Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

By Xia Zhang, Binlin Zhang, Dušan D. Repovš. 2016

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Abstract This paper is concerned with the following fractional Schrodinger 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de93f9931a568dd3e320d5716488b9850077490e