Electronic Journal of Differential Equations (Mar 2015)
Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term
Abstract
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.
