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.

Show me the PDF file (273 KB), TEX file, and other files for this article.

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

Return to the EJDE web page