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.

Show me the PDF file (305 KB), TEX file, and other files for this article.

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

Return to the EJDE web page