Regularity for Fully Nonlinear Elliptic Equations with Natural Growth in Gradient and Singular Nonlinearity

Abstract

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-α}&=0~\text{in}~Ω\\ u&=0~~\text{on}~~\partialΩ, \end{aligned} \right. \end{equation*} where $\hat{I}\text{c◯}$ is a bounded and $C^2$ smooth domain in ${\mathbb{R}}^N$ and $F$ has superlinear growth in gradient and $c(c)<-c_0$ for some positive constant $c_0.$ Here, we studies the boundary behaviour of the solutions to above equation and establishes the global regularity result similar to one established in [12,16] with linear growth in gradient.