Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 04, pp. 1-11.
Title: Positive solutions for elliptic equations with singular nonlinearity
Authors: Junping Shi (College of William and Mary, Williamsburg, VA, USA)
Miaoxin Yao (Tianjin Univ. China)
Abstract:
We study an elliptic boundary-value problem with singular
nonlinearity via the method of monotone iteration scheme:
$$\displaylines{
-\Delta u(x)=f(x,u(x)),\quad x \in \Omega,\cr
u(x)=\phi(x),\quad x \in \partial \Omega ,
}$$
where $\Delta$ is the Laplacian operator, $\Omega$ is a bounded domain
in $\mathbb{R}^{N}$, $N \geq 2$, $\phi \geq 0$ may take the value $0$
on $\partial\Omega$, and $f(x,s)$ is possibly singular near $s=0$.
We prove the existence and the uniqueness of positive solutions
under a set of hypotheses that do not make neither monotonicity nor
strict positivity assumption on $f(x,s)$, which improvements of
some previous results.
Submitted August 15, 2004. Published January 2, 2005.
Math Subject Classifications: 35J25, 35J60.
Key Words: Singular nonlineararity; elliptic equation; positive solution;
monotonic iteration.