Nonlocal problems at nearly critical growth. (May 2016) - Mosconi, Sunra; Squassina, Marco
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.