SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS*

doi:10.1007/s10473-023-0311-4

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1175-1194.doi: 10.1007/s10473-023-0311-4

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*

Xuemei Zhang^†, Shikun Kan

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

收稿日期:2022-01-20 修回日期:2022-06-14 出版日期:2023-06-25 发布日期:2023-06-06
通讯作者: ^† Xuemei Zhang, E-mail: zxm74@sina.com
作者简介:Shikun Kan, E-mail: kskmath@163.com
基金资助:
Beijing Natural Science Foundation (1212003).

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*

Xuemei Zhang^†, Shikun Kan

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received:2022-01-20 Revised:2022-06-14 Online:2023-06-25 Published:2023-06-06
Contact: ^† Xuemei Zhang, E-mail: zxm74@sina.com
About author:Shikun Kan, E-mail: kskmath@163.com
Supported by:
Beijing Natural Science Foundation (1212003).

摘要/Abstract

摘要： Let $\Omega$ denote a smooth, bounded domain in $ \mathbb{R}^N (N\geq 2)$. Suppose that $g$ is a nondecreasing $C^1$ positive function and assume that $b(x)$ is continuous and nonnegative in $\Omega$, and that it may be singular on $\partial\Omega$. In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the $p$-Laplacian problem
$ \Delta_p u=b(x)g(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } { dist}(x,\partial \Omega)\rightarrow 0$.
The estimates of such solutions are also investigated. Moreover, when $b$ has strong singularity, the nonexistence of boundary blow-up (radial) solutions and infinitely many radial solutions are also considered.

关键词: singular $p$-Laplacian equation, boundary blow-up, sub-supersolution method, existence, nonexistence and estimates, sufficient and necessary conditions

Abstract: Let $\Omega$ denote a smooth, bounded domain in $ \mathbb{R}^N (N\geq 2)$. Suppose that $g$ is a nondecreasing $C^1$ positive function and assume that $b(x)$ is continuous and nonnegative in $\Omega$, and that it may be singular on $\partial\Omega$. In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the $p$-Laplacian problem
$ \Delta_p u=b(x)g(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } { dist}(x,\partial \Omega)\rightarrow 0$.
The estimates of such solutions are also investigated. Moreover, when $b$ has strong singularity, the nonexistence of boundary blow-up (radial) solutions and infinitely many radial solutions are also considered.

Key words: singular $p$-Laplacian equation, boundary blow-up, sub-supersolution method, existence, nonexistence and estimates, sufficient and necessary conditions

Xuemei Zhang, Shikun Kan. SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(3): 1175-1194.

Xuemei Zhang, Shikun Kan. SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1175-1194.

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SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^*

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