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

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We consider the nonlinear and nonlocal problem A₁/₂u=|u|2-2uin Ω, u=0 on ∂Ωwhere A₁/₂ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, Ω is a bounded smooth domain in ⁿ, 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∈ⁿ= s.t.R₁<|x|<R₂}⊂Ω and 0∉Ω, and invariant under a group Γ of orthogonal transformations of ⁿ without fixed points. We establish the existence of positive and multiple sign changing solutions in 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.