The nontrivial solutions for fractional Schrödinger–Poisson equations with magnetic fields and critical or supercritical growth

Abstract

In this paper, we study the following fractional Schrödinger–Poisson equation with magnetic field (−Δ)Asu+V(x)u+(|x|2t−3∗|u|2)u=f(x,|u|2)u+λ|u|p−2uinR3,where λ>0, s∈(34,1), t∈(0,1), p≥2s∗=63−2s, (−Δ)As is the fractional magnetic Laplacian, V:R3→R is a positive continuous potential, A:R3→R3 is a smooth magnetic potential. We mainly prove that the above equation has a nontrivial solution for small λ>0 and p>2s∗.