Semilinear elliptic degenerate equations with critical exponent
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_ | 1866913272338317312 |
|---|---|
| author | Tri, N. M. Tuan, D. A. |
| author_facet | Tri, N. M. Tuan, D. A. |
| contents | In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation
\begin{gather}
-\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=|x|^{2k}u^p+f(x,y,u) \quad\text{ in }Ω, \
u=0 \quad\text{ on }\partialΩ,\label{equ0}
\end{gather} where $Ω$ is a bounded domain with smooth boundary in $\mathbb{R}^2, Ω\cap \{x=0\}\ne \emptyset,$ $k\in\mathbb N,$ $f(x,y,0)=0,$ and $p=(4+5k)/k$ is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semilinear elliptic degenerate equations with critical exponent Tri, N. M. Tuan, D. A. Analysis of PDEs 35B33, 35J61, 35J70, 46E30, 46E35 In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=|x|^{2k}u^p+f(x,y,u) \quad\text{ in }Ω, \ u=0 \quad\text{ on }\partialΩ,\label{equ0} \end{gather} where $Ω$ is a bounded domain with smooth boundary in $\mathbb{R}^2, Ω\cap \{x=0\}\ne \emptyset,$ $k\in\mathbb N,$ $f(x,y,0)=0,$ and $p=(4+5k)/k$ is the critical exponent. Recently, the equation (1) was investigated in [12] for the subcritical case based on a new result obtained in [17] on embedding theorem of weighted Sobolev spaces. In the critical case considered in this paper we will essentially use the optimal functions and constants found in [17] |
| title | Semilinear elliptic degenerate equations with critical exponent |
| topic | Analysis of PDEs 35B33, 35J61, 35J70, 46E30, 46E35 |
| url | https://arxiv.org/abs/2403.12828 |