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

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.