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In this paper, we consider the following free boundary problem $$(P)\left\{\begin{array}{ll} \Delta u = \lambda \phi(x)\Sum_{i=1}^n H(u-\mu_i )&\quad \mbox{ in }\ \Omega=\Omega_2\setminus \overline{\Omega}_1, \\[0.3cm]u =0&\quad \mbox{ on } \partial \Omega_2, \\[0.3cm]u =M &\quad \mbox{ on } \partial\Omega_1. \end{array} \right. $$ The domain $\Omega$ is a convex ring where$\Omega_1$ and $\Omega_2$ are bounded convex domain in $\mathbb{R}^N,$ $N\geq2,$ $H$ is the Heaviside step function, $\lambda, M,\mu_i$ $(i=1,..,n)$ are agiven positive real parameters and $\phi$ is a given function . We show thatunder suitable conditions, there exists a solution to problem $(P)$ with aconvex level set. We derive also an interesting result for the convexity of theset which is delimited by the free boundary in obstacle problem when thenonlinearity is discontinuous. At last, a detailed analysis is given by aconstruction of function $\phi$ which show that there are more solutions.

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