Geodesic
Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions
Electronic journal of differential equations, Tome 2010 (2010)
We study the nonlinear elliptic problem with discontinuous nonlinearity

$\displaylines{ -\Delta u = f(u)H(u-\mu ) \quad\hbox{in } \Omega, \cr u =h \quad \hbox{on }\partial \Omega, }$

where $H$ is the Heaviside unit function, $f,h$ are given functions and $\mu$ is a positive real parameter. The domain $\Omega$ is the unit ball in $\mathbb{R}^n$ with $n\geq 3$. We show the existence of a positive solution $u$ and a hypersurface separating the region where $-\Delta u=0$ from the region where $-\Delta u=f(u)$. Our method relies on the implicit function theorem and bifurcation analysis.
Classification : 34R35, 35J25
Keywords: Green function, maximum principle, bifurcation, free boundary problem 
@article{EJDE_2010__2010__a63,
     author = {Bensid,  Sabri and Bouguima,  Sidi Mohammed},
     title = {Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1196.35219},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a63/}
}
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Bensid,  Sabri; Bouguima,  Sidi Mohammed. Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a63/