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

Abstract

This paper is concerned with the existence of solutions to the following fractional Schrödinger type equations: , where the primitive of the nonlinearity is of superquadratic growth near infinity in and the potential 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.

1. 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 , , and is a continuous function with some proper growth conditions. Here is the so-called fractional Laplacian operator of order , which can be characterized as , with being the usual Fourier transform in ( is the pseudodifferential operator with symbol ).

We need the following assumptions on :

() is bounded from below.

() For any , there exists such thatwhere meas denotes the Lebesgue measure in .

More recently, Di Nezza et al. [2] have proved that can be reduced to the standard Laplacian as . When , problem (1) is the classical Schrödinger equationsOver the past several decades, the existence and multiplicity of nontrivial solutions to problem (3) have been studied extensively by numerous researches using a variety of methods and techniques; see, for instance, [3–7].

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.

We need the following conditions on and its primitive :

()   and there exist constant and such that where denotes the critical Sobolev exponent, that is, .

()  , for all , and uniformly on .

() There exists a constant such thatwhere .

, .

Our main theorem of this work reads as follows.

Theorem 1. Suppose that , , and hold. Then problem (1) possesses infinitely many nontrivial solutions.

Remark 2. Conditions like and have been given in [1], but there is required. As shown in [10], the condition due to [11] is somewhat weaker than the condition that is nondecreasing in for all .

Remark 3. It follows from that there exists a constant such that for all . Let for all and consider the following new fractional Schrödinger equationsThen, problem (1) and problem (6) are equivalent. Evidently, the hypotheses , and still hold for and provided that those hold for and . Hence, we can assume without loss of generality that for all in .

2. The Proof of Main Result

In this section, is a fixed number. We denote by the usual norm of the space . 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 (is dense in ), that is, the rapidly decreasing function on , which will be used later. If , the fractional Laplacian acts on as where the symbol 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 byThe inner product and the norm are defined as

Under the assumptions and , we have the following lemma due to [10].

Lemma 4. The Hilbert space is compactly embedded in for .

Definition 5. A weak solution to problem (1) is a function such that

The energy functional associated with problem (1) is defined by By Lemma 4 and conditions and , we can prove that Φ is well defined and withTherefore weak solutions of (1) correspond to critical points of .

Let be a Banach space with the norm and let be a sequence of subspace of with for each . Further, , the closure of the direct sum of all . Set for . Consider a family of functionals defined by

The following variant Fountain theorem comes from Zou [12].

Theorem 6. Assume that the functional defined above satisfies the following:
   maps bounded sets to bounded sets uniformly for , and for all .
   for all or as .
There exists such thatThenwhere . Moreover, for almost every , there exists a sequence such that

In order to use Theorem 6 to prove the main result, we define the functionals , , and on the working space by for all and . We choose an orthonormal basis of and let for all . Obviously . In order to complete the proof of our theorem, we need the following lemmas.

Lemma 7. Assume that , , and hold. Then there exists a positive integer and a sequence as such thatwhere for all .

Proof. From the definition of and , there holdswhere is a constant. LetSince is compactly embedded into both and , we haveFrom (30) and (31), it follows thatFrom (32), there exists a positive integer such that For each , let . Then as , since . It is immediate to check thatHence, the proof of Lemma 7 is complete.

Lemma 8. Assume that , , , and hold. Then for the positive integer and the sequence obtained in Lemma 7, there exist for each , such thatwhere for all .

Proof. To begin with, we carry out that for any finite dimensional subspace there exists a constant such thatAssume for contradiction that, for any , there exists such thatLet for all . Then for all , andUp to a subsequence, if necessary, we can say that in for some since is of finite dimension. Evidently, . By Lemma 4 and the equivalence of any two norms on , we conclude thatSince , we see that there exists a constant satisfyingFor any , we define the setsLet . Then for sufficiently large , from (39) and (41), one can easily see thatSo that, for large enough, we obtain This is impossible, and (37) holds.
Since is finite dimensional for each , we deduce from this and (37) thatwhere for all and . By , for each , there exists a constant such thatSo, by (45), (46), and , for any and , we havefor all with . Let We see that The proof of Lemma 8 is complete.

Proof of Theorem 1. Now the assumption and the definition of imply that maps bounded sets to bounded sets uniformly for . From , we obtain for all . Therefore, the condition of Theorem 6 holds. Clearly, the condition holds by the definitions of and . Moreover, Lemmas 7 and 8 imply that the condition holds for all , where is given in Lemma 7. Thus, it follows from Theorem 6 that, for each , a.e. , there exists a sequence such thatwhere with and .

Furthermore, the proof of Lemma 7 shows thatwhere and as .

For each , (50) shows that we can choose (depending on ) and can get the corresponding sequences satisfying

We prove the following 2 claims.

Claim 1. For each given above, the sequence has a convergent subsequence.
Throughout this paragraph, for the simplicity of notation, we denote by throughout this paragraph. Indeed, by (53), we may letfor some . With the aid of Riesz Representation Theorem, and can be written as and , respectively. HenceFrom Lemma 4, (53), (54), and (55), we see immediately that the right-hand side of (55) converges strongly in , so that in , and then the proof of Claim 1 is complete.
Therefore, by Claim 1, up to subsequences, we may assumeand hence from (50) and (52) we deduce that

Claim 2. For each , the sequence in (56) is bounded.
Throughout this paragraph, for notational simplicity, we will write for all .
Otherwise, up to subsequences, we may assumeBy Lemma 4 and (58), we can prove thatUp to subsequences, if necessary, we haveIf , then there exists a sequence such thatFor any , setting , by (59) and , we obtain thatThus, for large enough, there holds These yield immediately . Since and in (57), we obtain in (61) for large enough, which shows thatHence, from (64) and , we deduce that where is given in . By (57), this is impossible.
Since , of course has positive measure. Equation (58) shows thatFrom (58), (66), (), and Fatou’s lemma, it is easy to see thatThis contradiction shows that cannot hold and concludes the proof of Claim 2.

For each , using the same arguments in the proof of Claim 1, one can also prove that the sequence has a strong convergent subsequence with the limit being just the critical point of by Claim 2 and (57). Clearly, for all . These imply that 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.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11471187 and 11571197).