关于一类半线性退化椭圆方程的存在性结果

An Existence Result for a Class of Semilinear Degenerate Elliptic Equations

本文研究下列半线性退化椭圆Ｄｉｒｉｃｈｌｅｔ问题：这里Ｘ＝｛Ｘ１，…，Ｘｍ｝是一组满足Ｈｏｒｍａｎｄｅｒ条件的实光滑向量场．假设它们在区域的边界附近还满足一些附加条件，以及ｆ∈Ｃ∞〔Ω×Ｒ×Ｒｍ），并且 ｚｆ（ｘ，ｚ，ξ）≥０，ｓｉｇｎＸｆ（ｘ，ｚ，０）≥μ＞－∞，ｃ（ｘ）≥ｃ０＞０和ｆ（ｘ，ｚ，ξ）关于变量ξ满足一定的增长条件．我们证明了当边界是无穷可微时，上述岸线性Ｄｉｒｉｃｈｌｅｔ问题的光滑解的存在性和唯一性．

In this work,we study the semilinear degenerate elliptic Dirichlet problem,∑=1,X Xju + cu + f(x,u,Xu)=0,in Ω;u =,on Ω,where X = {X1,…,Xm}is a system of real smooth vector fields which satisfies the Hormander's condition. Assume that X1,…,Xm satisfies some supplementary conditions on the boundary Ω, f ∈C∞(Ω×R×Rm),zf(x,z,ξ)≥ 0, sign zf(x,z,0)≥μ>-∞,c(x)≥ c0> 0. With some growth hypothesis on f(x,z,ξ) in the variables ξ,we have proved the existence and the uniqueness of solution u ∈C∞(Ω) of the above semililinear Dirichlet problem,if ∈C∞(Ω).