Poisson方程的非线性扰动

Nonlinear perturbation of Poisson equation

考虑Poisson方程的非线性扰动的Dirichlet问题-Δu =g(x) +λh(x,u,Du) x∈Ω ( 1 )u｜ Ω =0 ( 2 )其中λ∈R ,Ω是Rn 中具有C2 ,α 边界的有界区域 ,n∈N ,α∈ ]0 ,1 [.用截断函数法和Schauder不动点定理得到定理 设g∈Cα( Ω) ,h∈Cα( Ω×R×Rn) ,则存在δ >0 ,使得当 ｜λ｜<δ时 ,问题 ( 1 ) ,( 2 )在C2 ,α( Ω)

Consider Dirichlet boundary value problem of the perturbation of Poisson e quation-Δu=g(x)+λh(x, u, Du)\ \ \ x∈Ω(1) u|\-\{Ω\}=0(2)\%where\% λ∈R, Ω (R\+n)\% is a bounded domain with\% C\+\{2,α\} \%boundary , n∈N, α∈ ]0, 1[. With the cut function method and Schauder's fixed point theorem, th e following theorem is proved.\;Theorem\ Assume that\% g∈C\+2() and h∈C\+2(×R×R\+n)\%. The n there es ists \%δ\%>0 such that for every \%|λ|<δ, \%problem (1), (2) has at least one solution in\% C\+\{2,α\}().\%\%