Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

Abstract

In this paper, we study the fractional Schrödinger equation {(−Δ)su+u=a(x)|u|p−2u+b(x)|u|q−2u,u∈Hs(RN),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} (-\\Delta )^{s}u+u=a(x) \\vert u \\vert ^{p-2}u+b(x) \\vert u \\vert ^{q-2}u, \\\\ u\\in H^{s}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where (-Delta )^{s} denotes the fractional Laplacian of order sin (0,1), N>2s, 2< p< q<2^{*}_{s}, and 2^{*}_{s} is the fractional critical Sobolev exponent. The weight potentials a or b is a sign-changing function and satisfies some valid assumptions. We obtain the existence of infinitely many solutions to the problem by the Nehari manifold.