含分数阶 $p$-Laplacian 算子基尔霍夫方程解的存在性及其渐近行为

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 434-449.

含分数阶 $p$ -Laplacian 算子基尔霍夫方程解的存在性及其渐近行为

孟笑莹(),陆璐^*()

中南财经政法大学统计与数学学院武汉 430073

收稿日期:2024-07-15 修回日期:2024-10-15 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 陆璐 E-mail:mxy922@163.com;lulu@zuel.edu.cn
作者简介:孟笑莹，E-mail: mxy922@163.com
基金资助:
国家自然科学基金(11771127)

Existence and Asymptotic Behavior of Solutions for Kirchhoff Equations Involving the Fractional $p$ -Laplacian

Xiaoying Meng(),Lu Lu^*()

School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073

Received:2024-07-15 Revised:2024-10-15 Online:2025-04-26 Published:2025-04-09
Contact: Lu Lu E-mail:mxy922@163.com;lulu@zuel.edu.cn
Supported by:
NSFC(11771127)

摘要/Abstract

摘要：

该文主要考虑一类含分数阶 $p$ -Laplacian 算子基尔霍夫型方程正规化解的存在性和渐近行为. 利用能量估计技巧, 当 $q\in (p,\ p+\frac{2sp^2}{N})$ 时, 该文得到了含分数阶 $p$ -Laplacian 算子基尔霍夫型方程正规化解的存在性与非线性项指数 $q$ 和预定值 $c$ (其中 $\int_{\mathbb{R}^N}|u|^p{\rm d}x=c^p$ ) 的一个完整的分类. 当 $q\geq p+\frac{2sp^2}{N}$ 时, 该文得到了方程山路型正规化解的存在性. 进一步探讨了正规化解关于 $c$ 的渐近行为.

关键词: 基尔霍夫方程, 正规化解, 存在性, 渐近行为

Abstract:

In this paper, we are interested in the existence of normalized solutions for some fractional Kirchhoff equations with $p$ -Laplacian operator. For the existence and nonexistence of normalized solutions, using the method of energy estimates, we give a complete classification with respect to nonlinear term exponent $q$ and an explicit threshold value of $c$ (with $\int_{\mathbb{R}^N}|u|^p{\rm d}x=c^p$ ) in the range $q\in (p,p+\frac{2sp^2}{N})$ . We also derive some existence of mountain pass type normalized solutions on the $L^2$ manifold in the range $q\geq p+\frac{2sp^2}{N}$ . Furthermore, some asymptotic behaviors with respect to $c$ were also given.

Key words: Kirchhoff equation, normalized solutions, existence, asymptotic behavior

中图分类号:

O175.25

孟笑莹,陆璐. 含分数阶 $p$ -Laplacian 算子基尔霍夫方程解的存在性及其渐近行为[J]. 数学物理学报, 2025, 45(2): 434-449.

Xiaoying Meng,Lu Lu. Existence and Asymptotic Behavior of Solutions for Kirchhoff Equations Involving the Fractional $p$ -Laplacian[J]. Acta mathematica scientia,Series A, 2025, 45(2): 434-449.

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