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

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.