On existence and nonexistence of positive solutions of an elliptic system with coupled terms

Yayun Li; Yutian Lei

doi:10.3934/cpaa.2018083

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2018, Volume 17, Issue 5: 1749-1764. Doi: 10.3934/cpaa.2018083

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-Laplacian equations with mixed nonlinearities Next Article Positive radial solutions of a nonlinear boundary value problem

On existence and nonexistence of positive solutions of an elliptic system with coupled terms

Yayun Li^1, and
Yutian Lei^2,

1.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu, 210023, China
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

Received: April 2017

Revised: December 2017

Published: April 2018

This research was supported by NSF (11471164, 11671209) of China.

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Abstract

This paper is concerned with the elliptic system

$\begin{cases}-{\triangle u} = (q+1)u^qv^{p+1},~~ u>0~ in~ R^n,\\-{\triangle v} = (p+1)v^pu^{q+1},~~ v>0~in ~R^n,\end{cases}$

where $ n ≥ 3 $, $ p,q>0 $ and $ \max\{p,q\} ≥ 1 $. We discuss the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case, the necessary and sufficient conditions of classification in the critical case, and the Joseph-Lundgren-type condition for existence of local stable solutions.

Keywords:

Mathematics Subject Classification: Primary: 35J47; Secondary: 45G15.

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References

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