We consider the boundary value problem \(\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _\gamma u = \lambda u + \left| u \right| ^{2^*_\gamma -2}u ~& \text {in}~ \Omega \\ u = 0 ~& \text {on}~ \partial \Omega , \end{array}\right. } \end{aligned}\) where \(\lambda >0\) , \(\Omega \) is an open bounded domain in \(\mathbb {R}^N\) , \(N \ge 3\) , with \(0\in \overline{\Omega }\) , while \(\varDelta _\gamma \) is the Grushin operator \(\begin{aligned} \varDelta _ \gamma u(z) = \varDelta _x u(z) + | x |^{2\gamma } \varDelta _y u (z) \quad (\gamma \ge 0). \end{aligned}\) We prove a multiplicity and bifurcation result for this problem, extending the results of Cerami, Fortunato and Struwe in [7] and of Fiscella, Molica Bisci and Servadei in [10].
