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 integral(Omega) vertical bar del vertical bar(2)) Delta u = upsilon f(x, u), in Omega, u=0, on partial derivative Omega 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. (C) 2013 Elsevier Masson SAS. All rights reserved.