Abstract
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 solutionIntroduction
In this paper we consider the following fractional Schrödinger equation:
where , , stands for the fractional Laplacian of order α, and the potential is a continuous function satisfying(V) .
The nonlinearity is a continuous function, satisfying the subcritical condition.
(H1) There exist , and such that
where is the fractional critical exponent.Recently, there have appeared plenty of works on the fractional Schrödinger equations; for example, see [[1]–[11]] and the references therein. In [[1]], Shang and Zhang considered the critical fractional Schrödinger equation
where ε and λ are positive parameters, V and f satisfy (V) and (H1), respectively. They obtained the result that (1.2) has a nonnegative ground state solution and investigated the relation between the number of solutions and the topology of the set where V attains its minimum for all sufficiently large λ and small ε. In [[2]], Shang et al. considered the existence of nontrivial solutions for (1.2) with , where .In [[8]], Hua and Yu studied the critical fractional Laplacian equation
where , , is a bounded domain. They obtained the result that the problem (1.3) possesses a nontrivial ground state solution for any .In [[7]], 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,
(g2) ,
(g3) ,
(g4) for some such that .
Inspired by the mentioned papers, we first establish a compact embedding lemma via a fractional GagliardoNirenberg 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).
Preliminary results
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
endowed with the norm where is the socalled Gagliardo seminorm of u. Let be the Schwartz space of rapidly decaying functions in , for any and , and let be defined as The symbol P.V. stands for the Cauchy principal value, and is a dimensional constant that depends on N and α, precisely given by .Indeed, the fractional Laplacian can be viewed as a pseudodifferential operator of symbol , as stated in the following.
Lemma 2.1
(see [[12]])
Letandbe the fractional Laplacian operator defined by (2.1). Then for any,
where ℱ is the Fourier transform, i.e.,Now we can see that an alternative definition of the fractional Sobolev space via the Fourier transform is as follows:
It can be proved that As a result, the norms on , are all equivalent.In this paper, in view of the presence of potential , we consider its subspace
We define the norm in E by where . Moreover, by [[6]], E is a Hilbert space with the inner product Note that by (2.2) and (2.3), together with the condition (V), we know that is equivalent to the norm The corresponding inner product is Throughout out this paper, we will use the norm in E.Definition 2.2
We say that is a weak solution of (1.1), if
Lemma 2.3
Eis continuously embedded intoforand compactly embedded intofor.
Lemma 2.4
Eis compactly embedded intoforwith.
Proof
By [[4]], 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 GagliardoNirenberg inequality, see [[13], Corollary 2.3]. Let , , and . Then
with Note that the dimension , we can take , and then whence as . Consequently, from (2.5) we have Furthermore, note that (2.4); we see that Then by (2.6) and , we find Therefore, E is compactly embedded into for with , as required. This completes the proof. □The functional associated with (1.1) is defined by
where .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 .
Remark 2.5
(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 AmbrosettiRabinowitz condition (see [[6], ()]):
(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 [[14]] we have (H4) and (H5) imply (H2).
Lemma 2.6
(see [[15], Lemma 1])
Let (V) and (H1) hold. Thenand its derivative
Moreover, the critical points ofJare weak solutions of (1.1).To complete the proofs of our theorems, we need the following critical point theorems in [[16]–[19]].
Definition 2.7
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 .
Lemma 2.8
Letbe a real reflexive Banach space, satisfies the () condition for anyandJis even. If for each sufficiently large, there existsuch that the following conditions hold:
(i) as,
(ii) ,
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 [[16]]. Let X and the subspace and be defined above. Consider the following functional defined by
Lemma 2.9
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
thenwhereand. Moreover, for a.e. , there exists a sequencesuch thatRemark 2.10
As mentioned in [[6]], 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)
Theorem 3.1
Assume that (V), (H1)(H4), and (H6) hold. Then (1.1) has infinitely many weak solutionssatisfying
Proof
We first prove that J satisfies the () condition for any . Let be a () sequence, i.e.,
which implies thatIn 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
Suppose that in E. Dividing by in both sides of (2.7), noting that , we obtain On the other hand, denote , by (H2), for all , and we find If , using Fatou’s lemma, we obtain This contradicts (3.4). Hence, has zero measure, i.e., a.e. in . Let such that Then we claim is bounded. If , ; if , . Therefore, is bounded when . If for n large enough Consequently, by (H3), noting that (3.1) and (3.2) hold, we have where is a positive constant. But fixing any , we let . Note that from (H1) we see that there exist , such that Then by (3.3) we have Then for n large enough, This also contradicts (3.5).Now the sequence is bounded, as required. Next, we verify that has a convergent subsequence. Without loss of generality, we assume that
Combining this with (H1) and the Hölder inequality, we see Consequently, with the fact that when . Therefore, we prove that J satisfies the () condition for any .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 [[19]], as for the fact that .
Now for with , we obtain
Hence,Next we shall prove that, for any finite dimensional subspace , we have
Suppose the contrary. For some sequence with , there is a such that for all . Put and then . Up to a subsequence, assume that weakly in E. Since , in E, a.e. on , and . Denote , then and for a.e. , . It follows from (2.7) that But, for large n, on account of F being nonnegative, (H2) and Fatou’s Lemma enable us to obtain as . This contradicts (3.10). Consequently, (3.9) holds, as required. Note that in Remark 2.10, and there exist positive constants such that Combining this and (3.8), we can take , and thus . Until now, we have proved the functional J satisfies all the conditions of Lemma 2.8. Hence, J has an unbounded sequence of critical values, i.e., there exists a sequence such that and as . This completes the proof. □We prove that there exists ( is determined in (H5)) such that
Indeed, by (3.6) we see This, together with (H1), implies that Clearly, (3.12) holds true with . In the following theorem, we make the following assumption instead of (V):(V′): , , where in (3.12), β in (H5).
Especially, by (V′), we obtain
Now, we define a class of functionals on E by
It is easy to know that for all and the critical points of correspond to the weak solutions of problem (1.1). Note that , where J is the functional defined in (2.7).Theorem 3.2
Assume that (V′), (H1), and (H4)(H6) hold. Then (1.1) possesses infinitely many weak solutions.
Proof
We first prove that there exist a positive integer and two sequences as such that
where and are defined in Remark 2.10.Step 1. We claim that (3.14) is true.
By (3.6) and (H4) we have
Since with , and from Theorem 3.1 we have Let as . Then there exists such that , . Therefore,Step 2. We show that (3.15) is true.
We apply the method in Lemma 2.6 of [[20]] to verify the claim. First, we prove that there exists such that
There would otherwise exist a sequence such that For each , let . Then , and Passing to a subsequence if necessary, we may assume in E for some since is of finite dimension. We easily find . Consequently, there exists a constant such that Indeed, if not, then we have which implies This leads to , contradicting . In view of and the equivalence of any two norms on , we have For every , denote and , where is defined by (3.20). Then for n large enough, by (3.20), we see Consequently, for n large enough, we find This contradicts (3.22). Therefore, (3.17) holds. For the ε given in (3.17), we let Then by (3.17), we find As is well known, (H5) implies (H2), and hence for any , there is a constant such that where ε is determined in (3.17). Therefore, Now for any , if we take , so is large enough, we haveStep 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
where Furthermore, we easily have , , where and as .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
Thus we obtainClaim 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
Therefore, is bounded in E. By Claim 1, we see that has a convergent subsequence, which converges to an element for all .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. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WD and JX obtained the results in a joint research. All the authors read and approved the final manuscript.
Acknowledgements
The authors are highly grateful for the referees’ careful reading of this paper and their comments. Supported by the NNSFChina (11371117), Shandong Provincial Natural Science Foundation (ZR2013AM009) and Hebei Provincial Natural Science Foundation (A2012402036).
References

