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

Abstract

In this article, we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator Δγp{\Delta }_{\gamma }^{p}. We extend to a generic p>1p\gt 1 a result, which was proved only when p=2p=2. When p≠2p\ne 2, the nonlinear operator −Δγp-{\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 use an abstract result based on a pseudo-index related to the Z2{{\mathbb{Z}}}_{2}-cohomological index that is applicable here. We provide a version of the Lions’ concentration-compactness principle for our operator.

Keywords

• p-grushin operator
• critical exponent
• 35j20
• 35j70