On nodal solutions with a prescribed number of nodes for a Kirchhoff-type problem

We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity |u|p-2u(3<4) in R3. By developing some useful analysis techniques and introducing a novel definition of the Nehari manifold for the auxiliary system of the equations, we show that, for any positive integer k, the problem has a sign-changing solution ukb changing signs exactly k times. Furthermore, the energy of ukb is strictly increasing in k, as well as some asymptotic behaviors of ukb are obtained. Our result is a complement of Deng (J Funct Anal 269:3500–3527, 2015), where the case 2<4 is left open.