A Morse lemma at infinity for nonlinear elliptic fractional equations

Abstract

In this paper, we consider the following nonlinear fractional critical equation with zero Dirichlet boundary condition Asu = Ku n+2s n−2s , u > 0 in Ω and u = 0 on ∂Ω, whereK is a positive function, Ω is a regular bounded domain of R, n ≥ 2 and As, s ∈ (0, 1) represents the spectral fractional Laplacian operator (−∆) in Ω with zero Dirichlet boundary condition. We prove a version of Morse lemmas at infinity for this problem. We also exhibit a relevant application of our novel result. More precisely, we characterize the critical points at infinity of the associated variational problem and we prove an existence result for s = 1 2 and n = 3. Mathematics Subject Classification (2010). Primary: 35J65; Secondary: 35R11, 58J20, 58C30.