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In this article, we study the following Kirchhoff equation:(0.1)−(a+b‖∇u‖L2(R3)2)Δu+V(∣x∣)u=f(u)inR3,-(a+b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2})\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},wherea,b&gt;0a,b\gt 0,VVis a positive radial potential function, andf(u)f\left(u)is an asymptotically cubic term. The nonlocal termb‖∇u‖L2(R3)2Δub\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta uis 3-homogeneous in the sense thatb‖∇tu‖L2(R3)2Δ(tu)=t3b‖∇u‖L2(R3)2Δub\Vert \nabla tu{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta \left(tu)={t}^{3}b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u, so it competes complicatedly with the asymptotically cubic termf(u)f\left(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 integerkk, equation ( 0.1 ) has a radial nodal solutionUk,4b{U}_{k,4}^{b}, which has exactlyk+1k+1nodal domains. Moreover, we show that the energy ofUk,4b{U}_{k,4}^{b}is strictly increasing inkk, and for any sequence{bn}→0+,\left\{{b}_{n}\right\}\to {0}_{+},up to a subsequence,Uk,4bn{U}_{k,4}^{{b}_{n}}converges strongly toUk,40{U}_{k,4}^{0}inH1(R3){H}^{1}\left({{\mathbb{R}}}^{3}), whereUk,40{U}_{k,4}^{0}also hask+1k+1nodal domains exactly and solves the classical Schrödinger equation:−aΔu+V(∣x∣)u=f(u)inR3.-a\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}.Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.

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