Nonlocal problems at nearly critical growth. (May 2016)

• Record Type: Journal Article
• Authors: Mosconi, Sunra; Squassina, Marco
• Is Part Of: Nonlinear analysis. Volume 136(2016)
• Page Start: 84
• Page End: 101
• Publication Date: 2016-05
• Subjects: 34K37 -- 58K05; Nonlinear nonlocal equation -- Critical embedding -- Nearly critical nonlinearities; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.02.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the asymptotic behavior of solutions to the nonlocal nonlinear equation ( − Δ p ) s u = | u | q − 2 u in a bounded domain Ω ⊂ R N as q approaches the critical Sobolev exponent p ∗ = N p / ( N − p s ) . We prove that ground state solutions concentrate at a single point x ̄ ∈ Ω ¯ and analyze the asymptotic behavior for sequences of solutions at higher energy levels. In the semi-linear case p = 2, we prove that for smooth domains the concentration point x ̄ cannot lie on the boundary, and identify its location in the case of annular domains.