Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator

Consider $$\{\begin{array}{l}F(D^{2}u,Du,u,x)=u^{-p}{v}^{-q},\text{ in }\mathrm{\Omega }\\ F(D^{2}v,Dv,v,x)=u^{-r}{v}^{-s},\text{ in }\mathrm{\Omega }\\ u,v>0\text{ in }\mathrm{\Omega }\\ u=v=0\text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is an open connected subset of ${\mathbb{R}}^N$ and $p,s$ are two non-negative and $q,r$ are positive real numbers. This article discuses the conditions in terms of the relations among $p,q,r$ and $s$ which lead to existence, uniqueness and non-existence of positive solutions to the system. Furthermore, we also have studied some regularity properties of solution of the system. These results are inspired by the study of Lane-Emden system of equations as in \cite{busca2002liouville,ghergu2010lane}.