Interior regularity of some weighted quasi-linear equations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909464487002112 |
|---|---|
| author | Castro, Hernán |
| author_facet | Castro, Hernán |
| contents | In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }Ω,\\ u\in H^{1,p}_{loc}&(Ω;wdx) \end{aligned} \right. \] where $\mathcal A$ and $\mathcal B$ are functions satisfying $\mathcal A(x,u,\nabla u)\sim \mathcal B(x,u,\nabla u)\sim w(|\nabla u|^{p-2}\nabla u+|u|^{p-2}u)$ for $p>1$ and a $p$-admissible weight function $w$.
We establish interior regularity results of weak solutions and use those results to obtain point-wise asymptotic estimates for solutions to \[ \left\{ \begin{aligned} -\mathrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u&&\text{in }Ω,\\ u\in D^{1,p}&(Ω,wdx) \end{aligned} \right. \] for a critical exponent $q>p$ in the sense of Sobolev. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07866 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interior regularity of some weighted quasi-linear equations Castro, Hernán Analysis of PDEs 35B45, 35B65, 35J62 In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }Ω,\\ u\in H^{1,p}_{loc}&(Ω;wdx) \end{aligned} \right. \] where $\mathcal A$ and $\mathcal B$ are functions satisfying $\mathcal A(x,u,\nabla u)\sim \mathcal B(x,u,\nabla u)\sim w(|\nabla u|^{p-2}\nabla u+|u|^{p-2}u)$ for $p>1$ and a $p$-admissible weight function $w$. We establish interior regularity results of weak solutions and use those results to obtain point-wise asymptotic estimates for solutions to \[ \left\{ \begin{aligned} -\mathrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u&&\text{in }Ω,\\ u\in D^{1,p}&(Ω,wdx) \end{aligned} \right. \] for a critical exponent $q>p$ in the sense of Sobolev. |
| title | Interior regularity of some weighted quasi-linear equations |
| topic | Analysis of PDEs 35B45, 35B65, 35J62 |
| url | https://arxiv.org/abs/2412.07866 |