Bubble solution for the critical Hartree equation in pierced domain
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909239462592512 |
|---|---|
| author | Ghimenti, Marco Huang, Xiaomeng Pistoia, Angela |
| author_facet | Ghimenti, Marco Huang, Xiaomeng Pistoia, Angela |
| contents | In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -Δu=\left(\int_{Ω_\varepsilon}\frac{u^{2_μ^*}}{|x-y|^μ}dy\right)u^{2_μ^*-1}, &\text{ in } Ω_\varepsilon, \\ u=0, &\text{ on } \partialΩ_\varepsilon, \end{cases} \end{align*} where $2_μ^*=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\geq 5$, $0<μ<4$ with $μ$ sufficiently close to $0$, $Ω_\varepsilon:=Ω\backslash B(0,\varepsilon)$ and $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, which contains the origin, and $\varepsilon$ is a positive parameter. As $\varepsilon$ goes to zero, we construct bubble solution which blows up at the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bubble solution for the critical Hartree equation in pierced domain Ghimenti, Marco Huang, Xiaomeng Pistoia, Angela Analysis of PDEs In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -Δu=\left(\int_{Ω_\varepsilon}\frac{u^{2_μ^*}}{|x-y|^μ}dy\right)u^{2_μ^*-1}, &\text{ in } Ω_\varepsilon, \\ u=0, &\text{ on } \partialΩ_\varepsilon, \end{cases} \end{align*} where $2_μ^*=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $N\geq 5$, $0<μ<4$ with $μ$ sufficiently close to $0$, $Ω_\varepsilon:=Ω\backslash B(0,\varepsilon)$ and $Ω$ is a bounded smooth domain in $\mathbb{R}^N$, which contains the origin, and $\varepsilon$ is a positive parameter. As $\varepsilon$ goes to zero, we construct bubble solution which blows up at the origin. |
| title | Bubble solution for the critical Hartree equation in pierced domain |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.02438 |