Kirchhoff-类椭圆方程的单边全局分歧和保号解的存在性问题  

Unilateral Global Bifurcation and One-Sign Solutions for Kirchhoff-Type Elliptic Equations

作　　者：沈文国 SHEN Wen-guo(College of General Education,Guangdong University of Science and Technology,Dongguan 523083,China)

机构地区：[1]广东科技学院通识教育学院,广东东东莞523083

出　　处：《数学的实践与认识》2024年第7期186-191,共6页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11561038)。

摘　　要：将研究下列问题单边全局分歧及保号解的存在性{-M(∫δΩ|Δ↓|^(2)dx)△u=λa(x)f(u),x∈Ω,u=0,x∈δΩ,其中Ω是一个R^(N)中有界并且在其边界上光滑的区域,M(·)∈C(R^(+)),λ是一个参数,a(x)∈C(Ω,(0,+∞)),f∈C(R,R),对于s≠0,满足sf(s)>0.当f_(0)(0,+∞)或f_(∞)■(0,+∞)(其中f_(0)=lim_(|s|→0)f(s)/s,f_(∞)=lim_(|s|→+∞)f(s)/s.),λ≠0属于给定区间时,研究上述K-类方程保号解的存在性.我们用单边全局分歧技巧和连通序列集取极限的方法获得主要结果.In this paper,we study the unilateral global bifurcation and one-sign solutions for the following problems:{-M(∫δΩ|Δ↓|^(2)dx)△u=λa(x)f(u),x∈Ω,u=0,x∈δΩ,whereΩis a bounded domain in R^(N)with a smooth boundaryδΩ,M is a continuous function on R^(+),λis a parameter,a(x)∈C(Ω,(0,+∞)),f∈C(R,R)with sf(s)>0 for s≠0.We give the intervals for the parameterλ≠0 which ensure the existence of positive solutions for the above Kirchhoff type equations if fo&(O,oo)or fo&(O,co),f_(∞)■(0,+∞)or f_(∞)■(0,∞)where f_(0)=lim_(|s|→0)f(s)/s,f_(∞)=lim_(|s|→+∞)f(s)/s.We use Global bifurcation techniques and the approximation of connected components to prove our main results.

关 键 词：K-类方程 单边全局分歧 保号解 非线性项在零点和无穷远处非渐进增长 

分 类 号：O175.25[理学—数学]