A Fractional Magnetic System with Critical Nonlinearities

Abstract

In the present paper, we investigate a fractional magnetic system involving critical concave–convex nonlinearities with Laplace operators. Specifically, (−Δ)Asu1=λ1|u1|q−2u1 + 2α1α1+β1|u1|α1−2u1|u2|β1 in Ω, (−Δ)Asu2=λ2|u2|q−2u2+2β1α1+β1|u2|β1−2u2|u1|α1 in Ω, u1=u2=0 in Rn∖Ω, where Ω is a bounded set with Lipschitz boundary ∂Ω in Rn, 1<q<2<ns with s∈(0,1), λ1, λ2 are two real positive parameters, α1>1,β1>1, α1+β1=2s∗=2nn−2s, 2s∗ is the fractional critical Sobolev exponent, and (−Δ)As is a fractional magnetic Laplace operator. By using Lusternik–Schnirelmann’s theory, we prove the existence result of infinitely many solutions for the magnetic fractional system.