Existence and asymptotic behavior of a unique solution to a singular Dirichlet boundary-value problem with a convection term

Wan, Haitao

Electronic journal of differential equations, Tome 2015 (2015)

Résumé

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.

Détail

Classification : 35A01, 35B40, 35J25
Keywords: singular Dirichlet problem, karamata regular variation theory, convection term, boundary asymptotic behavior, global asymptotic behavior