Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations

This paper is concerned with the existence of solutions to the following fractional Schrödinger type equations: (−󳵻)su + V(x)u = f(x, u), x ∈ RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing. By using variant Fountain theorems, a sufficient condition is obtained for the existence of infinitely many nontrivial high energy solutions.


Introduction and the Main Result
In this work, under the assumptions that  satisfies some weaker conditions than those in [1] and the primitive of  satisfies a more general superquadratic condition near infinity, we study the existence of infinitely many nontrivial high energy solutions to the following fractional Schrödinger equations: where  ∈ (0, 1),  > 2, and  :   ×  →  is a continuous function with some proper growth conditions.Here (−)  is the so-called fractional Laplacian operator of order  ∈ (0, 1), which can be characterized as (−)   = F −1 (|| 2 F), with F being the usual Fourier transform in   ((−)  is the pseudodifferential operator with symbol ||  ).
The nonlinear equations involving fractional Laplacian, which is a powerful tool for the descriptions of physics, probability, and finance, have attracted the attention of many researchers and have been successfully applied in various fields; see, for instance, [1,8,9] and the reference therein.
Our main theorem of this work reads as follows.

The Proof of Main Result
In this section,  ∈ (0, 1) is a fixed number.We denote by ‖ ⋅ ‖  the usual norm of the space   (  ).  ( = 1, 2, . ..) or  denotes some positive constants.
In the light of finite differences, the nonhomogeneous Sobolev space   (  ) is defined by It is a Hilbert space, when endowed with the scalar product given by The corresponding norm is therefore The space   (  ) is also denoted by the Fourier transform.Indeed, it is defined as follows: This space has a Hilbert structure when endowed with the scalar product so that the corresponding norm is To illustrate the relationship of the above two norms, let us start from the concept of Schwartz function S (is dense in   (  )), that is, the rapidly decreasing  ∞ function on   , which will be used later.If  ∈ S, the fractional Laplacian (−)  acts on  as where the symbol P.V. represents the principle value of the integral and the constant (, ) depends only on the space dimension  and on the order .We can write an integral expression for (, ) in the form In [2], the authors have proved where is the Gagliardo (semi)norm.Moreover, by the Plancherel formula in Fourier analysis, it is easy to show that Hence, the norms on   (  ) which was defined above are all equivalent.
For our problem (1), the Hilbert space  is defined by The inner product and the norm are defined as Under the assumptions ( 1 ) and ( 2 ), we have the following lemma due to [10].
The energy functional associated with problem (1) is defined by By Lemma 4 and conditions ( 1 ) and ( 2 ), we can prove that Φ is well defined and Φ ∈  1 () with Therefore weak solutions of (1) correspond to critical points of Φ.
( 3 ) There exists   >   > 0 such that Then where In order to use Theorem 6 to prove the main result, we define the functionals , , and Φ  on the working space  by for all  ∈  and  ∈ [1,2].We choose an orthonormal basis {  :  ∈ N} of  and let   = span{  } for all  ∈ N.
Obviously Φ 1 = Φ.In order to complete the proof of our theorem, we need the following lemmas.
Proof.From the definition of Φ  () and ( 1 ), there holds where  1 is a constant.Let Since  is compactly embedded into both  2 (  ) and   (  ), we have From ( 30) and (31), it follows that From (32), there exists a positive integer  1 such that For each  ≥  1 , let   = (16 1    ()) Up to a subsequence, if necessary, we can say that V  → V 0 in  for some V 0 ∈  since  is of finite dimension.Evidently, ‖V 0 ‖ = 1.By Lemma 4 and the equivalence of any two norms on , we conclude that Since V 0 ̸ = 0, we see that there exists a constant  0 > 0 satisfying For any  ∈ N, we define the sets Let Λ 0 = { ∈   : |V 0 ()| ≥  0 }.Then for sufficiently large , from (39) and (41), one can easily see that So that, for  large enough, we obtain This is impossible, and (37) holds.Since   is finite dimensional for each  ∈ N, we deduce from this and (37) that where So, by (45), (46), and ( 2 ), for any  ∈ N and  ∈ [1, 2], we have for all  ∈   with ‖‖ ≥   /  .Let We see that The proof of Lemma 8 is complete.
Furthermore, the proof of Lemma 7 shows that where   = max ∈  Φ 1 () and   fl If  0 = 0, then there exists a sequence {  } ⊂ [0, 1] such that For any  > 0, setting   = √ 4 This contradiction shows that  0 ̸ = 0 cannot hold and concludes the proof of Claim 2.
For each  ≥  1 , using the same arguments in the proof of Claim 1, one can also prove that the sequence {   } ∞ =1 has a strong convergent subsequence with the limit   being just the critical point of Φ = Φ 1 by Claim 2 and (57).Clearly, Φ(  ) ∈ [  ,   ] for all  ≥  1 .These imply that Φ have infinitely many nontrivial critical points of since   → +∞ as  → ∞.Therefore, problem (1) admits infinitely many nontrivial solutions and the proof of Theorem 1 is complete.
is somewhat weaker than the condition that (, )/|| is nondecreasing in  for all  ∈   .), ( 2 ), and ( 1 )-( 4 ) still hold for  and  provided that those hold for  and .Hence, we can assume without loss of generality that () ≥ 1 for all  ∈   in ( 1 ).