Asymptotic symmetries for fractional operators. (December 2015) - Grumiau, C.; Squassina, M.; Troestler, C.
Abstract: In this paper, we study equations driven by a non-local integrodifferential operator L K with homogeneous Dirichlet boundary conditions. More precisely, we study the problem { − L K u + V ( x ) u = | u | p − 2 u, in  Ω, u = 0, in  R N ∖ Ω, where 2 < p < 2 s ∗ = 2 N N − 2 s, Ω is an open bounded domain in R N for N ⩾ 2 and V is a L ∞ potential such that − L K + V is positive definite. As a particular case, we study the problem { ( − Δ ) s u + V ( x ) u = | u | p − 2 u, in  Ω, u = 0, in  R N ∖ Ω, where ( − Δ ) s denotes the fractional Laplacian (with 0 < s < 1 ). We give assumptions on V, Ω and K such that ground state solutions (resp. least energy nodal solutions) respect the symmetries of some first (resp. second) eigenfunctions of − L K + V, at least for p close to 2 . We study the uniqueness, up to a multiplicative factor, of those types of solutions. The results extend those obtained for the local case.