Wolff potentials and nonlocal equations of Lane-Emden type
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916252061007872 |
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| author | Nguyen, Quoc-Hung Ok, Jihoon Song, Kyeong |
| author_facet | Nguyen, Quoc-Hung Ok, Jihoon Song, Kyeong |
| contents | We consider nonlocal equations of the type \[ (-Δ_{p})^{s}u = μ\quad \text{in }Ω, \] where $Ω\subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $μ$ is a Radon measure on $Ω$, $0<s<1$ and $1<p<n/s$. Especially, we extend the existence, regularity and Wolff potential estimates for SOLA (Solutions Obtained as Limits of Approximations), established by Kuusi, Mingione, and Sire (Comm. Math. Phys. 337:1317--1368, 2015), to the strongly singular case $1<p\le2-s/n$. Moreover, using Wolff potentials and Orlicz capacities, we present both a sufficient and a necessary conditions for the existence of SOLA to nonlocal equations of the type \[ (-Δ_{p})^{s}u = P(u) + μ\quad \text{in }Ω, \] where $P(\cdot)$ is either a power function or an exponential function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11747 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wolff potentials and nonlocal equations of Lane-Emden type Nguyen, Quoc-Hung Ok, Jihoon Song, Kyeong Analysis of PDEs We consider nonlocal equations of the type \[ (-Δ_{p})^{s}u = μ\quad \text{in }Ω, \] where $Ω\subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $μ$ is a Radon measure on $Ω$, $0<s<1$ and $1<p<n/s$. Especially, we extend the existence, regularity and Wolff potential estimates for SOLA (Solutions Obtained as Limits of Approximations), established by Kuusi, Mingione, and Sire (Comm. Math. Phys. 337:1317--1368, 2015), to the strongly singular case $1<p\le2-s/n$. Moreover, using Wolff potentials and Orlicz capacities, we present both a sufficient and a necessary conditions for the existence of SOLA to nonlocal equations of the type \[ (-Δ_{p})^{s}u = P(u) + μ\quad \text{in }Ω, \] where $P(\cdot)$ is either a power function or an exponential function. |
| title | Wolff potentials and nonlocal equations of Lane-Emden type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.11747 |