Electron. J. Diff. Equ., Vol. 2015 (2015), No. 57, pp. 1-33.

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

Haitao Wan

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.

Submitted December 9, 2014. Published March 6, 2015.
Math Subject Classifications: 35A01, 35B40, 35J25.
Key Words: Singular Dirichlet problem; Karamata regular variation theory; convection term; boundary asymptotic behavior; global asymptotic behavior.

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Haitao Wan
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: wht200805@163.com, Phone: +8618954556896

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