Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains

Abstract

We consider the nonlinear and nonlocal problem $$ A_{1/2}u=|u|^{2^\sharp-2}u\ \text{in Ω, \quad u=0 \text{on} \partialΩ$$where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, $Ω$ is a bounded smooth domain in $\R^n$, $n\ge 2$ and $2^{\sharp}=2n/(n-1)$ is the critical trace-Sobolev exponent. We assume that $Ω$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{x\in\R^n\ \text{= s.t.}\ R_1<|x|<R_2\}\subsetΩ$ and $0\notinΩ$, and invariant under a group $Γ$ of orthogonal transformations of $\R^n$ without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if $R_1/R_2$ is arbitrary and the minimal $Γ$-orbit of $Ω$ is large enough, or if $R_1/R_2$ is small enough and $Γ$ is arbitrary.