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arXiv · 2601.19874

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

Karan Rathore·Mohan Mallick·Ram Baran Verma
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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}.

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Karan Rathore, Mohan Mallick, Ram Baran Verma. 2026-01-27. Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator. https://arxiv.org/abs/2601.19874

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