Unilateral Global Bifurcation and One-Sign Solutions for Kirchhoff-Type Elliptic Equations
In this paper,we study the unilateral global bifurcation and one-sign solutions for the following problems:{-M(∫Ω|▽u|2dx)△u=λa(x)f(u),in Ω,u=0,on ∂Ω,where Ω is a bounded domain in(R)N with a smooth boundary ∂Ω,M is a continuous function on(R)+,λ is a parameter,a(x)∈ C((Ω),(0,+oo)),f ∈ C((R),(R))with sf(s)＞0 for s ≠ 0.We give the intervals for the parameter λ≠0 which ensure the existence of positive solutions for the above Kirchhoff type equations if f0(∈)(0,∞)or f∞(∈)(0,∞),where f0=lim|s|→0f(s)/s,f∞=lim|s|→+∞ f(s)/s.We use Global bifurcation techniques and the approximation of connected components to prove our main results.
Kirchhoff-type equationsunilateral global bifurcationone-sign solutionsnon-asymptotic nonlinearity at 0 or ∞
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