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

TL;DR

The paper addresses bifurcation and multiplicity for critical problems involving the degenerate $p$-Grushin operator by developing a variational framework that does not rely on linear eigenspaces (which fail when $p\neq 2$). It introduces an abstract critical-point theorem based on the $\mathbb{Z}_2$-cohomological index and a pseudo-index, together with a Lions-type concentration-compactness principle adapted to the $p$-Grushin setting, and constructs a variational sequence of eigenvalues via index theory. Using these tools, the authors prove bifurcation and multiplicity results: for $\lambda$ in appropriate bands relative to the eigenvalues $\{\lambda_k\}$ there exist $m$ pairs of nontrivial solutions, with convergence to zero as $\lambda$ approaches $\lambda_{k+1}$. They also obtain a global existence interval for $\lambda$. This work extends known $p=2$ results to all $p>1$ and provides a robust framework for critical problems with degenerate, non-uniformly elliptic operators, with potential implications for degenerate variational PDEs.

Abstract

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

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• proof
• Proposition 4.1
• Proposition 4.2
• Theorem 5.1