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.
Keywordsfractional Laplacian subcritical nonlinearity fountain theorem weak solution
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).
- Shang X, Zhang J: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 2014, 27: 187-207. 10.1088/0951-7715/27/2/187MathSciNetView ArticleGoogle Scholar
- Shang X, Zhang J, Yang Y:On fractional Schrödinger equation in with critical growth. J. Math. Phys. 2013., 54: 10.1063/1.4835355Google Scholar
- Cabré X, Sire Y: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2014, 31: 23-53. 10.1016/j.anihpc.2013.02.001View ArticleGoogle Scholar
- Chang X: Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys. 2013., 54: 10.1063/1.4809933Google Scholar
- Chang X, Wang Z: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 2013, 26: 479-494. 10.1088/0951-7715/26/2/479MathSciNetView ArticleGoogle Scholar
- Secchi S:Ground state solutions for nonlinear fractional Schrödinger equations in . J. Math. Phys. 2013., 54: 10.1063/1.4793990Google Scholar
- Secchi, S: On fractional Schrödinger equations in ℝN without the Ambrosetti-Rabinowitz condition (2012) , [http://arxiv.org/abs/arXiv:1210.0755]Google Scholar
- Hua Y, Yu X: On the ground state solution for a critical fractional Laplacian equation. Nonlinear Anal. 2013, 87: 116-125. 10.1016/j.na.2013.04.005MathSciNetView ArticleGoogle Scholar
- Barrios B, Colorado E, de Pablo A, Sánchez U: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 2012, 252: 6133-6162. 10.1016/j.jde.2012.02.023View ArticleGoogle Scholar
- Autuori G, Pucci P:Elliptic problems involving the fractional Laplacian in . J. Differ. Equ. 2013, 255: 2340-2362. 10.1016/j.jde.2013.06.016MathSciNetView ArticleGoogle Scholar
- Dipierro S, Pinamonti A: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ. 2013, 255: 85-119. 10.1016/j.jde.2013.04.001MathSciNetView ArticleGoogle Scholar
- Di Nezza E, Palatucci G, Valdinoci E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 2012, 136: 521-573. 10.1016/j.bulsci.2011.12.004MathSciNetView ArticleGoogle Scholar
- Hajaiej H, Yu X, Zhai Z: Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms. J. Math. Anal. Appl. 2012, 396: 569-577. 10.1016/j.jmaa.2012.06.054MathSciNetView ArticleGoogle Scholar
- Wu X:Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in . Nonlinear Anal., Real World Appl. 2011, 12: 1278-1287. 10.1016/j.nonrwa.2010.09.023MathSciNetView ArticleGoogle Scholar
- Ye Y, Tang C:Multiple solutions for Kirchhoff-type equations in . J. Math. Phys. 2013., 54: 10.1063/1.4819249Google Scholar
- Zou W: Variant fountain theorems and their applications. Manuscr. Math. 2001, 104: 343-358. 10.1007/s002290170032View ArticleGoogle Scholar
- Struwe M: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, New York; 2000.Google Scholar
- Ambrosetti A, Rabinowitz P: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
- Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleGoogle Scholar
- Zhang Q, Xu B: Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential. J. Math. Anal. Appl. 2011, 377: 834-840. 10.1016/j.jmaa.2010.11.059MathSciNetView ArticleGoogle Scholar
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