Perturbation In A Free Boundary Problem In R~3

In this paper, we use the perturbation method to consider the following free boundary problem: Whereλ,εare constants andΩis the unit ball in R3. The bounded increasing nonnegative function g satisfies the Lipschitz conditions, and h satisfies: h(0,ρ)= h(2π,ρ).(?)ρ[0,π], h<1,(?)θ∈[0,2π].Under some conditions, we prove that there is a positive constantε0, if 0<ε<ε0 there is at least one solution for above problem and some results for the solution asεtend to zero.