Asymptotic symmetries for fractional operators. (December 2015)

• Record Type: Journal Article
• Authors: Grumiau, C.; Squassina, M.; Troestler, C.
• Is Part Of: Nonlinear analysis. Volume 26(2015)
• Page Start: 351
• Page End: 371
• Publication Date: 2015-12
• Subjects: Fractional Laplacian -- Mountain Pass solutions -- Symmetries; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.06.001
• Languages: English
• ISSNs: 1468-1218

Abstract

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.