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

TL;DR

This paper investigates the existence, uniqueness, and non-existence of positive solutions for a class of Lane-Emden systems involving fully nonlinear elliptic operators with singular nonlinearities, analyzing conditions based on parameter relations.

Contribution

It provides new criteria for solution existence and regularity in fully nonlinear elliptic Lane-Emden systems with singular terms, extending previous studies to more general operators.

Findings

Derived conditions for existence and non-existence of solutions.

Established regularity properties of solutions.

Extended Lane-Emden analysis to fully nonlinear operators.

Abstract

Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~\Omega\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~\Omega\\ u,v>0~~\text{in}~~\Omega\\ u=v=0~\quad~\text{on}~~\partial\Omega, \end{cases} \] where $\Omega$ 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}.