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

Abstract

Assume that $f(s)=F^{\prime }(s)$ 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 $\HH^n$ must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of $\HH^n$ which are invariant under a cohomogeneity one subgroup of the group of isometries of $\HH^n$. We also prove existence of these one-dimensional solutions.

24 pages