Existence and multiplicity results for the fractional magnetic Schrödinger equations with critical growth

Abstract

In this paper, we study the following critical fractional Schrödinger equations with the magnetic field: ε2s(−Δ)A/εsu+V(x)u=λf(|u|)u+|u|2s*−2uinRN, where ɛ and λ are positive parameters and V:RN→R and A:RN→RN are continuous electric and magnetic potentials, respectively. Under a global assumption on the potential V, by applying the method of Nehari manifold, Ekeland’s variational principle, and Ljusternick–Schnirelmann theory, we show the existence of ground state solution and multiplicity of non-negative solutions for the above equation for all sufficiently large λ and small ɛ. In this problem, f is only continuous, which allows us to study larger classes of nonlinearities.