Electron. J. Diff. Equ., Vol. 2010(2010), No. 56, pp. 1-16.

Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions

Sabri Bensid, Sidi Mohammed Bouguima

Abstract:
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.

Submitted February 22, 2010. Published April 19, 2010.
Math Subject Classifications: 34R35, 35J25.
Key Words: Green function; maximum principle; bifurcation; free boundary problem.

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Sabri Bensid
Department of Mathematics, Faculty of Sciences
University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria
email: edp_sabri@yahoo.fr
Sidi Mohammed Bouguima
Department of Mathematics, Faculty of Sciences
University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria
email: bouguima@yahoo.fr

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