Symmetry and monotonicity of singular solutions for the Hartree equation
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915439705063424 |
|---|---|
| author | Cai, Ying Gu, Guangze Jevnikar, Aleks |
| author_facet | Cai, Ying Gu, Guangze Jevnikar, Aleks |
| contents | In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-Δu\!=\Big( \int_{\mathbb{R}^N\setminus Γ}\frac{F(u(y))}{|x-y|^μ}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminusΓ,$$ where $F$ is the primitive of $f$ and $Γ$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07893 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry and monotonicity of singular solutions for the Hartree equation Cai, Ying Gu, Guangze Jevnikar, Aleks Analysis of PDEs 35A15, 35J20, 35R11, 47G20 In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-Δu\!=\Big( \int_{\mathbb{R}^N\setminus Γ}\frac{F(u(y))}{|x-y|^μ}dy \Big)f (u(x)), \quad x\in \mathbb{R}^N\setminusΓ,$$ where $F$ is the primitive of $f$ and $Γ$ is the singular set. Under suitable assumptions, we prove that $u$ is symmetric and monotone with respect to the singular set by using moving plane methods. Furthermore, we complement this study by showing the existence, for a model problem, of a singular solution with the desired properties. |
| title | Symmetry and monotonicity of singular solutions for the Hartree equation |
| topic | Analysis of PDEs 35A15, 35J20, 35R11, 47G20 |
| url | https://arxiv.org/abs/2508.07893 |