A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

doi:10.1007/s10473-022-0205-x

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (2): 502-510.doi: 10.1007/s10473-022-0205-x

A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

雷春雨¹, 廖家锋², 储昌木³, 索洪敏³

1. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China;
2. College of Mathematics Education, China West Normal University, Nanchong 637002, China;
3. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China

收稿日期:2020-03-23 修回日期:2021-03-17 出版日期:2022-04-25 发布日期:2022-04-22
通讯作者: Jiafeng LIAO,E-mail:liaojiafeng@163.com E-mail:liaojiafeng@163.com
作者简介:Chunyu LEI,E-mail:leichygzu@sina.cn;Changmu CHU,E-mail:gzmychuchangmu@sina.com;Hongmin SUO,E-mail:gzmysxx88@sina.com
基金资助:
The first author is supported by Natural Science Foundation of Guizhou Minzu University ([2018]5773-YB03); the second author is supported by Fundamental Research Funds of China West Normal University (18B015) and Innovative Research Team of China West Normal University (CXTD2018-8); the third author is supported by National Natural Science Foundation of China (11861021); the fourth author is supported by National Natural Science Foundation of China (11661021).

A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

Chunyu LEI¹, Jiafeng LIAO², Changmu CHU³, Hongmin SUO³

1. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China;
2. College of Mathematics Education, China West Normal University, Nanchong 637002, China;
3. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China

Received:2020-03-23 Revised:2021-03-17 Online:2022-04-25 Published:2022-04-22
Supported by:
The first author is supported by Natural Science Foundation of Guizhou Minzu University ([2018]5773-YB03); the second author is supported by Fundamental Research Funds of China West Normal University (18B015) and Innovative Research Team of China West Normal University (CXTD2018-8); the third author is supported by National Natural Science Foundation of China (11861021); the fourth author is supported by National Natural Science Foundation of China (11661021).

摘要/Abstract

摘要： We consider the logarithmic elliptic equation with singular nonlinearity \begin{equation*} \begin{cases} \Delta u+u\log u^2 +\frac{\lambda}{u^\gamma}=0, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ ($N\geq3$) is a bounded domain with a smooth boundary, $0<\gamma<1$ and $\lambda$ is a positive constant. By using a variational method and the critical point theory for a nonsmooth functional, we obtain the existence of two positive solutions. This result generalizes and improves upon recent results in the literature.

关键词: Logarithmic elliptic equation, singular nonlinearity, positive solutions, variational method

Abstract: We consider the logarithmic elliptic equation with singular nonlinearity \begin{equation*} \begin{cases} \Delta u+u\log u^2 +\frac{\lambda}{u^\gamma}=0, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ ($N\geq3$) is a bounded domain with a smooth boundary, $0<\gamma<1$ and $\lambda$ is a positive constant. By using a variational method and the critical point theory for a nonsmooth functional, we obtain the existence of two positive solutions. This result generalizes and improves upon recent results in the literature.

Key words: Logarithmic elliptic equation, singular nonlinearity, positive solutions, variational method

中图分类号:

35A15

雷春雨, 廖家锋, 储昌木, 索洪敏. A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY[J]. 数学物理学报(英文版), 2022, 42(2): 502-510.

Chunyu LEI, Jiafeng LIAO, Changmu CHU, Hongmin SUO. A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY[J]. Acta mathematica scientia,Series B, 2022, 42(2): 502-510.

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A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

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