Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior

Existence and bifurcation of positive solutions to a Kirchhoff type equation{−(a+b∫Ω|∇u|2)Δu=νf(x,u),in Ω,u=0,on ∂Ω are considered by using topological degree argument and variational method. Here f is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. The new results fill in a gap of recent research about the Kirchhoff type equation in bounded domain, and in our results the nonlinearity may be resonant near zero or infinity.