1-维次线性p-Laplacian方程的无穷多周期解

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1462-1472.

1-维次线性p-Laplacian方程的无穷多周期解

王学蕾()

^{山东农业大学信息科学与工程学院山东泰安 271018}

收稿日期:2021-03-26 出版日期:2022-10-26 发布日期:2022-09-30
作者简介:王学蕾, E-mail: wangxl@sdau.edu.cn
基金资助:
国家自然科学基金(11671287);国家自然科学基金(61573228)

Large Multiple Periodic Solutions for the 1-Dimensional Sub-Linear p-Laplacian Equation

Xuelei Wang()

^{Department of Mathematics, College of Information Science and Engineering, Shandong Agricultural University, Shandong Taian 271018}

Received:2021-03-26 Online:2022-10-26 Published:2022-09-30
Supported by:
the NSFC(11671287);the NSFC(61573228)

摘要/Abstract

摘要：

该文研究1-维p-Laplacian方程 $({\left|x'\right|}^{p-2} x')'+f(t, x)=0$ end{document}周期解的存在性和多解性, 其中 $f(t, x)$ 满足原点附近的次线性条件, 即 $\lim\limits_{\mid x\mid \rightarrow 0} \frac{f(t, x)}{\mid x\mid^{p-2}x}= 0$ .得到的存在性结果可以应用于经典方程 $x''+f(t, x)=0$ . 证明方法基于Poincaré-Birkhoff扭转定理.

关键词: Hamiltonian系统, 周期解, Poincaré-Birkhoff扭转定理, 盘旋性质

Abstract:

In this paper, we obtain existence and multiplicity of periodic solutions for 1-dimensional p-Laplacian equation $(|x'|^{p-2}x')'+f(t, x)=0$ , where $f\in C(\mathbb{R} \times\mathbb{R} , \mathbb{R} )$ is \pi$-periodic in the first variable and satisfies the assumption $\frac{f(t, x)}{\mid x\mid^{p-2}x}\rightarrow 0$ , as $\mid x\mid \rightarrow 0$ . The new existence results can be applied to situations in which the more classical equation $x''+f(t, x)=0$ . Proofs are based on Poincaré-Birkhoff twist theorem.

Key words: Hamiltonian systems, Periodic solution, Poincaré-Birkhoff twist theorem, Spiral property

中图分类号:

O175.14

王学蕾. 1-维次线性p-Laplacian方程的无穷多周期解[J]. 数学物理学报, 2022, 42(5): 1462-1472.

Xuelei Wang. Large Multiple Periodic Solutions for the 1-Dimensional Sub-Linear p-Laplacian Equation[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1462-1472.

参考文献 16

1	Manásevich R , Zanolin F . Time mapping and multiplicity of solutions for the one-dimensional p-Laplacian. Nonlinear Analysis TMA, 1993, 21, 269- 291 doi: 10.1016/0362-546X(93)90020-S
2	Del Pino M A , Manásevich R . Infinitely many 2π-periodic solutions for a problem arising in nonlinear elasticity. J.Differential Equations, 1993, 103, 260- 277 doi: 10.1006/jdeq.1993.1050
3	Amster P , Napoli P . Landesman-Lazer type conditions for a system of p-Laplacian like operators. J Math Anal Appl, 2007, 326, 1236- 1243 doi: 10.1016/j.jmaa.2006.04.001
4	Del Pino M A , Manásevich R . Multiple solutions for the p-Laplacian under global nonresonance. Proc Amer Math Soc, 1991, 112, 131- 138
5	Del Pino M A , Drábek P , Manásevich R . The Fredholm alternative at the first eigen-value for the one-dimensional p-Laplacian. J Differential Equations, 1999, 151, 386- 419 doi: 10.1006/jdeq.1998.3506
6	Del Pino M A , Manásevich R , Elgueta M . A homotopic deformation along p of a Leray-Schauder degree results and existence for $(\|u\|^{p-2}u')'+f(t, u)=0, u(0)=u(1)=0, p>1.$ . J Differential Equations, 1989, 80, 1- 13 doi: 10.1016/0022-0396(89)90093-4
7	Drábek P , Takac P . A counterexample to the Fredholm alternative for the p-Laplacian. Proc Amer Math Soc, 1999, 127, 1079- 1087 doi: 10.1090/S0002-9939-99-05195-3
8	Drábek P , Robinson S B . Resonance problems for the the one-dimensional p-Laplacian. Proc Amer Math Soc, 2000, 128, 755- 765
9	Manásevich R , Zanolin F . Time mapping and multiplicity of solutions for the one-dimensional p-Laplacian. Nonlinear Analysis TMA, 1992, 18, 79- 92 doi: 10.1016/0362-546X(92)90048-J
10	Zhang M . Nonuniform nonresonance of semilinear differential equations. J Differential Equations, 2000, 166, 33- 50 doi: 10.1006/jdeq.2000.3798
11	Xiong M , Wu S , Liu J . Periodic solutions for the 1-dismensional p-Laplacian equation. J Math Anal Appl, 2007, 325, 879- 888 doi: 10.1016/j.jmaa.2006.02.027
12	Chu K D , Hai D D . Positive solutions for the one-dimensional singular superlinear p-laplacian problem. Communications on Pure and Applied Analysis, 2020, 19 (1): 241- 252 doi: 10.3934/cpaa.2020013
13	Jacobowitz H . Periodic solutions of $x''+f(x, t)=0$ via the Poincaré -Birkhoff theorem. J Differential Equations, 1976, 20, 37- 52 doi: 10.1016/0022-0396(76)90094-2
14	相福香. 一维p-次线性Laplacian方程的无穷多次调和解. 硕士学位论文. 苏州: 苏州大学, 2009
	Xiang F X. Infinitely Mang Subharmonic Solutions for One-dimensional p-sublinear Laplacian Equations. Master's Degree Thesis. Suzhou: Soochow University, 2009
15	Hale J. Ordinary Differential Equations. Mineola New York: Dover Publications, INC, 2009
16	丁同仁. 常微分方程定性方法的应用. 北京: 高等教育出版社, 2004,
	Ding T . Applications of Qualitative Methods of Ordinary Differential Equations. Beijing: Higher Education Press, 2004,

