Radial symmetry and monotonicity of the positive solutions for [formula omitted]-Hessian equations

Abstract

This paper is the first attempt to design the direct method of moving planes for k-Hessian equations. Various maximum principles such as Maximum principle for anti-symmetric functions, Narrow region principle and Decay at infinity are established. Finally, the radial symmetry and monotonicity of the positive solutions to k-Hessian equations are proved in the unit sphere and the whole space.