Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2407.06905 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914863227338752 |
|---|---|
| author | Chen, Wenjing Wang, Zexi |
| author_facet | Chen, Wenjing Wang, Zexi |
| contents | In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem \begin{align*}
\left\{
\begin{array}{ll}
-Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{6-α}(y)}{|x-y|^α}dy\Big)u^{5-α}+λu,
\ \ &\mbox{in}\ Ω,
u=0,
\ \ &\mbox{on}\ \partial Ω,
\end{array}
\right.
\end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $α\in (0,3)$, $6-α$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and $λ$ is a real positive parameter. By applying the reduction argument, we find and characterize a positive value $λ_0$ such that if $λ-λ_0>0$ is small enough, then the above problem admits a solution, which blows up and concentrates at the critical point of the Robin function as $λ\rightarrow λ_0$. Moreover, we consider the above problem under zero Neumann boundary condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three Chen, Wenjing Wang, Zexi Analysis of PDEs In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem \begin{align*} \left\{ \begin{array}{ll} -Δu=\displaystyle\Big(\int\limits_Ω\frac{u^{6-α}(y)}{|x-y|^α}dy\Big)u^{5-α}+λu, \ \ &\mbox{in}\ Ω, u=0, \ \ &\mbox{on}\ \partial Ω, \end{array} \right. \end{align*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^3$, $α\in (0,3)$, $6-α$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and $λ$ is a real positive parameter. By applying the reduction argument, we find and characterize a positive value $λ_0$ such that if $λ-λ_0>0$ is small enough, then the above problem admits a solution, which blows up and concentrates at the critical point of the Robin function as $λ\rightarrow λ_0$. Moreover, we consider the above problem under zero Neumann boundary condition. |
| title | Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.06905 |