Symmetry of solutions to higher and fractional order semilinear equations on hyperbolic spaces

We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space ${\mathbb{H}}^n$ as well as on the entire space ${\mathbb{H}}^n$. Applying the Helgason-Fourier analysis techniques on ${\mathbb{H}}^n$, we develop a moving plane approach for integral equations on ${\mathbb{H}}^n$. We also establish the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtain symmetry to solutions of some semilinear equations involving fractional order derivatives.