PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION*

doi:10.1007/s10473-023-0310-5

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1161-1174.doi: 10.1007/s10473-023-0310-5

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

Bo Chen¹, Zhengmao Chen², Junhui Xie^3,†

1. School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
3. School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China

收稿日期:2021-11-11 修回日期:2022-10-18 出版日期:2023-06-25 发布日期:2023-06-06
通讯作者: ^† Junhui Xie, E-mail: smilexiejunhui@hotmail.com
作者简介:Bo Chen, E-mail: BoChenmath@outlook.com; Zhengmao Chen, E-mail: zhengmaochen@aliyun.com
基金资助:
The first author and the third author were supported by the National Natural Science Foundation of China (11761030) and the Cultivation Project for High-Level Scientific Research Achievements of Hubei Minzu University (PY20002). The second author was supported by the China Postdoctoral Science Foundation (2021M690773).

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

Bo Chen¹, Zhengmao Chen², Junhui Xie^3,†

1. School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China;
3. School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China

Received:2021-11-11 Revised:2022-10-18 Online:2023-06-25 Published:2023-06-06
Contact: ^† Junhui Xie, E-mail: smilexiejunhui@hotmail.com
About author:Bo Chen, E-mail: BoChenmath@outlook.com; Zhengmao Chen, E-mail: zhengmaochen@aliyun.com
Supported by:
The first author and the third author were supported by the National Natural Science Foundation of China (11761030) and the Cultivation Project for High-Level Scientific Research Achievements of Hubei Minzu University (PY20002). The second author was supported by the China Postdoctoral Science Foundation (2021M690773).

摘要/Abstract

摘要： In the present paper, we consider the problem
$\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1)
where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.

关键词: Harmonic-Mappings type equation, positive solution, existence, regularity, $\alpha$-concavity

Abstract: In the present paper, we consider the problem
$\begin{equation} \left\{\begin{array}{ll}\label{0001} -\Delta u=u^{\beta_1}|\nabla u|^{\beta_2}, &\ \ { in} \ \Omega,\\ u=0,&\ \ { on} \ \partial{\Omega},\\ u>0,&\ \ { in} \ {\Omega},\\ \end{array}\right. \end{equation}$ $ \ \ \ \ \ $ (0.1)
where $\beta_1,\beta_2>0$ and $\beta_1+\beta_2<1$, and $\Omega $ is a convex domain in $ \mathbb{R}^{n} $. The existence, uniqueness, regularity and $\frac{2-\beta_{2}}{1-\beta_1-\beta_2}$-concavity of the positive solutions of the problem (0.1) are proven.

Key words: Harmonic-Mappings type equation, positive solution, existence, regularity, $\alpha$-concavity

Bo Chen, Zhengmao Chen, Junhui Xie. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*[J]. 数学物理学报(英文版), 2023, 43(3): 1161-1174.

Bo Chen, Zhengmao Chen, Junhui Xie. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1161-1174.

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PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*

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