of variational methods, problems of the form (1.1) have been extensively studied in the past decades. There are many works adopting various assumptions on V and f ; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein.
Besides (V), in [11,12], it is also assumed that inf V > 0, so that λ 1 > 0. Then the energy functional with respect to problem (1.1) has mountain pass geometry. In this work, we are interested in the case where the Schrödinger operator -+ V possesses a nontrivial negative space, which leads to more difficulty in verifying the compactness conditions. To the best of our knowledge, there are not many results in this case.
In this paper, we do not assume any compactness conditions on the potential function V . It is well known that the main difficulty in studying (1.1) in R N is the lack of compactness. This difficulty can be avoided for (1.1) in bounded domains or if the potential function V possesses some compactness conditions. For example, if lim |x|→∞ V (x) = ∞ or u is radially symmetric, we can get some compactness embedding, and then the Palias-Smale condition can be proved. We refer to [16] in this direction.
Denote F(x, t) := t 0 f (x, s) ds, 2 * := 2N N-2 , and p := p p-1 , the conjugate exponent of p. We make the following assumptions on the nonlinearity f .
(f 1 ) f ∈ C 1 (R N × R), and there exist constants p ∈ (2, 2 * ) and c > 0 such that |f (x,t)| |t| = 0 for every l > 0. (f 5 ) There exist a, b > 0 and α ∈ (0, α * ) such that for x ∈ R N and t = 0, where α * = min{p , (2 * -1)p -2 * }. Then we have the following two results. Remark 1.4 To produce critical points of the variational functional of (1.1), we will eventually encounter the compactness problem. For this issue, we introduced assumption (f 4 ). It is easy to see that if a : R N → R is continuous, lim |x|→∞ a(x) = 0, and p ∈ (2, 2 * ), then Remark 1.5 Most papers concerned with the superlinear Schrödinger equations involve the following classical condition of Ambrosetti and Rabinowitz: (AR) There exists μ > 2 such that 0 < μF(x, t) ≤ tf (x, t) for all x ∈ R N and t = 0. Condition (AR) plays a crucial role in proving the boundedness of Palias-Smale or Cerami sequences. Instead, we introduce a new condition (f 5 ), and we will illustrate a general technique to establish the boundedness of Cerami sequences. It is well known that many superlinear nonlinearities such as f (x, t) = t ln 1 + |t| do not satisfy condition (AR). Note that 1 a|t| α +b → 0 as |t| → ∞, which indicates that (f 5 ) is somewhat weaker than (AR). Note also that (2 * -1)p -2 * > 0 whenever p < 2 * . So the parameter α ∈ (0, α * ) is available. It is also worth pointing out that (f 5 ) is not a superlinear condition. Indeed, there are asymptotically linear functions satisfying (f 5 ).

Preliminaries
We denote by E := H 1 (R N ) the usual Sobolev space. Define the functional : E → R by Our assumptions on V and f stated above imply that the Schrödnger operator -+ V is selfadjoint and semibounded in L 2 (R N ) and ∈ C 1 (E, R). A direct computation gives that, for all u, v ∈ E, It is well known that the critical points of are solutions of problem (1.1). By (V) 0 is not an eigenvalue of -+ V . If λ 1 > 0,we easily see that has the mountain pass geometry. This case is simple, and we omit it here. In view of Remark 1.1, we arrange the eigenvalues (counted with multiplicity) of -+ V as and denote by e j the corresponding eigenfunction of λ j . Let E -= span{e 1 , . . . , e } and E + = (E -) ⊥ . From (V) we deduce that E = E -⊕ E + , where Eand E + are the negative and positive eigenspaces of the operator -+ V , and that dim where u = u -+ u + with u -∈ Eand u + ∈ E + . Then (·, ·) is an inner product on E. Therefore E is a Hilbert space with the norm · := √ (·, ·). We easily see that For any s ∈ [2, 2 * ], the imbedding E → L s (R N ) is continuous. Consequently, there exists a constant τ s > 0 such that where | · | s denotes the L s norm. We next recall some abstract critical point theorems, which will be used in the proofs of our main results.

Definition 2.1
Let E be a Banach space, and let ∈ C 1 (E, R). Given c ∈ R, a sequence {u n } ⊂ E is called a Cerami sequence of at level c (shortly, a (C) c sequence) if We say that satisfies the Cerami condition at level c (shortly, condition (C) c ) if every (C) c sequence of contains a convergent subsequence. If satisfies condition (C) c for every c ∈ R, then we say that satisfies the Cerami condition (shortly, condition (C)).
Obviously, condition (C) is weaker than the Palais-Smale condition. However, as was shown in [17], the deformation theorem is still valid under the Cerami condition. Thus we have the following theorems. Theorem 2.2 (Linking theorem [18]) Let E = E -⊕ E + be a Banach space with dim E -< ∞. Let R > r > 0, and let φ ∈ E + \{0}. Define If ∈ C 1 (E, R) satisfies condition (C) and then has a nontrivial critical point.
For the proofs of Theorems 1.2-1.3, we will use the following fountain theorem, which is a generalization of the classical fountain theorem of Bartsch [19] (see also [10]). For k ∈ N, let [20]) Suppose that the functional ∈ C 1 (E, R) is even and satisfies condition (C). Suppose that for every k ≥ k 0 for some constant k 0 > 0, there exist ρ k > r k > 0 such that

Theorem 2.3 (Fountain theorem
Then has a sequence of critical points {u k } such that (u k ) → ∞.
Proof It follows from (f 1 ) and (f 2 ) that, for given ε > 0, there is a constant C ε > 0 such that For u ∈ E + , we have where τ 2 and τ p are constants in (2.4). Let ε = 1 4τ 2 . Since p > 2, we can fix some r small enough such that inf u∈E + , u =r (u) > 0.