Renormalized Solutions for Quasilinear Elliptic Equations with Robin Boundary Conditions, Lower-Order Terms, and $L^1$ Data
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Apaza, Juan A. de Souza, Manassés |
| author_facet | Apaza, Juan A. de Souza, Manassés |
| contents | In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_α|\nabla u|^{p-2} \nabla u)+\vartheta_γb|\nabla u|^{p-1}+\vartheta_γc|u|^{r-1} u & =f \vartheta_α& & \text { in } Ω, \\ \vartheta_α|\nabla u|^{p-2} \nabla u \cdot ν+\vartheta_β|u|^{p-2} u & =g \vartheta_β& & \text { on } \partial Ω. \end{aligned}\right. \end{equation} Here, $Ω$ is an open subset of $\mathbb{R}^N$ with a Lipschitz boundary, where $N\geq 2$ and $1 < p < N$. We define $\vartheta_a(x) = (1 + |x|)^a$ for $a \in (-N, (p-1)N)$, and the constants $α, β, γ, r$ satisfy suitable conditions. Additionally, $f$ and $g$ are measurable functions, while $b$ and $c$ belong to a Lorentz space. Our approach also allows us to establish stability results for renormalized solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12399 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renormalized Solutions for Quasilinear Elliptic Equations with Robin Boundary Conditions, Lower-Order Terms, and $L^1$ Data Apaza, Juan A. de Souza, Manassés Analysis of PDEs Functional Analysis 35J62, 35A35, 35J25 In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_α|\nabla u|^{p-2} \nabla u)+\vartheta_γb|\nabla u|^{p-1}+\vartheta_γc|u|^{r-1} u & =f \vartheta_α& & \text { in } Ω, \\ \vartheta_α|\nabla u|^{p-2} \nabla u \cdot ν+\vartheta_β|u|^{p-2} u & =g \vartheta_β& & \text { on } \partial Ω. \end{aligned}\right. \end{equation} Here, $Ω$ is an open subset of $\mathbb{R}^N$ with a Lipschitz boundary, where $N\geq 2$ and $1 < p < N$. We define $\vartheta_a(x) = (1 + |x|)^a$ for $a \in (-N, (p-1)N)$, and the constants $α, β, γ, r$ satisfy suitable conditions. Additionally, $f$ and $g$ are measurable functions, while $b$ and $c$ belong to a Lorentz space. Our approach also allows us to establish stability results for renormalized solutions. |
| title | Renormalized Solutions for Quasilinear Elliptic Equations with Robin Boundary Conditions, Lower-Order Terms, and $L^1$ Data |
| topic | Analysis of PDEs Functional Analysis 35J62, 35A35, 35J25 |
| url | https://arxiv.org/abs/2401.12399 |