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SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

Part of: Elliptic equations and systems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  18 December 2014

HAIYANG HE
HAIYANG HE*
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing, (Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, P.R. China e-mail: [email protected]
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Abstract

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(0.1)

\begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation}
in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

MSC classification

Primary: 58J05: Elliptic equations on manifolds, general theory
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 519 - 541
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

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