Regularity for fully nonlinear elliptic equations with natural growth in gradient and singular nonlinearity

In this article, we consider the following boundary value problem:$\{\begin{array}{cc}F(x,u,Du,D^{2}u)+c\left(x\right)u+p\left(x\right)u^{-\alpha }& =0\text{in}\Omega ,\\ u& =0\text{on}\partial \Omega ,\end{array}$ where Ω is a bounded and $C^2$ smooth domain in $R^N$. The operator F is proper and has superlinear growth in gradient. This study examines the boundary behavior of the solutions to the above equation and establishes a global regularity result similar to that established in [11], [15], which involves linear growth in the gradient.