Geodesic
Positive solutions for elliptic equations with singular nonlinearity
Electronic journal of differential equations, Tome 2005 (2005)
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.
Classification : 35J25, 35J60
Keywords: singular nonlineararity, elliptic equation, positive solution, monotonic iteration 
@article{EJDE_2005__2005__a78,
     author = {Shi,  Junping and Yao,  Miaoxin},
     title = {Positive solutions for elliptic equations with singular nonlinearity},
     journal = {Electronic journal of differential equations},
     year = {2005},
     volume = {2005},
     zbl = {1129.35344},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a78/}
}
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Shi,  Junping; Yao,  Miaoxin. Positive solutions for elliptic equations with singular nonlinearity. Electronic journal of differential equations, Tome 2005 (2005). http://geodesic.mathdoc.fr/item/EJDE_2005__2005__a78/