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

We consider the boundary value problem \cases{ -\Delta_\gamma u = \lambda u + \left\vert u \right\vert^{2^*_\gamma-2}u &in $\Omega$\cr u = 0 &on $\partial\Omega$,\cr } where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, while $\Delta_\gamma$ is the Grushin operator $$\Delta_ \gamma u(z) = \Delta_x u(z) + \vert x \vert^{2\gamma} \Delta_y u (z) \quad (\gamma\ge 0).$$ 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.