In this paper we are concerned with the fractional Schrödinger equation , , where , , stands for the fractional Laplacian of order α, V is a positive continuous potential, and f is a continuous subcritical nonlinearity. We obtain the existence of infinitely many weak solutions for the above problem by the fountain theorem in critical point theory.
Keywords:fractional Laplacian; subcritical nonlinearity; fountain theorem; weak solution
In this paper we consider the following fractional Schrödinger equation:
The nonlinearity is a continuous function, satisfying the subcritical condition.
(H1) There exist , and such that
Recently, there have appeared plenty of works on the fractional Schrödinger equations; for example, see [–] and the references therein. In [], Shang and Zhang considered the critical fractional Schrödinger equation], Shang et al. considered the existence of nontrivial solutions for (1.2) with , where .
In [], Hua and Yu studied the critical fractional Laplacian equation
In [], Secchi investigated the existence of radially symmetric solutions for (1.1) replacing by , where g satisfies the following conditions.
(g1) is of class for some , and odd,
(g4) for some such that .
Inspired by the mentioned papers, we first establish a compact embedding lemma via a fractional Gagliardo-Nirenberg inequality. Then by virtue of the fountain theorem in critical point theory, we get two existence results of infinitely many weak solutions for (1.1).
In this section we offer some preliminary results which enable us to obtain the main existence theorems. First, we collect some useful facts of the fractional order Sobolev spaces.
For any , the fractional Sobolev space is defined by
Indeed, the fractional Laplacian can be viewed as a pseudo-differential operator of symbol , as stated in the following.
Letandbe the fractional Laplacian operator defined by (2.1). Then for any,
Now we can see that an alternative definition of the fractional Sobolev space via the Fourier transform is as follows:
In this paper, in view of the presence of potential , we consider its subspace], E is a Hilbert space with the inner product
We say that is a weak solution of (1.1), if
Eis continuously embedded intoforand compactly embedded intofor.
Eis compactly embedded intoforwith.
By [], we know E is compactly embedded into , i.e., if there exists a sequence and such that weakly in E, passing to a subsequence if necessary, we have strongly in . Therefore, we only consider . In order to do this, we need the following fractional Gagliardo-Nirenberg inequality, see [, Corollary 2.3]. Let , , and . Then
The functional associated with (1.1) is defined by
Now, we list our assumptions on f and F.
(H2) uniformly for .
(H3) There exist and such that and
(H4) for .
(H5) There exist , such that
(H6) for all .
(1) Let , for all and . Then (H1), (H2), (H4), and (H6) hold. Moreover, we easily have and , so (H3) is satisfied.
However, we can see that does not satisfy the Ambrosetti-Rabinowitz condition (see [, ()]):
(AR) there is a constant such that
Indeed, is impossible for all and .
(2) Let and , for all and . Then (H1), (H2), (H4), and (H6) hold. Moreover, from , and (H5) holds.
Note that from Theorem 4 in [] we have (H4) and (H5) imply (H2).
(see [, Lemma 1])
Let (V) and (H1) hold. Thenand its derivative
Let be a real Banach space, . We say that J satisfies the () condition if any sequence such that and as has a convergent subsequence.
Let X be a Banach space equipped with the norm and , where for any . Set and .
Letbe a real reflexive Banach space, satisfies the () condition for anyandJis even. If for each sufficiently large, there existsuch that the following conditions hold:
then the functionalJhas an unbounded sequence of critical values, i.e., there exists a sequencesuch thatandas.
In the following, we will introduce a variant fountain theorem by Zou []. Let X and the subspace and be defined above. Consider the following -functional defined by
If the functionalsatisfies
(T1) maps bounded sets to bounded sets uniformly for, and, moreover, for all,
(T2) for all; moreover, oras,
(T3) there existsuch that
As mentioned in [], E is a Hilbert space. Let be an orthonormal basis of E and define , , and , . Clearly, with for all .
Existence of weak solutions for (1.1)
Assume that (V), (H1)-(H4), and (H6) hold. Then (1.1) has infinitely many weak solutionssatisfying
We first prove that J satisfies the () condition for any . Let be a () sequence, i.e.,
In what follows, we shall show that is bounded. Otherwise, up to a subsequence, is unbounded in E, and we may assume that as . We define the sequence by , . Clearly, and for any n. Going over, if necessary, to a subsequence, we may assume that
Now the sequence is bounded, as required. Next, we verify that has a convergent subsequence. Without loss of generality, we assume that
Clearly, by (H6). It remains to prove that the conditions (i) and (ii) of Lemma 2.8 hold. Let with , where is defined in Remark 2.10. Then by Lemma 3.8 of [], as for the fact that .
Now for with , we obtain
Next we shall prove that, for any finite dimensional subspace , we have
We prove that there exists ( is determined in (H5)) such that
(V′): , , where in (3.12), β in (H5).
Especially, by (V′), we obtain
Now, we define a class of functionals on E by
Assume that (V′), (H1), and (H4)-(H6) hold. Then (1.1) possesses infinitely many weak solutions.
We first prove that there exist a positive integer and two sequences as such that
Step 1. We claim that (3.14) is true.
By (3.6) and (H4) we have
Step 2. We show that (3.15) is true.
We apply the method in Lemma 2.6 of [] to verify the claim. First, we prove that there exists such that
Step 3. Clearly, implies that maps bounded sets to bounded sets uniformly for . In view of (H6), for all . Thus the condition (T1) of Lemma 2.9 holds. Besides, the condition (T2) of Lemma 2.9 holds for the fact that as and since . Evidently, Step 1 and Step 2 imply that the condition (T3) of Lemma 2.9 also holds for all . Consequently, Lemma 2.9 implies that for any and a.e. , there exists a sequence such that
Claim 1. possesses a strong convergent subsequence in E, a.e. and . In fact, by the boundedness of , passing to a subsequence, as , we may assume in E. By the method of Theorem 3.1, we easily prove that strongly in E.
Thus, for each , we can choose such that for the sequence we have obtained a convergent subsequence, and passing again to a subsequence, we may assume
Claim 2. is bounded in E and has a convergent subsequence with the limit for all . For convenience, we set for all . Consequently, (3.12) and (H5) imply that
Hence, passing to the limit in (3.24), we see and , and . Since as , we get infinitely many nontrivial critical points of . Therefore (1.1) possesses infinitely many nontrivial solutions by Lemma 2.9. This completes the proof. □
The authors declare that they have no competing interests.
WD and JX obtained the results in a joint research. All the authors read and approved the final manuscript.
The authors are highly grateful for the referees’ careful reading of this paper and their comments. Supported by the NNSF-China (11371117), Shandong Provincial Natural Science Foundation (ZR2013AM009) and Hebei Provincial Natural Science Foundation (A2012402036).
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