On the Phragmén-Lindelöf and the superposition principles for the $p$-Laplacian
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914813047734272 |
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| author | Lamberti, Pier Domenico Moroz, Vitaly |
| author_facet | Lamberti, Pier Domenico Moroz, Vitaly |
| contents | We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-Δ_p u-V|u|^{p-2}u=0\quad\text{in $Ω$},$$ where $Ω$ is a radially symmetric domain in ${\mathbb{R}}^N$ and $V(x)\ge 0$ is a continuous potential such that the solutions of the equation satisfy the comparison principle on bounded subdomains of $Ω$. In this work we establish a superposition principle and then use it to develop a version of a Phragmén-Lindelöf comparison principle in the case $p\ge 2$. Moreover, by applying this principle to the case of Hardy-type potentials we recover and improve a number of known lower and upper estimates for sub and supersolutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Phragmén-Lindelöf and the superposition principles for the $p$-Laplacian Lamberti, Pier Domenico Moroz, Vitaly Analysis of PDEs 35J60, 35B53, 35B40 We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-Δ_p u-V|u|^{p-2}u=0\quad\text{in $Ω$},$$ where $Ω$ is a radially symmetric domain in ${\mathbb{R}}^N$ and $V(x)\ge 0$ is a continuous potential such that the solutions of the equation satisfy the comparison principle on bounded subdomains of $Ω$. In this work we establish a superposition principle and then use it to develop a version of a Phragmén-Lindelöf comparison principle in the case $p\ge 2$. Moreover, by applying this principle to the case of Hardy-type potentials we recover and improve a number of known lower and upper estimates for sub and supersolutions. |
| title | On the Phragmén-Lindelöf and the superposition principles for the $p$-Laplacian |
| topic | Analysis of PDEs 35J60, 35B53, 35B40 |
| url | https://arxiv.org/abs/2405.16705 |