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

Abstract

We investigate the equation (Formula presented.)where (Formula presented.) corresponds to the fractional Laplacian on hyperbolic space for (Formula presented.) and (Formula presented.) 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 (Formula presented.) at any point of the two hemispheres (Formula presented.) and which is strictly increasing with respect to the signed distance to a totally geodesic hyperplane (Formula presented.). 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 (Formula presented.) is close to one.