Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 56, pp. 1-16.

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

Authors: Sabri Bensid  (Univ. of Tlemcen, Algeria)
         Sidi Mohammed Bouguima (Univ. of Tlemcen, Algeria)

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.