On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity

Abstract

Abstract In this paper, we consider a class of fractional Choquard equations with indefinite potential ( − Δ ) α u + V ( x ) u = [ ∫ R N M ( ϵ y ) G ( u ) | x − y | μ d y ] M ( ϵ x ) g ( u ) , x ∈ R N , $$ (-\Delta )^{\alpha}u+V(x)u= \biggl[ \int _{{\mathbb{R}}^{N}} \frac{M(\epsilon y)G(u)}{ \vert x-y \vert ^{\mu}}\,\mathrm{d}y \biggr]M( \epsilon x)g(u), \quad x\in {\mathbb{R}}^{N}, $$ where α ∈ ( 0 , 1 ) $\alpha \in (0,1)$ , N > 2 α $N> 2\alpha $ , 0 < μ < 2 α $0<\mu <2\alpha $ , ϵ is a positive parameter. Here ( − Δ ) α $(-\Delta )^{\alpha}$ stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.

Keywords

• Concentration
• Ground state solutions
• Fractional Choquard equation