含有Hénon项的非线性椭圆系统稳定解的Liouville定理
PDF下载 (34)黄宇鑫,甘怡清,胡良根.含有Hénon项的非线性椭圆系统稳定解的Liouville定理[J].宁波大学学报(理工版),2024,37(2):86-95.DOI:10.20098/j.cnki.1001-5132.2023.0719
Huang Yuxin,Gan Yiqing,Hu Lianggen.Liouville theorems for stable solutions of a nonlinear elliptic system with Hénon term[J].Journal of Ningbo University(Natural Science & Engineering Edition),2024,37(2):86-95.DOI:10.20098/j.cnki.1001-5132.2023.0719
| Title: | Liouville theorems for stable solutions of a nonlinear elliptic system with Hénon term |
| 作者: | 黄宇鑫, 甘怡清, 胡良根 |
| Author(s): | Huang Yuxin, Gan Yiqing, Hu Lianggen |
| 关键词: | Liouville定理; 稳定解; Pohozaev恒等式; 单调公式; 爆缩序列 |
| Keywords: | Liouville theorem; stable solutions; Pohozaev identity; monotonicity formula; blowing down sequence |
| 分类号: | O186 |
| DOI: | 10.20098/j.cnki.1001-5132.2023.0719 |
| 文献标识码: | A |
| 摘要: | 本文主要研究含有Hénon项的k-耦合椭圆系统 , 其中k是一个固定的正整数, , , , 为实对称矩阵. 首先使用Pohozaev恒等式构造单调公式, 并发现其等价关系. 当矩阵B严格余正时, 联合使用Pohozaev恒等式、单调公式和爆缩序列方法证明了(无论是正的还是变号的)稳定解的Liouville定理 |
| Abstract: | In this paper, we focus on the k-coupled elliptic system with Hénon term in , where k is a fixed positive integer, , , , and is a real symmetric matrix. We first use Pohozaev identity to construct monotonicity formula and find their equivalent relation. For the case that the matrix B is strictly co-positive, we obtain Liouville theorems of stable solutions (whether positive or sign-changing), by the use of Pohozaev identity, monotonicity formula of solution together with a blowing down sequence |
| 参考文献 /References: | [1].Frantzeskakis D J. Dark solitons in atomic Bose-Einstein condensates: from theory to experiments[J]. Journal of Physics A: Mathematical and Theoretical, 2010, 43(21):213001. [2].Kivshar Y S, Luther-Davies B. Dark optical solitons: physics and applications[J]. Physics Reports, 1998, 298(2/3):81-197. [3].Dancer E N, Wei J C, Weth T. A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system[J]. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 2010, 27(3):953-969. [4].Tavares H, Terracini S, Verzini G, et al. Existence and nonexistence of entire solutions for non- cooperative cubic elliptic systems[J]. Communications in Partial Differential Equations, 2011, 36(11):1988-2010. [5].Quittner P, Souplet P. Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications[J]. Communications in Mathematical Physics, 2012, 311(1):1-19. [6].Wei J C, Weth T. Radial solutions and phase separation in a system of two coupled Schrödinger equations[J]. Archive for Rational Mechanics and Analysis, 2008, 190(1):83-106. [7].Le P, Rahal B. On stable and finite Morse index solutions to quasilinear Schrödinger equations[J]. Nonlinear Differential Equations and Applications, 2022, 29(4):46. [8].Yang H, Zou W M. Stable and finite Morse index solutions of a nonlinear elliptic system[J]. Journal of Mathematical Analysis and Applications, 2019, 471(1/2): 147-169. [9].Le P. Stable and finite Morse index solutions of a nonlinear Schrödinger system[J]. Nonlinear Differential Equations and Applications, 2021, 28(4):39. [10].Dancer E N, Weth T. Liouville-type results for non- cooperative elliptic systems in a half-space[J]. Journal of the London Mathematical Society, 2012, 86(1):111-128. [11].Souplet P. The proof of the Lane-Emden conjecture in four space dimensions[J]. Advances in Mathematics, 2009, 221(5):1409-1427. [12].Bahri A, Lions P L. Solutions of superlinear elliptic equations and their Morse indices[J]. Communications on Pure and Applied Mathematics, 1992, 45(9):1205-1215. [13].Farina A. On the classification of solutions of the Lane-Emden equation on unbounded domains of [J]. Journal de Mathématiques Pures et Appliquées, 2007, 87(5):537-561. [14].Dávila J, Dupaigne L, Wang K L, et al. A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem[J]. Advances in Mathematics, 2014, 258:240-285. [15].Hu L G. A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system[J]. Advances in Differential Equations, 2017, 22(1/2):49-76. [16].Ao W W, Lai S S, Luo S P. Classification of finite Morse index solutions to the polyharmonic Hénon equation[J]. Calculus of Variations and Partial Differential Equations, 2023, 62:19. [17].Fazly M, Hu Y Y, Yang W. On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand- Liouville equation[J]. Calculus of Variations and Partial Differential Equations, 2021, 60:26. [18].Luo S P, Wei J C, Zou W M. Monotonicity formula and classification of stable solutions to polyharmonic Lane-Emden equations[J]. International Mathematics Research Notices, 2022, 2022(21):16902-16953. [19].Wang K L. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations[J]. Electronic Research Archive, 2021, 29(6):3805-3816. [20].Dancer E N, Du Y H, Guo Z M. Finite Morse index solutions of an elliptic equation with supercritical exponent[J]. Journal of Differential Equations, 2011, 250(8):3281-3310. |
| 备注/Memo: | 收稿日期: 2023−07−25. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/ 基金项目: 宁波市自然科学基金（2023J127）. 第一作者: 黄宇鑫, 硕士研究生, 主要研究方向: 非线性泛函分析. Email: 2289862186@qq.com *通信作者: 胡良根, 教授, 主要研究方向: 非线性泛函分析. Email: hulianggen@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
