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

We consider the nonlinear and nonlocal problemA1/2u=|u|^2^-2u in Ω, u=0 on ∂Ωwhere A1/2 represents the square root of the Laplacian in a bounded domainwith zero Dirichlet boundary conditions, Ω is a bounded smooth domain in Rⁿ, n≥ 2 and 2^=2n/(n-1) is the critical trace-Sobolev exponent. We assume that Ω is annular-shaped, i.e., there exist R₂>R₁>0 constants such that\ x ∈ Rⁿ s.t. R₁ < |x| < R₂ \⊂Ω and 0∉Ω, andinvariant under a group Γ of orthogonal transformations of Rⁿ without fixedpoints.We establish the existence of positive and multiple sign changing solutionsin the two following cases:if R₁/R₂ is arbitrary and the minimal Γ-orbit of Ω is large enough,or if R₁/R₂ is small enough and Γ is arbitrary.