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

TL;DR

This paper establishes bifurcation and multiplicity results for a critical nonlinear problem involving the degenerate $p$-Grushin operator, extending previous results from the special case $p=2$ to general $p>1$ using advanced abstract critical point theory.

Contribution

It extends bifurcation and multiplicity results to the $p$-Grushin operator for all $p>1$, introducing new abstract theorems and a concentration-compactness principle for this operator.

Findings

Proved bifurcation and multiplicity results for the $p$-Grushin operator.

Developed a new abstract critical point theorem based on a pseudo-index.

Established a version of Lions' Concentration-Compactness Principle for the operator.

Abstract

In this article we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator $\Delta_\gamma^p$. We extend to a generic $p>1$ a result which was proved only when $p=2$. When $p\neq 2$, the nonlinear operator $-\Delta_\gamma^p$ 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.