The critical fractional Ambrosetti–Prodi problem

Abstract

In this paper we focus on the following nonlocal problem with critical growth: (-Δ)su=λu+u+2s∗-1+f(x)inΩ,u=0inRN\\Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} (-\\Delta )^{s} u = \\lambda u + u_{+}^{2^{*}_{s}-1} + f(x) &{} \ ext{ in } \\Omega ,\\\\ u=0 &{} \ ext{ in } \\mathbb {R}^{N}\\setminus \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where sin (0, 1), N>2s, Omega subset mathbb {R}^{N} is a smooth bounded domain, lambda >0, (-Delta )^{s} is the fractional Laplacian, f= te_{1}+h where tin mathbb {R}, e_{1} is the first eigenfunction of (-Delta )^{s} with homogeneous Dirichlet boundary datum, and hin L^{infty }(Omega ) is such that int _{Omega } h e_{1}, dx=0. According to the interaction of the nonlinear term with the spectrum of (-Delta )^{s}, we establish some existence and multiplicity results for the above problem by means of variational methods.