Symmetry of solutions to higher and fractional order semilinear equations on hyperbolic spaces

TL;DR

This paper proves that solutions to certain higher and fractional order semilinear equations on hyperbolic spaces are radially symmetric and nonincreasing, using Helgason-Fourier analysis and a moving plane method, extending symmetry results to fractional and singular cases.

Contribution

It develops a new moving plane approach for integral equations on hyperbolic spaces and extends symmetry results to fractional order and singular semilinear equations.

Findings

Solutions are radially symmetric and nonincreasing on hyperbolic spaces.

The method applies to equations with singular terms and fractional derivatives.

Symmetry results are established for Euclidean space equations as well.

Abstract

We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space $\mathbb{H}^n$. Applying the Helgason-Fourier analysis techniques on $\mathbb{H}^n$, we develop a moving plane approach for integral equations on $\mathbb{H}^n$. We also establish the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtain symmetry to solutions of some semilinear equations involving fractional order derivatives.