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 $p^*=Np/(N-ps)$. 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.