On a semilinear elliptic systems in Hyperbolic space

TL;DR

This paper studies semilinear elliptic systems in Hyperbolic space, establishing decay and symmetry of solutions, and demonstrating the existence of nontrivial solutions, contrasting with Euclidean space results.

Contribution

It introduces existence results and symmetry properties for solutions of elliptic systems in Hyperbolic space, which differ from Euclidean space cases.

Findings

Established decay estimates for solutions.

Proved symmetry properties of solutions.

Demonstrated existence of nontrivial solutions in Hyperbolic space.

Abstract

In this paper, we consider systems of semilinear elliptic equations \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space $\mathbb{H}^{N}$. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space $\mathbb{R}^N$, we prove that there exists a nonnegative nontrivial solution of problem.