Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in $\mathbb{R}^N$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912344065441792 |
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| author | Joseph, Anumol Sarkar, Abhishek |
| author_facet | Joseph, Anumol Sarkar, Abhishek |
| contents | We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = λK(x) |u|^{p-2} u \hspace{.1cm} \mbox { in } \hspace{.1cm} \mathbb{R}^N , \end{equation*} where $λ\in \mathbb{R}$, $p>1$, $K : \mathbb{R}^N \rightarrow \mathbb{R}$, $L : \mathbb{R}^N \rightarrow \mathbb{R}^+$ are locally integrable functions. The weight function $K$ is allowed to change sign, provided it remains positive on a set of nonzero measure. We establish the existence, regularity, and asymptotic behavior of the principal eigenfunctions. We also prove local and global antimaximum principles for a perturbed version of the problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in $\mathbb{R}^N$ Joseph, Anumol Sarkar, Abhishek Analysis of PDEs 35J92, 35P30, 35B40, 35J62, 35A15 We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = λK(x) |u|^{p-2} u \hspace{.1cm} \mbox { in } \hspace{.1cm} \mathbb{R}^N , \end{equation*} where $λ\in \mathbb{R}$, $p>1$, $K : \mathbb{R}^N \rightarrow \mathbb{R}$, $L : \mathbb{R}^N \rightarrow \mathbb{R}^+$ are locally integrable functions. The weight function $K$ is allowed to change sign, provided it remains positive on a set of nonzero measure. We establish the existence, regularity, and asymptotic behavior of the principal eigenfunctions. We also prove local and global antimaximum principles for a perturbed version of the problem. |
| title | Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in $\mathbb{R}^N$ |
| topic | Analysis of PDEs 35J92, 35P30, 35B40, 35J62, 35A15 |
| url | https://arxiv.org/abs/2504.17325 |