一类非线性椭圆边值问题解的存在性

目前 ,对 s——拉普拉斯算子△s的研究是较为活跃的数学课题 .原因在于算子 -△s与许多物理现象有关 .比如 :反射扩散问题 ,石油提取问题等等 .基于此因 ,在文 3]的基础上 ,我们将继续研究以下非线性边值问题在 Ls(Ω) ,( 1 2 nn+1 )中解的存在条件 .-△su +g( x,u) =f几乎处处在Ω中-〈 ,| u|s- 2 u〉 =0几乎处处在Γ上其中 f∈Ls( Ω)给定 ,Ω Rn( n 1 ) ,△su=div( | u|s- 2 u) ,g∶Ω× R→ R满足 Caratheodory条件 .本文把文 3]关于非线性边值问题 @在 Lp( Ω) ( 2 p<+∞ )空间中解的存在性的研究推广到 Ls( Ω) ( 1 2 nn+1 )空间中 .

The Existence of Solution of Nonlinear Elliptic Boundary Value Problem

In recent years, many mathematicians haved studied the problems involving the s -Laplacian operator △ s. It arises from a variety of physical phenomena such as reaction-diffusion problems, oil extracting problems, etc. In this paper, we continue study the abstract results on the existence of a solution u∈L s(Ω) under some conditions of the nonlinear boundary value problem:HL(2:1,Z;2,Z]-△ su+g(x,u)=fa.e.\ in Ω -〈,｜u｜ s-2 u〉=0a.e.\ on Γwhere f∈L s(Ω)(12nn+1) is given, △ su=div(｜u｜ s-2 u),ΩR n(n1), g∶Ω×R→R satisfies Carathedory′s conditions. Our results extend the solution in L p(Ω)(2p<+∞) of nonlinear boundary value problem @ in Wei-He 3] to a more general case in L s(Ω)(12nn+1).