Electron. J. Diff. Equ., Vol. 2015 (2015), No. 19, pp. 1-18.

Boundary behavior of solutions to a singular Dirichlet problem with a nonlinear convection

Bo Li, Zhijun Zhang

Abstract:
In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem
$$\displaylines{ -\Delta u=b(x)g(u)+\lambda|\nabla u|^q+\sigma, \quad u>0, \; x \in \Omega,\cr u\big|_{\partial \Omega}=0, }$$
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $q\in (0, 2]$, $\sigma>0$, $\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$, $\lim_{s \to 0^+}g(s)=\infty$, $g$ is decreasing on $(0, s_0)$ for some $s_0>0$, $b \in C_{\rm loc}^{\alpha}({\Omega})$ for some $\alpha\in (0, 1)$, is positive in $\Omega$, but may be vanishing or singular on the boundary. We show that $\lambda |\nabla u|^q$ does not affect the first expansion of classical solutions near the boundary.

Submitted December 3, 2014. Published January 20, 2015.
Math Subject Classifications: 35J65, 35B05, 35J25, 60J50.
Key Words: Semilinear elliptic equation; singular Dirichlet problem; nonlinear convection term; classical solution; boundary behavior.

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Bo Li
School of mathematics and statistics
Lanzhou University
Lanzhou 730000, Gansu, China
email: libo_yt@163.com
Zhijun Zhang
School of Mathematics and Information Science
Yantai University
Yantai 264005, Shandong, China
email: chinazjzhang2002@163.com, zhangzj@ytu.edu.cn

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