# Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains - Mathematics > Analysis of PDEs

Abstract: We consider the nonlinear and nonlocal problem $$A {1-2}u=|u|^{2^\sharp-2}u\\text{in \Omega, \quad u=0 \text{on} \partial\Omega$$where $A {1-2}$represents the square root of the Laplacian in a bounded domain with zeroDirichlet boundary conditions, $\Omega$ is a bounded smooth domain in $\R^n$,$n\ge 2$ and $2^{\sharp}=2n-n-1$ is the critical trace-Sobolev exponent. Weassume that $\Omega$ is annular-shaped, i.e., there exist $R 2>R 1>0$ constantssuch that \$\{x\in\R^n\ \text{= s.t.}\ R 1<|x|

Author: Antonio Capella Kort

Source: https://arxiv.org/