On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity

Abstract

In this paper, we consider a class of fractional Choquard equations with indefinite potential (−Δ)αu+V(x)u=[∫RNM(ϵy)G(u)|x−y|μdy]M(ϵx)g(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (-\\Delta )^{\\alpha}u+V(x)u= \\biggl[ \\int _{{\\mathbb{R}}^{N}} \\frac{M(\\epsilon y)G(u)}{ \\vert x-y \\vert ^{\\mu}}\\,\\mathrm{d}y \\biggr]M( \\epsilon x)g(u), \\quad x\\in {\\mathbb{R}}^{N}, $$\\end{document} where alpha in (0,1), N> 2alpha , 0<mu <2alpha , ϵ is a positive parameter. Here (-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.