Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties

TL;DR

This paper studies layer solutions for the fractional Laplacian on hyperbolic space, proving existence, uniqueness, symmetry, and qualitative properties, with extensions to multilayer solutions near the classical case.

Contribution

It introduces new existence and uniqueness results for layer solutions of fractional Laplacians on hyperbolic space, including symmetry and qualitative analysis.

Findings

Existence of heteroclinic layer solutions in hyperbolic space.

Uniqueness of solutions under certain conditions.

Symmetry and qualitative properties of solutions.

Abstract

We investigate the equation $$(-\Delta_{\mathbb H^n})^{\gamma} w=f(w)\quad in \mathbb H^{n},$$ where $(-\Delta_{\mathbb H^n})^\gamma$ corresponds to the fractional Laplacian on hyperbolic space for $\gamma \in (0,1)$ and $f$ is a smooth nonlinearity that typically comes from a double well potential. We prove the existence of heteroclinic connections in the following sense; a so-called layer solution is a smooth solution of the previous equation converging to $\pm 1$ at any point of the two hemispheres $S_\pm \subset \partial_\infty \mathbb H^n$ and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane $\Pi.$ We prove that under additional conditions on the nonlinearity uniqueness holds up to isometry. Then we provide several symmetry results and qualitative properties of the layer solutions. Finally, we consider the multilayer case, at least when $\gamma$ is close to one.