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.