POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

doi:10.1007/s10473-021-0522-5

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (5): 1764-1776.doi: 10.1007/s10473-021-0522-5

POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

刘玲君¹, 石飞林²

1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

收稿日期:2019-08-30 修回日期:2020-11-04 出版日期:2021-10-25 发布日期:2021-10-21
通讯作者: Feilin SHI E-mail:shifeilin1116@163.com
作者简介:Lingjun LIU,E-mail:liulingjun@amss.ac.cn
基金资助:
Feilin Shi was supported by National Natural Science Foundation of China (11801167) and Hunan Provincial Natural Science Foundation of China (2019JJ50387).

POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

Lingjun LIU¹, Feilin SHI²

1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

Received:2019-08-30 Revised:2020-11-04 Online:2021-10-25 Published:2021-10-21
Contact: Feilin SHI E-mail:shifeilin1116@163.com
Supported by:
Feilin Shi was supported by National Natural Science Foundation of China (11801167) and Hunan Provincial Natural Science Foundation of China (2019JJ50387).

摘要/Abstract

摘要： In this paper, we will study the nonlocal and nonvariational elliptic problem \begin{equation} \left\{\begin{array}{ll}\label{eq0.1} -(1+a||u||_q^{\alpha q})\Delta u=|u|^{p-1}u+h(x,u,\nabla u) & \mbox{in}\ \ \Omega,\\ u=0 & \mbox{on}\ \ \partial\Omega,\\ \end{array} (0.1)\right. \end{equation} where $ a>0, \alpha>0, 1< q< 2^*, p\in(0,2^*-1)\setminus\{1\}$ and $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ $(N\geq 2)$. Under suitable assumptions about $h(x,u,\nabla u)$, we obtain \emph{a priori} estimates of positive solutions for the problem (0.1). Furthermore, we establish the existence of positive solutions by making use of these estimates and of the method of continuity.

关键词: positive solutions, nonvariational elliptic problem, a priori estimates

Abstract: In this paper, we will study the nonlocal and nonvariational elliptic problem \begin{equation} \left\{\begin{array}{ll}\label{eq0.1} -(1+a||u||_q^{\alpha q})\Delta u=|u|^{p-1}u+h(x,u,\nabla u) & \mbox{in}\ \ \Omega,\\ u=0 & \mbox{on}\ \ \partial\Omega,\\ \end{array} (0.1)\right. \end{equation} where $ a>0, \alpha>0, 1< q< 2^*, p\in(0,2^*-1)\setminus\{1\}$ and $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ $(N\geq 2)$. Under suitable assumptions about $h(x,u,\nabla u)$, we obtain \emph{a priori} estimates of positive solutions for the problem (0.1). Furthermore, we establish the existence of positive solutions by making use of these estimates and of the method of continuity.

Key words: positive solutions, nonvariational elliptic problem, a priori estimates

中图分类号:

35A05

刘玲君, 石飞林. POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM[J]. 数学物理学报(英文版), 2021, 41(5): 1764-1776.

Lingjun LIU, Feilin SHI. POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM[J]. Acta mathematica scientia,Series B, 2021, 41(5): 1764-1776.

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POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

POSITIVE SOLUTIONS OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

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