一类非线性二阶边值问题正解的存在性与多解性  

Existence and multiplicity of positive solutions for a class of second-order boundary value problems

作　　者：马满堂 贾凯军 MA Man-Tang;JIA Kai-Jun(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]西北师范大学数学与统计学院

出　　处：《四川大学学报（自然科学版）》2019年第6期1014-1018,共5页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(11671322)

摘　　要：本文考虑非线性二阶边值问题{q(t)u′(t))′+f(u′(t))=0,t∈(0,1),q(0)u′(0)=0,cu(1)+dq(1)u′(1)=0正解的存在性及多解性,其中f:(-∞,0]→[0,∞),q:[0,1]→(0,∞)为连续函数,c>0,d≥0为常数.当非线性项f满足超线性增长或次线性增长的条件时,本文证明该问题至少存在一个正解.当非线性项f满足f0:=lim s→0-f(s)/s=f∞:=lim s→-∞f(s)/s=0或f 0:=lim s→0-f(s)/s=f∞:=lim s→-∞f(s)s=∞的条件时,本文证明该问题至少存在两个正解.主要结果的证明基于锥上的不动点定理.In this paper,existence and multiplicity of positive solutions of the nonlinear second-order boundary value problems{(q(t)u′(t))′+f(u′(t))=0,t∈(0,1),q(0)u′(0)=0,cu(1)+dq(1)u′(1)=0 are considered,where f:(-∞,0]→[0,∞),q:[0,1]→(0,∞)are continuous functions,c>0,d≥0 are constants.When the nonlinear term f satisfies superlinear growth condition or sublinear growth condition,we show that there exists at least one positive solution to the problem.When the nonlinear term f satisfies f 0:=lim s→0-f(s)s=f∞:=lim s→-∞f(s)/s=0 or f 0:=lim s→0-f(s)/s=f∞:=lim s→-∞f(s)/s=∞,we show that there are at least two positive solutions to the problem.The proof is based on the fixed point theorem on cones.

关 键 词：正解 存在性 多解性 锥 不动点指数 

分 类 号：O175.8[理学—数学]