Symmetry properties for nonnegative solutions of non-uniformly elliptic equations in the hyperbolic space

We are interested in monotonicity and symmetry properties for nonnegative solutions of elliptic equations defined in geodesic balls of the hyperbolic space which is the simplest example of manifold with negative curvature. More precisely, let B be a geodesic ball in H-n and let u is an element of W-1,W-p(B)boolean AND L-infinity (B) be a sufficiently regular solution of Delta(p)u + f(u) = 0 in B with boundary condition u = 0, where Delta(p) is the p-Laplace Beltrami operator with p > 2. Then we prove local or global symmetry results for nonnegative solutions according to the assumptions about the zeros of the nonlinearity f(s), which is merely continuous. (C) 2015 Elsevier Inc. All rights reserved.