Nonlocal problems at nearly critical growth

Abstract

We study the asymptotic behavior of solutions to the nonlocal nonlinear equation $(-{\Delta }_p{)}^{s}u=|u{|}^{q-2}u$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$ as $q$ approaches the critical Sobolev exponent $p^{\ast }=Np/(N-ps)$. We prove that ground state solutions concentrate at a single point $\bar{x}\in \overline{\Omega }$ 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 $\bar{x}$ cannot lie on the boundary, and identify its location in the case of annular domains.

18 pages