一类带有Neumann边界的奇异拟线性椭圆方程解的存在性

应用变分方法中的极值理论来研究Neumann边界问题{ -div(｜x｜α｜▽u｜p-2▽u)=｜x｜βup(α,β)-1-λ｜x｜γup-1+｜x｜μq-1,u(x)＞0,x∈Ω｜▽u｜p-2?u/?u=0, x∈?Ω其中Ω是RN(N≥3)中具有C2光滑边界的有界区域,0 ∈Ω,n表示(e)Ω的单位外法向向量,且1＜p＜N,α＜0,β＜0,使得p(α,β)(△)p(N+β)/N-p+α＞P,γ＞α-p,P＜q＜p(α,μ).对于参数α,β,γ及μ的不同范围,建立上述方程解的存在性结果.其中对参数不同范围的讨论对解的存在性所起到的至关重要的作用.

The Existience of Solutions about a Kind of Quasilinear Elliptic Equations with Neumann Boundary and Singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN(N≥3),0∈Ω,and n denote the unit outward normal to▽Ω.The Neumann boundary problems is concerned:-div(|x | α |u | p-2 u) = |x | βup(α,β)-1-λ |x | γup-1 + |x | μuq-1,u(x) > 0,x∈Ω;|▽u | p-2 ▽u ▽n = 0,x∈▽Ω { 。 where1 < p < N,and α < 0,β < 0,such that p(α,β) p(N + β) N-p + α > p,γ > α-p,p < q < p(α,μ).For various parameters α,β,γ and μ,some existence results of the solutions in the case 0∈Ω is established.The scopes of the parameters α,β,γ and μ are discussed wihch play an important role in the existence of solutions.