Gradient estimate for solutions to Δbu = 0 and Δbu+auln⁡u+bu=0 on pseudohermitian manifold

Abstract

In this paper, we will establish fundamental $C^1$ estimate for solutions to ${\Delta }_{b}u=0$ and $C^0$ and $C^1$ estimates for solutions to a class of nonlinear elliptic equations of the form ${\Delta }_{b}u+au\ln u+bu=0$ on pseudohermitian manifold. The proof consists of combining the Nash-Moser iteration and Saloff-Coste type Sobolev inequalities.