Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials. (September 2016) - Zhang, Xia; Zhang, Binlin; Repovš, Dušan
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.