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

Abstract In this paper, we show that nontrivial solutions to a class of higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{H}}^{n}$ as well as on the entire <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{H}}^{n}$ . Applying the Helgason-Fourier analysis techniques on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{H}}^{n}$ , we develop a moving plane approach for integral equations on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{H}}^{n}$ . We also establish the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtain the symmetry property of solutions to some semilinear equations involving fractional order derivatives.