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

Abstract

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 $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, while ${\Delta }_{\gamma }$ is the Grushin operator $${\Delta }_{\gamma }u(z)={\Delta }_{x}u(z)+|x{|}^{2\gamma }{\Delta }_{y}u(z)(\gamma \ge 0).$$ 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.