A free boundary problem for a discontinuous semilinear elliptic equations in convex ring

In this paper, we consider the following free boundary problem $$(P)\{\begin{array}{ll}\mathrm{\Delta }u=\lambda \phi (x){\text{Sum}}_{i=1}^{n}H(u-{\mu }_i)& in\mathrm{\Omega }={\mathrm{\Omega }}_2\setminus {\bar{\mathrm{\Omega }}}_1,\\ u=0& on\mathrm{\partial }{\mathrm{\Omega }}_2,\\ u=M& on\mathrm{\partial }{\mathrm{\Omega }}_1\mathrm{.}\end{array}$$ The domain $\mathrm{\Omega }$ is a convex ring where ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are bounded convex domain in ${\mathbb{R}}^N,$ $N\ge 2,$ $H$ is the Heaviside step function, $\lambda ,M,{\mu }_i$ $(i=1,\mathrm{.}\mathrm{.},n)$ are a given positive real parameters and $\phi$ is a given function . We show that under suitable conditions, there exists a solution to problem $(P)$ with a convex level set. We derive also an interesting result for the convexity of the set which is delimited by the free boundary in obstacle problem when the nonlinearity is discontinuous. At last, a detailed analysis is given by a construction of function $\phi$ which show that there are more solutions.