全空间$\mathbb{R} ^N$上双相问题径向解的多解性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 433-446.

全空间 $\mathbb{R} ^N$ 上双相问题径向解的多解性

葛斌^*(),袁文硕()

哈尔滨工程大学数学科学学院哈尔滨 150001

收稿日期:2021-08-05 修回日期:2022-04-25 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 葛斌，E-mail:gebin791025@hrbeu.edu.cn
作者简介:袁文硕，E-mail: yuanwenshuo@hrbeu.edu.cn
基金资助:
国家自然科学基金(11201095);中央高校基本科研业务费(3072022TS2402);黑龙江省博士后科研启动基金(LBH-Q14044);黑龙江省自然科学基金留学回国基金(LC201502)

Existence and Multiplicity of Radial Solutions for Double Phase Problem on the Entire Space $\mathbb{R} ^N$

Ge Bin(),Yuan Wenshuo()

College of Mathematical Sciences, Harbin Engineering University, Harbin 150001

Received:2021-08-05 Revised:2022-04-25 Online:2023-04-26 Published:2023-04-17
Supported by:
National Natural Science Foundation of China(11201095);Fundamental Research Funds for the Central Universities(3072022TS2402);Postdoctoral Research Startup Foundation of Heilongjiang(LBH-Q14044);Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province(LC201502)

摘要/Abstract

摘要： 该研究涉及以下双相问题 $\begin{array}{ll}-{\rm div}(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u) +|u|^{p-2}u+\mu(x)|u|^{q-2}u =\lambda f(x,u),\;x\in \mathbb{R} ^N, &\\ \end{array}$ 其中 1 < p < q < N, $\frac{q}{p}\leq 1+\frac{\alpha}{N}$ , $\lambda$ 是实参数, $0\leq\mu\in C^{0,\alpha}(\mathbb{R} ^N)$ , $\alpha\in(0,1]$ 且 $f: \mathbb{R} ^N\times\mathbb{R} \rightarrow \mathbb{R}$ 满足 Carathéodory 条件. 目的是通过利用抽象的临界点理论来确定参数 $\lambda$ 的精确正区间, 使该问题允许至少一个或两个非平凡径向对称解.

关键词: 双相算子, 径向对称解, 临界点定理, 变分方法

Abstract: This study is concerned with the following double phase problem $\begin{array}{ll}-{\rm div}(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u) +|u|^{p-2}u+\mu(x)|u|^{q-2}u =\lambda f(x,u),\;x\in \mathbb{R} ^N, &\\ \end{array}$ where 1 < p < q < N, $\frac{q}{p}\leq 1+\frac{\alpha}{N}$ , $\lambda$ is a real parameter, $0\leq\mu\in C^{0,\alpha}(\mathbb{R} ^N)$ with $\alpha\in(0,1]$ and $f: \mathbb{R} ^N\times\mathbb{R} \rightarrow \mathbb{R}$ satisfies a Carathéodory condition. The aim is to determine the precise positive interval of $\lambda$ for which the problem admits at least one or two nontrivial radially symmetric solutions by applying abstract critical point results.

Key words: Double phase operator, Radially symmetric solutions, Critical point theorems, Variational methods

中图分类号:

O175

葛斌, 袁文硕. 全空间 $\mathbb{R} ^N$ 上双相问题径向解的多解性[J]. 数学物理学报, 2023, 43(2): 433-446.

Ge Bin, Yuan Wenshuo. Existence and Multiplicity of Radial Solutions for Double Phase Problem on the Entire Space $\mathbb{R} ^N$ [J]. Acta mathematica scientia,Series A, 2023, 43(2): 433-446.

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