Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator
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| Format: | Preprint |
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2025
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| _version_ | 1866912193921941504 |
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| author | Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone |
| author_facet | Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator Malanchini, Paolo Bisci, Giovanni Molica Secchi, Simone Analysis of PDEs 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. |
| title | Bifurcation and multiplicity results for critical problems involving the $p$-Grushin operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.11013 |