关键词: 边值问题, 线形算子, 隐函数定理, 特征值

Abstract:

In this paper, the authors  use the perturbation method to consider the following free boundary problem
{ -Δu=λ+εg(u))H(u-1),  x ∈Ω, 
 ul ∂Ω=0. 
where λ is a constant and Ω is the unit ball in R³, H(t)={ 1,    t ≥ 0,  0,   t<0. 
Under some conditions, the authors  prove that there is a positive constant ε₀ >0, if 0< ε < ε₀ there is at least one solution for above problem and some results for the solution as ε tend to zero.

Key words: Boundary problem, Linear operator, Implicit theorem, Eigenvalue