Perturbation analysis in a free boundary problem arising in tumor growth model

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.