Multiple entire solutions of fractional Laplacian Schrödinger equations

Abstract: We consider the semi-linear fractional Schrödinger equation (−∆)su + V(x)u = f (x, u), x ∈ RN , u ∈ H s(RN), where both V(x) and f (x, u) are periodic in x, 0 belongs to a spectral gap of the operator (−∆)s + V and f (x, u) is subcritical in u. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of f , which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions.


Introduction
We consider the following semi-linear fractional Schrödinger equation Here P.V. stands for the Cauchy principal value and the positive constant C(N, s) depends only on N and s, which is not essential in our problem and we will omit it for simplicity of notation. For fractional Laplacian operators and fractional spaces, the reader can refer to [4] and [8]. The authors in [1] raise the following assumption (AR) of the nonlinear term to study a semi-linear elliptic boundary value problem (AR): There exists µ > 2 such that 0 < µF( By a direct integration of (AR), one can deduce the existence of positive constants A, B such that F(x, t) ≥ A|t| µ − B for any t ∈ R. We first recall some main results of the particular case s = 1, namely the standard Laplacian case of (1.1). The existence of a nontrivial solution to (1.1) has been obtained in [2,3,7,15,23,29,31] under (AR) and some other standard assumptions of f . The authors of [21] introduce the following more natural super-quadratic condition to replace (AR) (SQ) : lim |t|→∞ and obtain the existence of nontrivial solutions of (1.1) under (SQ) and some other standard assumptions of f by imposing some compact conditions on the potential function V.After that, condition (SQ) is also used in many papers, see [5,9,19,20,25,30,32,33]. In the definite cases where σ(−∆ + V) ⊂ (0, ∞), [20] obtains a ground state solution via a Nehari type argument for (1.1). The corresponding energy functional of (1.1) in the case s = 1 is Recall that E = E − ⊕ E + corresponds to the spectral decomposition of −∆ + V with respect to the positive and negative part of the spectrum, and u = u − + u + ∈ E − ⊕ E + . (See Section 2 for more details.) The following set has been introduced in [22] By definition, M contains all nontrivial critical points of I. The authors of [25] develop an ingenious approach to find ground state solutions of (1.1). Their approach transforms, by a direct and simple reduction, the indefinite variational problem to a definite one, resulting in a new minimax characterization of the corresponding critical value. More precisely, they establish the following two propositions by introducing the strictly monotonicity assumption (Mo) |t| is strictly increasing on (−∞, 0) and on (0, ∞). In [24], the author obtains nontrivial and ground solutions of Schrödinger equation (1.1) under weaker conditions than those of [25]. Via deformation arguments jointed with the notion of Cerami sequence(See Section 2 for concrete definition), [9] establishes the following proposition. Proposition 1.3. ( [9]) Assume that V and f satisfy (V 1 ), (F 1 ), (F 2 ), (S Q) and the following condition G(x, t) → +∞ as |t| → ∞ uniformly in x ∈ R N , and there exists c 2 , r 0 > 0 and ν > max{1, N 2 } such that f (x, t) t ν ≤ c 2 G(x, t) for all |t| ≥ r 0 and x ∈ R N .
Then (1.1) has a nontrivial solution. If, in addition, f (x, t) is odd in t, then (1.1) admits infinitely many pairs geometrically distinct solutions ±u.
In [26], the author obtains the existence of ground state solutions by non-Nehari manifold method for (1.1) with periodic and asymptotically periodic potential function V, under (SQ) and some other standard assumptions of f . Recently, under the weaker super-quadratic condition (SQ) and some other standard assumptions of f , the authors in [28] obtain the existence of nontrivial solution for (1.1) with periodic and non-periodic potential function V. The authors of [27] further obtain the existence of ground state solutions and infinitely many geometrically distinct solutions under (SQ) and non-strictly monotonicity condition (Mo) , and as a compensation, additional condition (F 0 ) or (F 0 ) is necessary. These conditions are defined as follows.
(SQ) : There exists a domain Ω ⊆ R N , such that lim |t|→∞ The following propositions are established in [27].
odd in t, then (1.1) admits infinitely many pairs geometrically distinct solutions ±u. Existence of nontrivial solutions to a strongly indefinite Choquard equation with critical exponent is obtained in [13]. We also want to mention that the existence and some quantitative properties of periodic solutions of fractional equation with double well potential in one-dimensional case are established in [10,12,14].
In this paper we will generalize the existence of nontrivial solutions in [9] by replacing (SQ), (DL) by the weaker conditions (F 3 ) and (F 4 ), and generalize the existence of infinitely many geometrically different solutions in [27] by replacing (Mo) , (SQ) by the weaker conditions (F 3 ) and (F 4 ).
The corresponding energy functional of (1.1) is It is easy to verify that Φ s is C 1 (H s (R N ), R) and From (F 1 ) and (F 2 ), for any given > 0, there exists C > 0 such that The following conditions are required to arrive at our results.
An example that satisfies the conditions (F 1 )-(F 4 ), but does not satisfy (SQ) is The followings are our main results.

