Electron. J. Diff. Eqns., Vol. 2005(2005), No. 04, pp. 1-11.

Positive solutions for elliptic equations with singular nonlinearity

Junping Shi, Miaoxin Yao

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.

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Junping Shi
Department of Mathematics, College of William and Mary
Williamsburg, VA 23187, USA
Department of Mathematics, Harbin Normal University
Harbin, Heilongjiang, China
email: shij@math.wm.edu
Miaoxin Yao
Department of Mathematics, Tianjin University
and Liu Hui Center for Applied Mathematics
Nankai University & Tianjin University
Tianjin, 300072, China

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