Symmetry for Solutions of Two-phase Semilinear Elliptic Equations on Hyperbolic Space

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 Δ u = f ( u ) on hyperbolic space H n must reduce to functions of one variable provided they admit asymptotic boundary values on S n − 1 = ∂ ∞ H n which are invariant under a cohomogeneity one subgroup of the group of isometries of H n . We also prove existence of these one-dimensional solutions.