Solutions of stationary Kirchhoff equations involving nonlocal operators with critical nonlinearity in RN

Abstract

In this paper, we consider the existence and multiplicity of solutions for fractional Schrödinger equations with critical nonlinearity in R^N. We use the fractional version of Lions' second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS_c) condition holds locally. Under suitable assumptions, we prove that it has at least one solution and, for any m ∈ N, it has at least m pairs of solutions. Moreover, these solutions can converge to zero in some Sobolev space as ε → 0.