article

Title: Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

Authors: Isabeau Birindelli and Rafe Mazzeo

Issue: Volume 58 (2009), Issue 5, 2347-2368

Abstract:

$Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1, 1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space $\mathbb{H}^{n}$ must reduce to functions of one variable provided they admit asymptotic boundary values on $S^{n-1} = \partial_{\infty}\mathbb{H}^{n}$ which are invariant under a cohomogeneity one subgroup of the group of isometries of $\mathbb{H}^{n}$. We also prove existence of these one-dimensional solutions.$