Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications
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| Format: | Preprint |
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2024
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| _version_ | 1866913554296209408 |
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| author | da Silva, João Vitor Prazeres, Disson dos Ricarte, Gleydson Sá, Ginaldo |
| author_facet | da Silva, João Vitor Prazeres, Disson dos Ricarte, Gleydson Sá, Ginaldo |
| contents | In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$
\mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^αu_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^α$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications da Silva, João Vitor Prazeres, Disson dos Ricarte, Gleydson Sá, Ginaldo Analysis of PDEs 35B65, 35J62 In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$ \mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^αu_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^α$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models. |
| title | Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications |
| topic | Analysis of PDEs 35B65, 35J62 |
| url | https://arxiv.org/abs/2410.14862 |