Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials. (September 2016)

• Record Type: Journal Article
• Authors: Zhang, Xia; Zhang, Binlin; Repovš, Dušan
• Is Part Of: Nonlinear analysis. Volume 142(2016)
• Page Start: 48
• Page End: 68
• Publication Date: 2016-09
• Subjects: 35A15 -- 35J60 -- 46E35; Fractional Schrödinger equations -- Critical Sobolev exponent -- Ambrosetti–Rabinowitz condition -- Concentration compactness principle; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.04.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with the following fractional Schrödinger equations involving critical exponents: ( − Δ ) α u + V ( x ) u = k ( x ) f ( u ) + λ | u | 2 α ∗ − 2 u in R N, where ( − Δ ) α is the fractional Laplacian operator with α ∈ ( 0, 1 ), N ≥ 2, λ is a positive real parameter and 2 α ∗ = 2 N / ( N − 2 α ) is the critical Sobolev exponent, V ( x ) and k ( x ) are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity.