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

We consider the boundary value problem \cases{ -Δ_γu = λu + \left\vert u \right\vert^{2^*_γ-2}u &in $Ω$\cr u = 0 &on $\partialΩ$,\cr } where $\mathrm{\Omega }$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, while ${\mathrm{\Delta }}_{\gamma }$ is the Grushin operator $${\mathrm{\Delta }}_{\gamma }u(z)={\mathrm{\Delta }}_{x}u(z)+\mathrm{\mid }x{\mathrm{\mid }}^{2\gamma }{\mathrm{\Delta }}_{y}u(z)(\gamma \ge 0)\mathrm{.}$$ 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.