Symmetry for a quasilinear elliptic equation in Hyperbolic space

TL;DR

This paper proves that positive solutions to a p-Laplace equation in Hyperbolic space are radially symmetric, determines their decay rates, and explores their existence, advancing understanding of nonlinear PDEs in curved geometries.

Contribution

It establishes the radial symmetry, decay properties, and existence conditions for solutions of a p-Laplace equation in Hyperbolic space, linking geometric analysis with PDE theory.

Findings

Positive solutions are radially symmetric in Hyperbolic space.

Solutions exhibit sharp decay rates and gradient behavior.

The paper discusses conditions for the existence of solutions.

Abstract

In this article we establish the radial symmetry of positive solutions of a p- Laplace equation in the Hyperbolic space, which is the Euler Lagrange equation of the p- Poincare Sobolev inequality in the Hyperbolic space. We will also establish the sharp decay of solution and its gradient and also investigate the question of existence of solution.