  We study the existence and multiplicity of solutions of the following free
boundary problem $$ (P)\left\{ \begin{array}{rcll} \del u &=& \lam ( \eps
+(1-\eps ) H(u-\mu))~ \hspace{3mm}&\text{in}~\Omega (t)\\ u&=&
\overline{u}_{\infty}~\hspace{3mm} &\text{on } ~ \partial \Omega(t) \end{array}
\right. $$ where $\Omega(t) \subset \RR^3$ a regular domain at $t>0$,
$\eps,~\overline{u}_{\infty},~\lambda,~\mu$ are a positive parameters and $H$
is the Heaviside step function. \\The problem (P) has two free boundaries: the
outer boundary of $\Omega(t)$ and the inner boundary whose evolution is
implicit generated by the discontinuous nonlinearity $H$. The problem (P) arise
in tumor growth models as well as in other contexts such as climatology. First,
we show the existence and multiplicity of radial solutions of problem (P) where
$\Omega(t)$ is a spherical domain. Moreover, the bifurcation diagrams are
giving. Secondly, using the perturbation technic combining to local methods, we
prove the existence of solutions and characterize the free boundaries of
problem (P) near the corresponding radial solutions.
