Nodal solutions for a Kirchhoff type problem in RN

In this paper, we consider the existence of nodal solutions of the following Kirchhoff problem (KP)−(a+b∫RN|∇u|2dx)Δu+u=g(u),inRN,where a,b>0 and N≥3. We first prove that (KP) is equivalent to the following system with respect to (u,λ): (Fλ)−Δu+u=g(u),λ−a−bλN−22∫RN|∇u|2dx=0,inRN×R+.For every integer k>0, radial solutions of (KP) with exactly k nodes are obtained by dealing with the system (Fλ) under some suitable assumptions. Moreover, nonexistence results are proved if N≥4 and b is sufficiently large.