Shang, X, Zhang, J: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity. 27, 187–207 (2014). Publisher Full Text

Shang, X, Zhang, J, Yang, Y: On fractional Schrödinger equation in with critical growth. J. Math. Phys.. 54, (2013). Publisher Full Text

Cabré, X, Sire, Y: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 31, 23–53 (2014). Publisher Full Text

Chang, X: Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys.. 54, (2013). Publisher Full Text

Chang, X, Wang, Z: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 26, 479–494 (2013). Publisher Full Text

Secchi, S: Ground state solutions for nonlinear fractional Schrödinger equations in . J. Math. Phys.. 54, (2013). Publisher Full Text

Hua, Y, Yu, X: On the ground state solution for a critical fractional Laplacian equation. Nonlinear Anal.. 87, 116–125 (2013). Publisher Full Text

Barrios, B, Colorado, E, de Pablo, A, Sánchez, U: On some critical problems for the fractional Laplacian operator. J. Differ. Equ.. 252, 6133–6162 (2012). Publisher Full Text

Autuori, G, Pucci, P: Elliptic problems involving the fractional Laplacian in . J. Differ. Equ.. 255, 2340–2362 (2013). Publisher Full Text

Dipierro, S, Pinamonti, A: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ.. 255, 85–119 (2013). Publisher Full Text

Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math.. 136, 521–573 (2012). Publisher Full Text

Hajaiej, H, Yu, X, Zhai, Z: Fractional GagliardoNirenberg and Hardy inequalities under Lorentz norms. J. Math. Anal. Appl.. 396, 569–577 (2012). Publisher Full Text

Wu, X: Existence of nontrivial solutions and high energy solutions for SchrödingerKirchhofftype equations in . Nonlinear Anal., Real World Appl.. 12, 1278–1287 (2011). Publisher Full Text

Ye, Y, Tang, C: Multiple solutions for Kirchhofftype equations in . J. Math. Phys.. 54, (2013). Publisher Full Text

Zou, W: Variant fountain theorems and their applications. Manuscr. Math.. 104, 343–358 (2001). Publisher Full Text

Struwe, M: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, New York (2000)

Ambrosetti, A, Rabinowitz, P: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text

Zhang, Q, Xu, B: Multiplicity of solutions for a class of semilinear Schrödinger equations with signchanging potential. J. Math. Anal. Appl.. 377, 834–840 (2011). Publisher Full Text