A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior

Yichen Zhang; Meiqiang Feng; Yichen Zhang; Meiqiang Feng

doi:10.3934/era.2020075

Electronic Research Archive

2020, Volume 28, Issue 4: 1419-1438. doi: 10.3934/era.2020075

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A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior

Yichen Zhang ,
Meiqiang Feng ^,

School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China

Received: 01 April 2020 Revised: 01 June 2020 Published: 31 July 2020
35J60, 35J66, 35J92

We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by $ p $-Laplacian elliptic equations
$ \begin{align*} \left \{ \begin{array}{l} -\Delta_p z_1 = \lambda_1 g_1(z_2)\ \ {\rm in}\ \Omega,\\ -\Delta_p z_2 = \lambda_2 g_2(z_1)\ \ {\rm in}\ \Omega,\\ z_1 = z_2 = 0\ \ {\rm on}\ \ \partial \Omega, \end{array} \right. \end{align*} $
where $ \Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty $, $ \lambda_1 $ and $ \lambda_2 $ are positive parameters, $ \Omega $ is the open unit ball in $ \mathbb{R}^N,\ N\geq 2 $.
- p-Laplacian equatios,
- positive solutions,
- uniqueness and asymptotic behavior,
- eigenvalue theory in cones,
- Gustafson and Schmitt's fixed-point theorem
Citation: Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior[J]. Electronic Research Archive, 2020, 28(4): 1419-1438. doi: 10.3934/era.2020075

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Abstract

We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by $ p $-Laplacian elliptic equations

$ \begin{align*} \left \{ \begin{array}{l} -\Delta_p z_1 = \lambda_1 g_1(z_2)\ \ {\rm in}\ \Omega,\\ -\Delta_p z_2 = \lambda_2 g_2(z_1)\ \ {\rm in}\ \Omega,\\ z_1 = z_2 = 0\ \ {\rm on}\ \ \partial \Omega, \end{array} \right. \end{align*} $

where $ \Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty $, $ \lambda_1 $ and $ \lambda_2 $ are positive parameters, $ \Omega $ is the open unit ball in $ \mathbb{R}^N,\ N\geq 2 $.

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