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

TL;DR

This paper proves the existence of positive and sign-changing solutions for a critical exponent problem involving the half-Laplacian in annular domains, considering symmetry and geometric conditions.

Contribution

It introduces new existence results for solutions to a nonlocal critical problem in annular-shaped domains with symmetry considerations.

Findings

Existence of positive solutions under certain geometric conditions.

Existence of multiple sign-changing solutions.

Results depend on the ratio of radii and symmetry group properties.

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 zero Dirichlet 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. We assume that $\Omega$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{x\in\R^n\ \text{= s.t.}\ R_1<|x|<R_2\}\subset\Omega$ and $0\notin\Omega$, and invariant under a group $\Gamma$ 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 $\Gamma$-orbit of $\Omega$ is large enough, or if $R_1/R_2$ is small enough and $\Gamma$ is arbitrary.