Existence and regularity results for some Fully Non Linear singular or degenerate equation

TL;DR

This paper establishes existence, uniqueness, and regularity for a class of fully nonlinear singular or degenerate elliptic equations, extending previous results to more general operators and conditions.

Contribution

It generalizes known results by proving existence, uniqueness, and regularity for a broader class of singular fully nonlinear elliptic equations with specific conditions.

Findings

Proved existence and uniqueness of solutions.

Established regularity results for solutions.

Extended previous results to more general operators.

Abstract

In this article we prove existence, uniqueness and regularity for the singular equation \begin{eqnarray*} \begin{cases} |\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^{\alpha}u+p(x)u^{-\gamma}=0 \ \mbox{ in } \ \Omega\\ u>0 \ \mbox{ in } \ \Omega, \ u=0 \ \mbox{ on } \ \partial\Omega \end{cases} \end{eqnarray*} when $p$ is some continuous and positive function, $c$ and $h$ are continuous, $\alpha > -1$ and $F$ is Fully non linear elliptic. Some conditions on the first eigenvalue for the operator $|\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^{\alpha}u$ are required. The results generalizes the well known results of Lazer and Mac Kenna.