Infinitely many weak solutions for a fractional Schrödinger equation
© Dong et al.; licensee Springer 2014
Received: 22 February 2014
Accepted: 13 June 2014
Published: 12 July 2014
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.
The nonlinearity is a continuous function, satisfying the subcritical condition.
(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.
Indeed, the fractional Laplacian can be viewed as a pseudo-differential operator of symbol , as stated in the following.
E is continuously embedded intoforand compactly embedded intofor.
E is compactly embedded intoforwith.
Now, we list our assumptions on f and F.
(H2) uniformly for .
(H4) for .
(H6) for all .
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 [, ()]):
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 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 .
then the functional J has an unbounded sequence of critical values, i.e., there exists a sequencesuch thatandas.
If the functional satisfies
(T1) maps bounded sets to bounded sets uniformly for, and, moreover, for all,
(T2) for all; moreover, oras,
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)
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 .
Assume that (V′), (H1), and (H4)-(H6) hold. Then (1.1) possesses infinitely many weak solutions.
Step 1. We claim that (3.14) is true.
Step 2. We show that (3.15) is true.
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.
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 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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