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 \(|u|^{p-2}u(3< p < 4)\) in \(\mathbb {R}^3\) . 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 \(u_k^b\) changing signs exactly k times. Furthermore, the energy of \(u_k^b\) is strictly increasing in k, as well as some asymptotic behaviors of \(u_k^b\) are obtained. Our result is a complement of Deng (J Funct Anal 269:3500–3527, 2015), where the case \(2<p<4\) is left open.
