一类奇异非线性Dirichlet问题解的存在性

Existence of Solutions to a Singular Nonlinear Dirichlet Problem

主要采用上下解方法,并结合极大值原理证明了一类奇异非线性Dirichlet问题-Δu=b(x)g(u)+λa(x)f(u),u>0,x∈Ω,u|Ω=0解的存在性.其中Ω为Rn(n≥2)中的有界光滑区域,λ<0,g在0处有奇性,且g'(s)<0,s∈(0,∞),f∈C([0,∞),[0,∞))∩C1((0,∞)),b,a>0在Ω上局部Hlder连续.

By constructing a pair of sub- and super-solutions of nonlinear elliptic equation and combining with the maximum principle, we prove the existence of solutions to a singular nonlinear Dirichlet problem ： -△u=b（x）g（u）＋λa（x）f（u）,u〉0,x∈Ω,u｜ Ω=0. Here Ω is a bounded smooth domain in Rn（n≥2）, g is singular at zero, and g＇（s）〈0,s∈（0,∞） , and f∈C（[0,∞）,[0,∞））∩C1（（0,∞））,b,a〉0 is local Holder con- tinuous in Ω.