Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term
In this article, we consider the problem $$ -\Delta u =b(x)g(u)+ \lambda a(x)|\nabla u|^{q}+\sigma(x),\; u > 0,\; x\in \Omega,\quad u|_{\partial \Omega }= 0 $$ with $\lambda\in\mathbb{R}$, $q\in [0, 2]$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^{N}$. The weight functions $b, a,\...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-03-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/57/abstr.html |
| _version_ | 1818215725304643584 |
|---|---|
| author | Haitao Wan |
| author_facet | Haitao Wan |
| author_sort | Haitao Wan |
| collection | DOAJ |
| description | In this article, we consider the problem
$$
-\Delta u =b(x)g(u)+ \lambda a(x)|\nabla u|^{q}+\sigma(x),\;
u > 0,\; x\in \Omega,\quad u|_{\partial \Omega }= 0
$$
with $\lambda\in\mathbb{R}$, $q\in [0, 2]$ in a smooth bounded domain
$\Omega$ of $\mathbb{R}^{N}$. The weight functions
$b, a,\sigma$ belong to $C^{\alpha}_{\rm loc}(\Omega)$ satisfying
$b(x),a(x)>0$, $\sigma(x)\geq0$, $x\in \Omega$, which may vanish or
be singular on the boundary. $g\in C^1((0,\infty),(0,\infty))$
satisfies $\lim_{t\to 0^{+}}g(t)=\infty$. Our results
include the existence, uniqueness and the exact boundary asymptotic
behavior and global asymptotic behavior of the solution. |
| first_indexed | 2024-12-12T06:40:38Z |
| format | Article |
| id | doaj.art-1cb829ded8eb4ec9b20e4912707e96e4 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-12T06:40:38Z |
| publishDate | 2015-03-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-1cb829ded8eb4ec9b20e4912707e96e42022-12-22T00:34:21ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-03-01201557,133Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection termHaitao Wan0 Lanzhou Univ., Lanzhou, China In this article, we consider the problem $$ -\Delta u =b(x)g(u)+ \lambda a(x)|\nabla u|^{q}+\sigma(x),\; u > 0,\; x\in \Omega,\quad u|_{\partial \Omega }= 0 $$ with $\lambda\in\mathbb{R}$, $q\in [0, 2]$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^{N}$. The weight functions $b, a,\sigma$ belong to $C^{\alpha}_{\rm loc}(\Omega)$ satisfying $b(x),a(x)>0$, $\sigma(x)\geq0$, $x\in \Omega$, which may vanish or be singular on the boundary. $g\in C^1((0,\infty),(0,\infty))$ satisfies $\lim_{t\to 0^{+}}g(t)=\infty$. Our results include the existence, uniqueness and the exact boundary asymptotic behavior and global asymptotic behavior of the solution.http://ejde.math.txstate.edu/Volumes/2015/57/abstr.htmlSingular Dirichlet problemKaramata regular variation theoryconvection termboundary asymptotic behaviorglobal asymptotic behavior |
| spellingShingle | Haitao Wan Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term Electronic Journal of Differential Equations Singular Dirichlet problem Karamata regular variation theory convection term boundary asymptotic behavior global asymptotic behavior |
| title | Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term |
| title_full | Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term |
| title_fullStr | Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term |
| title_full_unstemmed | Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term |
| title_short | Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term |
| title_sort | existence and asymptotic behavior of a unique solution to a singular dirichlet boundary value problem with a convection term |
| topic | Singular Dirichlet problem Karamata regular variation theory convection term boundary asymptotic behavior global asymptotic behavior |
| url | http://ejde.math.txstate.edu/Volumes/2015/57/abstr.html |
| work_keys_str_mv | AT haitaowan existenceandasymptoticbehaviorofauniquesolutiontoasingulardirichletboundaryvalueproblemwithaconvectionterm |