Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916562845302784 |
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| author | Vuono, Domenico |
| author_facet | Vuono, Domenico |
| contents | We consider weak solutions of the equation $$-Δ_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } Ω,$$ where $H$ is in some cases called Finsler norm, $Ω$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and $a(\cdot,u)$, $f(\cdot,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06593 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term Vuono, Domenico Analysis of PDEs 35J62, 35B50, 35B51 We consider weak solutions of the equation $$-Δ_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } Ω,$$ where $H$ is in some cases called Finsler norm, $Ω$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and $a(\cdot,u)$, $f(\cdot,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator. |
| title | Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term |
| topic | Analysis of PDEs 35J62, 35B50, 35B51 |
| url | https://arxiv.org/abs/2501.06593 |