Electron. J. Diff. Equ., Vol. 2014 (2014), No. 183, pp. 1-12.

Exact boundary behavior of solutions to singular nonlinear Dirichlet problems

Bo Li, Zhijun Zhang

Abstract:
In this article we analyze the exact boundary behavior of solutions to the singular nonlinear Dirichlet problem
$$ -\Delta u=b(x)g(u)+\lambda a(x) f(u), \quad u>0, \quad x \in \Omega,\quad u|_{\partial \Omega}=0, $$
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $\lambda> 0$, $g\in C^1((0,\infty), (0,\infty))$, $\lim_{s \to 0^+}g(s)=\infty$, $b, a \in C_{\rm loc}^{\alpha}({\Omega})$, are positive, but may vanish or be singular on the boundary, and $f\in C([0, \infty), [0, \infty))$.

Submitted July 11, 2014. Published August 29, 2014.
Math Subject Classifications: 35J65, 35B05, 35J25, 60J50.
Key Words: Semilinear elliptic equation; singular Dirichlet problem; positive 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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