一类奇异p-Laplace方程边值问题的正解

Positive Solution of Some Singular p-Laplace Equations

利用 Leray- Schauder非线性抉择定理 ,在比较弱的条件 :(1 )存在 (0 ,+∞ )上的连续函数g(y)使得∫10 g(s) ds<+∞ ,且 0≤ f (t,y)≤ g(y) , (t,y)∈ (0 ,1 )× (0 ,+∞ ) ;(2 )存在正数 L>0 ,使得对于每一个 n≥ 1 ,都存在一个εn>0满足 q(t) f (t,y) >L , (t,y)∈ en× (0 ,εn]下 ,获得一维奇异 p- Laplace方程 (|y′|p-2 y′)′+q(t) f(t,y) =0 ,y(0 ) =y(1 ) =0 ,p >1的一个正解存在定理 .

An existence theorem about the positive solution of the BVP for one dimensional singular p Laplace equations (｜y′｜ p-2 y′)′+q(t)f(t,y)=0,y(0)=y(1)=0,p>1 was established under the conditions (1). There exists a continuous function g(y) on (0,+∞) such that ∫ 1 0g(s) d s<+∞ and 0≤f(t,y)≤g(y),(t,y)∈(0,1)×(0,+∞);(2)There exists a constant L>0,such that there is a constant ε n>0 , for all n≥1,which satisfies that q(t)f(t,y)>L,(t,y)∈e n×(0,ε n]. The result is obtained by using the nonlinear alternative of Leray Schauder.