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)|2 dx  ∆u = λf (x, u) in Ω, u = 0 on ∂Ω, where M is a continuous function. Under some natural hypotheses, we show that (λ1(a)M(0), 0) is a bifurcation point and there is a global continuum C emanating from (λ1(a)M(0), 0), where λ1(a) denotes the first eigenvalue of the above problem with f (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. © 2015 Elsevier Ltd. All rights reserved.