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
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.
Let and be 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
We say that is a weak solution of (1.1), if
Eis continuously embedded into for and compactly embedded into for .
Eis compactly embedded into for with .
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. Then and 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 .
Let be a real reflexive Banach space, satisfies the ( ) condition for any andJis even. If for each sufficiently large , there exist such that the following conditions hold:
(i) as ,
then the functionalJhas an unbounded sequence of critical values, i.e., there exists a sequence such that and as .
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 functional satisfies
(T1) maps bounded sets to bounded sets uniformly for , and, moreover, for all ,
(T2) for all ; moreover, or as ,
(T3) there exist such 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 solutions satisfying
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).
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 Gagliardo-Nirenberg 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 Schrödinger-Kirchhoff-type equations in . Nonlinear Anal., Real World Appl.. 12, 1278–1287 (2011). Publisher Full Text
Ye, Y, Tang, C: Multiple solutions for Kirchhoff-type 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
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 sign-changing potential. J. Math. Anal. Appl.. 377, 834–840 (2011). Publisher Full Text