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

Abstract

Abstract In this article, we study the following Kirchhoff equation: (0.1) − ( a + b ‖ ∇ u ‖ L 2 ( R 3 ) 2 ) Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 , -(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}, where a , b > 0 a,b\gt 0 , V V is a positive radial potential function, and f ( u ) f\left(u) is an asymptotically cubic term. The nonlocal term b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u is 3-homogeneous in the sense that b ‖ ∇ t u ‖ L 2 ( R 3 ) 2 Δ ( t u ) = t 3 b ‖ ∇ u ‖ L 2 ( R 3 ) 2 Δ u b\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 term f ( 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 integer k k , equation (0.1) has a radial nodal solution U k , 4 b {U}_{k,4}^{b} , which has exactly k + 1 k+1 nodal domains. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k , and for any sequence { b n } → 0 + , \left\{{b}_{n}\right\}\to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges strongly to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R 3 ) {H}^{1}\left({{\mathbb{R}}}^{3}) , where U k , 4 0 {U}_{k,4}^{0} also has k + 1 k+1 nodal domains exactly and solves the classical Schrödinger equation: − a Δ u + V ( ∣ x ∣ ) u = f ( u ) in R 3 . -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.