Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

Abstract

In this paper, we focus on the existence of solutions for the Choquard equation {−Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+λ|u|p−2u,x∈RN;u∈H1(RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} {-}\\Delta {u}+V(x)u=(I_{\\alpha }* \\vert u \\vert ^{\\frac{\\alpha }{N}+1}) \\vert u \\vert ^{ \\frac{\\alpha }{N}-1}u+\\lambda \\vert u \\vert ^{p-2}u,\\quad x\\in \\mathbb{R}^{N}; \\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where lambda >0 is a parameter, alpha in (0,N), Nge 3, I_{alpha }: mathbb{R}^{N}to mathbb{R} is the Riesz potential. As usual, alpha /N+1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if lambda >lambda _{*} for some given number lambda _{*} in three cases: (i) 2< p<frac{4}{N}+2, (ii) p=frac{4}{N}+2, and (iii) frac{4}{N}+2< p<2^{*}. Our result improves the previous related ones in the literature.