*Submitted Paper*

**Inserted:** 7 dec 2015

**Last Updated:** 7 dec 2015

**Pages:** 18

**Year:** 2015

**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.

**Keywords:**
Nonlinear nonlocal equation, critical embedding, nearly critical nonlinearities

**Download:**