Metrics

Viewed

Full text

144

HTML			PDF

Just accepted	Online first	Issue	Just accepted	Online first	Issue
0	0	1	0	0	143

From	Others	local

Times	3	141
Rate	2%	98%

Abstract

Just accepted	Online first	Issue

0	0	95

	From	Others

	Times	95
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	邓楠,冯美强. 电报方程的正双周期解: 存在性、唯一性、多重性和渐近性[J]. 数学物理学报, 2022, 42(5): 1360-1380.
[2]	姚绍文,李文洁,程志波. 三阶非线性微分方程周期解的非退化和存在唯一性[J]. 数学物理学报, 2022, 42(2): 454-462.
[3]	兰军. 一类二阶Duffing方程反周期解的存在性和多重性[J]. 数学物理学报, 2022, 42(2): 463-469.
[4]	王长有,李楠,蒋涛,杨强. 一类具有时滞及反馈控制的非自治非线性比率依赖食物链模型[J]. 数学物理学报, 2022, 42(1): 245-268.
[5]	杨静,柯昌成,魏周超. 一类连续和不连续分段线性系统的周期解研究[J]. 数学物理学报, 2021, 41(4): 1053-1065.
[6]	张清业,徐斌. 一类带局部非线性项的静态狄拉克方程的多重周期解[J]. 数学物理学报, 2021, 41(4): 1013-1023.
[7]	鲁世平,周诗乐,余星辰. 具有不确定奇性的Liénard方程周期正解的存在性[J]. 数学物理学报, 2021, 41(3): 686-701.
[8]	蒋婷婷,杜增吉. 带有脉冲和Holling-IV型功能反应函数的中立型捕食-食饵模型的周期解[J]. 数学物理学报, 2021, 41(1): 178-193.
[9]	曹忠威,文香丹,冯微,祖力. 一类具有随机扰动的非自治SIRI流行病模型的动力学行为[J]. 数学物理学报, 2020, 40(1): 221-233.
[10]	杨纪华,张二丽. 一类Hamiltonian系统的Abelian积分的零点[J]. 数学物理学报, 2019, 39(5): 1025-1032.
[11]	王超. 一类带有界回复力非线性Hill型方程的周期解[J]. 数学物理学报, 2019, 39(4): 761-772.
[12]	阳超,李润洁. 一类具有不连续捕获的Lasota-Wazewska模型周期解存在性及稳定性分析[J]. 数学物理学报, 2019, 39(4): 785-796.
[13]	童常青,郑静. 半线性Klein-Gordon方程的高频周期解[J]. 数学物理学报, 2019, 39(3): 484-500.
[14]	程志波,毕中华,姚绍文. 一类具有偏差变元的p-Laplacian Liénard型方程在吸引奇性条件下周期解的存在性[J]. 数学物理学报, 2019, 39(2): 277-285.
[15]	蓝桂杰,付盈洁,魏春金,张树文. 具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解[J]. 数学物理学报, 2018, 38(5): 984-1000.