A Semilinear Elliptic Problem with Critical Exponent and Potential Terms
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917652441595904 |
|---|---|
| author | Li, Haoyu Ma, Li |
| author_facet | Li, Haoyu Ma, Li |
| contents | This paper addresses the following problem. \begin{equation}
\left\{
\begin{array}{lr}
-Δu=λI_α*_Ωu+|u|^{2^*-2}u\mbox{ in }Ω,\nonumber
u\in H_0^1(Ω).\nonumber
\end{array}
\right. \end{equation} Here, $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N\geq3$, $2^*=\frac{2N}{N-2}$, $λ\in\mathbb{R}$, $λ\in(0,N)$, $I_α$ is the Riesz potential and \begin{align} I_α*_Ωu(x):=\int_Ω\frac{Γ(\frac{N-α}{2})}{Γ(\fracα{2})π^\frac{N}{2}2^α|x-y|^{N-α}} u(y)dy. \nonumber \end{align} We study the non-existence, existence and multiplicity results. Our argument combines Brezis-Nirenberg's method with the regularity results involving potential terms. Especially, we study the following nonlocal eigenvalue problem. \begin{equation}
\left\{
\begin{array}{lr}
-Δu=λI_α*_Ωu\mbox{ in }Ω,\nonumber
λ\in\mathbb{R},\,u\in H_0^1(Ω).\nonumber
\end{array}
\right. \end{equation} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Semilinear Elliptic Problem with Critical Exponent and Potential Terms Li, Haoyu Ma, Li Analysis of PDEs 35J20, 35B33, 35J61, 35J61, 45K05, 47A75, 35B50 This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -Δu=λI_α*_Ωu+|u|^{2^*-2}u\mbox{ in }Ω,\nonumber u\in H_0^1(Ω).\nonumber \end{array} \right. \end{equation} Here, $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N\geq3$, $2^*=\frac{2N}{N-2}$, $λ\in\mathbb{R}$, $λ\in(0,N)$, $I_α$ is the Riesz potential and \begin{align} I_α*_Ωu(x):=\int_Ω\frac{Γ(\frac{N-α}{2})}{Γ(\fracα{2})π^\frac{N}{2}2^α|x-y|^{N-α}} u(y)dy. \nonumber \end{align} We study the non-existence, existence and multiplicity results. Our argument combines Brezis-Nirenberg's method with the regularity results involving potential terms. Especially, we study the following nonlocal eigenvalue problem. \begin{equation} \left\{ \begin{array}{lr} -Δu=λI_α*_Ωu\mbox{ in }Ω,\nonumber λ\in\mathbb{R},\,u\in H_0^1(Ω).\nonumber \end{array} \right. \end{equation} |
| title | A Semilinear Elliptic Problem with Critical Exponent and Potential Terms |
| topic | Analysis of PDEs 35J20, 35B33, 35J61, 35J61, 45K05, 47A75, 35B50 |
| url | https://arxiv.org/abs/2404.18451 |