一类奇异非线性Dirichlet问题(英文)

A Singular Nonlinear Dirichlet Problem

构造新的精细上下解,结合摄动方法和估计理论,严格刻画了参数β对奇异Dirichlet问题-Δu=g(x)u-γ+λup,u>0,x∈Ω,u｜ Ω=0古典解的存在性、正则性和渐近行为的影响.其中Ω是RN(N≥1)中的有界区域,γ>0,λ≥0,p>0,g∈Cαloc(Ω),且在Ω上满足b0φβ1≤g≤b1φβ1,β∈R,b0,b1是正常数,φ1是通常的第一特征函数.

By constructing the new subsolution and supersolution, and the perturbed method, combining the estimate argument, we exactly show the parameter β how to affect existence, nonexistence, regularity, and the asymptotic behavior on Ω of the classical solutions to the singular boundary value problem - Δ u=g(x)u^(-γ)+λu^p,u>0,x∈Ω,u|_(Ω)=0, where Ω is a bounded domain with smooth boundary in R^N,γ>0,λ≥0,p>0,g∈C~α_(loc)(Ω) and b_0~β_1≤g≤b_1~β_1 on Ω, β∈R,b_0,b_1 are positive constants,φ_1 is the normal first eigenfunction.