Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems

Abstract

In this paper we consider the following critical nonlocal problem{−LKu=λu+|u|2⁎−2uin Ωu=0in Rn∖Ω, where s∈(0,1), Ω is an open bounded subset of Rn, n>2s, with continuous boundary, λ is a positive real parameter, 2⁎:=2n/(n−2s) is the fractional critical Sobolev exponent, while LK is the nonlocal integrodifferential operatorLKu(x):=∫Rn(u(x+y)+u(x−y)−2u(x))K(y)dy,x∈Rn, whose model is given by the fractional Laplacian −(−Δ)s.Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of −LK (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.