Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term

where a, b > 0, V is a positive radial potential function, and f (u) is an asymptotically cubic term. The nonlocal term-.. ()b. u.u L2 2 3 is 3-homogeneous in the sense that .. ().. () ()b. tu. tu = t b. u.u L L2 3 2 2 3 2 3, so it competes complicatedly with the asymptotically cubic term f (u), which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer k, equation (0.1) has a radial nodal solution U kb,4, which has exactly k + 1 nodal domains. Moreover, we show that the energy of U kb,4 is strictly increasing in k, and for any sequence {b}. 0+, n up to a subsequence, U kb,4n converges strongly to U k, 40 in H (-)1 3, where U k,40 also has k + 1 nodal domains exactly and solves the classical Schrodinger equation: - a.u + V(|x|) u = f (u) in 3. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.