Nonlocal problems at nearly critical growth

TL;DR

This paper investigates the asymptotic behavior of solutions to a nonlocal nonlinear PDE near critical growth, showing concentration phenomena and boundary behavior in specific cases.

Contribution

It provides new insights into the concentration points of solutions to nonlocal problems at nearly critical growth, including boundary and domain-specific results.

Findings

Solutions concentrate at a single point as q approaches critical

In smooth domains, concentration points do not lie on the boundary for p=2

Identifies concentration points in annular domains

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.