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,\...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-03-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/57/abstr.html |
| Summary: | 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. |
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| ISSN: | 1072-6691 |