A note on nonlinear critical problems involving the Grushin Subelliptic Operator: bifurcation and multiplicity results

We consider the boundary value problem -Δ_γ u = λ u + u 2^*_γ-2u &in = 0 &on∂Ω, where Ω is an open bounded domain in RN, N ≥ 3, while Δ_γ is the Grushin operator Δ_ γ u(z) = Δₓ u(z) + x 2γ Δy u (z) (γ≥ 0). We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe and of Fiscella, Molica Bisci and Servadei.