Global bifurcation and positive solution for a class of fully nonlinear problems

In this paper, we study global bifurcation phenomena for the following Kirchhoff type problem{−M(∫Ω|∇u(x)|2dx)Δu=λf(x,u)inΩ,u=0on∂Ω, where MM is a continuous function. Under some natural hypotheses, we show that (λ1(a)M(0),0)(λ1(a)M(0),0) is a bifurcation point and there is a global continuum CC emanating from (λ1(a)M(0),0)(λ1(a)M(0),0), where λ1(a)λ1(a) denotes the first eigenvalue of the above problem with f(x,s)=a(x)sf(x,s)=a(x)s. As an application of the above result, we study the existence of positive solution for this problem with asymptotically linear nonlinearity.