Existence Results for Fractional Choquard Equations with Critical or Supercritical Growth

Abstract

In this paper, we study the following fractional Choquard equation with critical or supercritical growth (-Δ)su+V(x)u=f(x,u)+λ|x|-μ∗|u|pp|u|p-2u,x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\ (-\\Delta )^su+V(x)u=f(x,u)+\\lambda \\left[ |x|^{-\\mu }*|u|^p\\right] p|u|^{p-2}u, \\quad x \\in {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where 0<s<1, (-Delta )^s denotes the fractional Laplacian of order s, N>2s, 0<mu <2s and pge 2_{mu ,s}^*:=frac{2N-mu }{N-2s}, which is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. Under some suitable conditions, we prove that the equation admits a nontrivial solution for small lambda >0 by variational methods, which extends results in Bhattarai in J. Differ. Equ. 263, 3197–3229 (2017).