关于非线性椭圆边值问题解的存在性的注

Remark on the Existence of Solution of Nonlinear Elliptic Boundary Value Problem

利用非线性增生映射值域的扰动理论,本文研究了与P拉普拉斯算子△p相关的非线性椭圆边值问题@在Ls(Ω)空间中解的存在性,其中2>sp>2nn+1且n1.@-Δpu+|u(x)|p-2u(x)+g(x,u(x))=fa.e.x∈Ω-〈υ,|u|p-2u〉=0a.e.x∈Γ其中f∈Ls(Ω)给定,ΩRn,n1,Δpu=div(|u|p-2u)为P拉普拉斯算子,υ为Γ的外法向导数,g∶Ω×R→R满足Caratheodory条件.本文所讨论的方程及所用的方法是对以往一些工作的补充和延续.

By using.the perturbation theories of ranges of nonlinear accretive mappings, we study the existence of a solution u∈L^s（Ω） of nonlinear elliptic boundary value problem @involving the P-Laplacian operator △p,where 2〉s≥p〉2n/n＋1andn≥1,@{-△pu＋｜u（x）＋g（x,u（x））=f a.e.x∈Ω -〈v,｜△↓u｜^p-2△↓u〉=a.e.x∈Г,where f∈L^s（Ω）is given,Ωbelong to R^n,n≥1.△pu=div（｜u｜^p-2△↓u｜）represents the P-Laplacian operator,v denotes the exterior normal derivative to Г,g：Ω×R→R satisfies Caratheodory＇s conditions. The equation discussed in this paper and the methods used here are continuity and complement to some previous works.