Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator ${\mathrm{\Delta }}_{\gamma }^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\mathrm{\ne }2$, the nonlinear operator $-{\mathrm{\Delta }}_{\gamma }^p$ has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem which is not based on linear subspaces. We also prove a new abstract result based on a pseudo-index related to the ${\mathbf{Z}}_2$-cohomological index that is applicable here. We provide a version of the Lions’ Concentration-Compactness Principle for our operator.