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Multiple nodal solutions of the Kirchhoff-type problem with a cubic term
Advances in Nonlinear Analysis ( IF 3.2 ) Pub Date : 2022-03-09 , DOI: 10.1515/anona-2022-0225
Tao Wang 1 , Yanling Yang 1 , Hui Guo 1
Affiliation  

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.

中文翻译:

具有三次项的基尔霍夫型问题的多节点解

在本文中,我们对以下 Kirchhoff 型问题感兴趣 (0.1) - 一种 + b R ñ 或者 d X Δ 或者 + v ( X ) 或者 = 或者 或者 R ñ , 或者 H ( R ñ ) , \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. 在哪里 一种 , b > 0 , ñ = a,b\gt 0,N=2 或3、势函数 v v 是径向的,从下方以正数为界。因为非本地 b 或者 大号 ( R ñ ) Δ 或者 乙| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u 是与非线性项复杂竞争的 3-齐次 或者 或者 | u\hspace{-0.25em}{| }^{2}你 . 这导致并非所有功能都在 H ( R ñ ) {H}^{1}\left({{\mathbb{R}}}^{N}) 可以投影到 Nehari 流形上,因此经典的 Nehari 流形方法不起作用。通过引入 Gersgorin Disk 定理和 Miranda 定理,通过极限方法和微妙的分析,我们证明了对于每个正整数 ķ ķ , 方程 (0.1) 承认径向节点解 或者 ķ , 4 b {U}_{k,4}^{b} 完全有 ķ ķ 不要给。此外,我们证明了能量 或者 ķ , 4 b {U}_{k,4}^{b} 严格增加 ķ ķ 对于任何序列 { b n } \左\{{b}_{n}\右\} b n 0 + , {b}_{n}\到 {0}_{+}, 直到一个子序列, 或者 ķ , 4 b n {U}_{k,4}^{{b}_{n}} 收敛到 或者 ķ , 4 0 {U}_{k,4}^{0} H ( R ñ ) {H}^{1}\left({{\mathbb{R}}}^{N}) , 这是一个径向节点解 ķ ķ 经典薛定谔方程的节点 - 一种 Δ 或者 + v ( X ) 或者 = 或者 或者 R ñ , 或者 H ( R ñ ) . \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. 我们的结果将存在性结果从超立方情况扩展到立方情况。
更新日期:2022-03-09
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