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

Paolo Malanchini; Giovanni Molica Bisci; Simone Secchi

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arXiv · 2501.11013

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

Paolo Malanchini·Giovanni Molica Bisci·Simone Secchi

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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.

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Paolo Malanchini, Giovanni Molica Bisci, Simone Secchi. 2025-01-19. Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator. https://arxiv.org/abs/2501.11013

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