On a free boundary problem

In this paper, we will be concerned with the existence of positive solutions of the following discontinuous problem: equation(P){−Δu=f(u)H(u−μ)in Ω,u=hon ∂Ω, where ΩΩ is the unit ball of Rn(n≥3) and HH is the Heaviside function. The problem is written in an equivalent integral equation form. The shape of the free boundary problem {x∈Ω/u(x)=μ}{x∈Ω/u(x)=μ} is studied for when μμ is a positive real parameter, ff and hh are given functions.