Ground state solution for a nonlinear fractional magnetic Schrödinger equation with indefinite potential

Abstract

This paper is concerned with the following nonlinear fractional Schrödinger equation with a magnetic field: ε2s(−Δ)A/εsu+V(x)u=f(|u|2)u inRN, where ɛ > 0 is a parameter, s ∈ (0, 1), N ≥ 3, V:RN→R and A:RN→RN are continuous potentials, and V may be sign-changing; the nonlinearity is superlinear with subcritical growth but without satisfying the Ambrosetti–Rabinowitz condition. Based on the Nehari manifold method, concentration-compactness principle, and variational methods, we prove the existence of a ground state solution for the above equation when ɛ is sufficiently small. Our results improve and extend the result of Ambrosio and d’Avenia [J. Differ. Equations 264, 3336–3368 (2018)].