Multiple nodal solutions of the Kirchhoff-type problem with a cubic term

Abstract

Abstract In this article, we are interested in the following Kirchhoff-type problem (0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a , b > 0 , N = 2 a,b\gt 0,N=2 or 3, the potential function V V is radial and bounded from below by a positive number. Because the nonlocal b ∣ ∇ u ∣ L 2 ( R N ) 2 Δ u b| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣ u ∣ 2 u | u\hspace{-0.25em}{| }^{2}u . This causes that not all function in H 1 ( R N ) {H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer k k , equation (0.1) admits a radial nodal solution U k , 4 b {U}_{k,4}^{b} having exactly k k nodes. 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 } \left\{{b}_{n}\right\} with b n → 0 + , {b}_{n}\to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R N ) {H}^{1}\left({{\mathbb{R}}}^{N}) , which is a radial nodal solution with exactly k k nodes of the classical Schrödinger equation − a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) . \left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case.