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

TL;DR

This paper investigates the regularity and boundary behavior of solutions to a class of fully nonlinear elliptic equations with natural gradient growth and singular nonlinearities, extending known regularity results to more complex growth conditions.

Contribution

It establishes global regularity results for solutions to fully nonlinear elliptic equations with natural gradient growth and singular nonlinearities, generalizing previous linear growth findings.

Findings

Solutions exhibit regularity up to the boundary.

Boundary behavior of solutions is characterized.

Global regularity similar to linear growth cases is proved.

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^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $\Omega$ 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.