Existence and regularity results for some fully nonlinear singular or degenerate equation

Abstract

In this article we prove existence, uniqueness and regularity for the singular equation $\{\begin{array}{ll}|\mathrm{\nabla }u{|}^{\alpha }\left(F\right(D^{2}u)+h(x)\cdot \mathrm{\nabla }u)+c\left(x\right)|u{|}^{\alpha }u+p\left(x\right)u^{-\gamma }=0& \text{in }\mathrm{\Omega }\\ u>0\text{in}\mathrm{\Omega },u=0& \text{on}\mathrm{\partial }\mathrm{\Omega }\end{array}$when p is some continuous and positive function, c and h are continuous, $\alpha >-1$ and F is Fully nonlinear elliptic. Some conditions on the first eigenvalue for the operator $-|\mathrm{\nabla }u{|}^{\alpha }\left(F\right(D^{2}u)+h(x)\cdot \mathrm{\nabla }u)-c\left(x\right)|u{|}^{\alpha }u$ are required. The results generalize the well known results of Lazer and McKenna.

Keywords:

• Viscosity solutions
• second-order elliptic equations
• nonlinear elliptic equations
• degenerate elliptic equations
• singular problem

2020 Mathematical subject classifications:

• 35D40
• 35J15
• 35J60
• 35J70
• 35J75

Acknowledgments

The author wishes to thank the anonymous referee for his remarks which permit to improve considerably this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).