Minimal L^p-Solutions to Singular Sublinear Elliptic Problems
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913188127178752 |
|---|---|
| author | May, Aye Chan Seesanea, Adisak |
| author_facet | May, Aye Chan Seesanea, Adisak |
| contents | We solve the existence problem for the minimal positive solutions $u\in L^{p}(Ω, dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=σu^q+μ\qquad \quad \text{in} \quad Ω, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in \partial_{\infty}Ω, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div} (\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $σ$ and data $μ$ are nonnegative Radon measures on an arbitrary domain $Ω\subset \mathbb{R}^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inqualities, and norm estimates in terms of generalized energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11352 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimal L^p-Solutions to Singular Sublinear Elliptic Problems May, Aye Chan Seesanea, Adisak Analysis of PDEs 35J61 We solve the existence problem for the minimal positive solutions $u\in L^{p}(Ω, dx)$ to the Dirichlet problems for sublinear elliptic equations of the form \[ \begin{cases} Lu=σu^q+μ\qquad \quad \text{in} \quad Ω, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in \partial_{\infty}Ω, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div} (\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $σ$ and data $μ$ are nonnegative Radon measures on an arbitrary domain $Ω\subset \mathbb{R}^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inqualities, and norm estimates in terms of generalized energy. |
| title | Minimal L^p-Solutions to Singular Sublinear Elliptic Problems |
| topic | Analysis of PDEs 35J61 |
| url | https://arxiv.org/abs/2310.11352 |