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 HnHn, which is the simplest example of manifold with negative curvature. More precisely, let B   be a geodesic ball in HnHn and let u∈W1,p(B)∩L∞(B)u∈W1,p(B)∩L∞(B) be a sufficiently regular solution of Δpu+f(u)=0Δpu+f(u)=0 in B   with boundary condition u=0u=0, where ΔpΔp is the p  -Laplace–Beltrami operator with p>2p>2. Then we prove local or global symmetry results for nonnegative solutions according to the assumptions about the zeros of the nonlinearity f(s)f(s), which is merely continuous.