Preliminaries
and |A s | be the spectral family and the absolute value of A s respectively, and |A s | For any u ∈ E s , it is easy to see that u = u − + u + and Under assumption (V s ), we can define an inner product and the corresponding norm By (V s ), E s = H s (R N ) with equivalent norms. Therefore E s embeds continuously in L p (R N ) for all 2 ≤ p ≤ 2 * s . Hence, there exists constant γ p > 0 such that u L p ≤ γ p u . By the definitions of Λ s and E + s we also have u 2 ≥ Λ s u 2 L 2 for any u ∈ E + s .
Let X be a real Hilbert space. Recall that a functional ψ ∈ C 1 (X, R) is said to be weakly sequentially lower semi-continuous if for any u n u in X one has ψ(u) ≤ lim inf n→∞ ψ(u n ), and ψ is said to be weakly sequentially continuous if lim n→∞ ψ (u n ), v = ψ (u), v for each v ∈ X. Let Ψ(u) = R N F(x, u) dx. By (F 1 )-(F 3 ), one can easily get that Ψ is weakly sequentially lower semi-continuous and Ψ is weakly sequentially continuous.
We introduce the following generalized linking theorem.
A sequence {u n } is called Cerami sequence (denoted also as (C e ) c -sequence) of the energy functional ϕ, if there exists constant c such that ϕ(u n ) → c and ϕ (u n ) (1 + u n ) → 0.

The existence of nontrivial solutions
In this section, we will prove Theorem 1.1 by applying Lemma 2.1.
shows that for any given > 0 the inequality |F(x, u)| ≤ |u| 2 holds for small |u|. So | R N F(x, u) dx| ≤ u 2 , and the conclusion follows if ρ is sufficiently small. Since Υ s (A s ) is purely continuous, for any given µ > Λ s , the space Y µ := (Υ s ) µ − (Υ s ) 0 L 2 is infinitely dimensional, where ((Υ s ) λ ) λ∈R denotes spectrum family of A s . By (2.5), for Λ s < µ < 2Λ s we have Suppose that (V s ) and (F 3 ) are satisfied. Then for any e ∈ Y µ , sup Φ s (E − s ⊕R + e) < ∞ and there is r e > 0 such that Φ s (u) ≤ 0 for any u ∈ E − s ⊕ R + e, u ≥ r e . Proof. Arguing indirectly, assume that for some sequence {u n } ⊆ E − s ⊕ R + e, e ∈ Y µ with u n → ∞ and Φ s (u n ) > 0. Setting v n = u n u n , then v n = 1. Hence there exists Here the strong convergence of {v + n } is due to the reason that R + e is finite dimensional. We have We claim that v 0. Suppose not, then where the first inequality use the fact that F ≥ 0. The above relation gives v − n → 0, hence 1 = v n 2 = v + n 2 + v − n 2 → 0, which is a contradiction, and the claim is true.
Hence, there exists a bounded set Ω ⊆ R N such that where the last inequality use the assumption (F 3 ). Here |Ω| denotes Lebesgue's measure of Ω. By the weak lower-semi continuity of the norm, we have a contradiction follows.
Lemma 3.4. Under the assumptions of (V s ), (F 2 ) and (F 4 ), any (C e ) c -sequence is bounded.
Passing to a subsequence, we may assume the existence of k n ∈ Z N such that Let us define v n (x) = u n (x + k n ), then (3.10) Since V(x) is 1-periodic in each of x 1 , x 2 , . . . , x N , then u n = v n and Passing to a subsequence, we have v n v in E s . Obviously, (3.10) implies thatv 0. By a standard argument, one has Φ s (v) = 0. We complete the proof of Theorem 1.1.

The existence of infinite many solutions
In this section, we give the proof of Theorem 1.2. We need to introduce some notations. For Proof. i) Assume b 1 = 0, then there is a sequence {u n } ⊂ K with u n → 0, and This and (1.2) yield that u n 2 ≤ u n 2 By this and Sobolev imbedding theorem we deduce u n 2−p ≤C , which contradicts with the assumption u n → 0.
ii) By we have b 2 ≥ 0. Assume b 2 = 0, then there is a sequence {u n } ⊂ K such that Φ s (u n ) → 0. Since {u n } is a (C e ) c=0 sequence, by lemma 3.4, u n is bounded. By Sobolev imbedding theorem, there exists C ≥ 0 such that u n 2 L 2 ≤ C. Note that By (4.3), for any 0 < a < b, we have as n → ∞. Similar as the derivation of (3.5), for any p ∈ (2, 2 * s ) we have Proof. The proof is rather similar as that of lemma 4.2 of [9]. The main differences are that the space E and the energy functional Φ there are replaced by E s and Φ s respectively. We omit it here.
Remark Theorem 1.1 shows that equation (1.1) has a nontrivial solution u ∈ E s , and so K ∅. We choose a subset Q of K such that Q = −Q (here −Q := {w : −w ∈ Q}) and each orbit O(u) ⊆ K has a unique representative in Q. It suffices to show that the set Q is infinite, so from now on we assume by contradiction that Q is a finite set.   For any c ≥ b 2 , as in [6,7,9,13,15], we let Plainly Q c ⊆ Q c for any c ≥ c ≥ b 2 . Following the argument of Proposition 1.55 in [6], we have the next lemma. To prove Theorem 1.2, we need to establish the following lemmas 4.6-4.10. The proofs of lemmas 4.6-4.10 are rather similar to the the proofs of lemmas 4.6-4.10 in [27]. The main differences are that the space E and the energy functional Φ there are replaced by E s and Φ s respectively. We omit them here.
Proof of Theorem 1.2 We can prove Theorem 1.2 by applying Lemmas 4.6-4.10. Since the proof is rather similar that of the second part of theorem 1.4 and 1.5 in [27], we omit it here.

Conclusions
In this paper we obtained the existence of nontrivial solutions to a semi-linear fractional Schrödinger equation by using a generalized linking theorem. Based on this existence results, infinitely many geometrically distinct solutions are further established under weaken conditions of the nonlinearity of the equation.