Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 19, pp. 1-18.
Title: Boundary behavior of solutions to a singular Dirichlet problem
with a nonlinear convection
Authors: Bo Li (Lanzhou Univ., Gansu, China)
Zhijun Zhang (Yantai Univ., Shandong, China)
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.