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

Karan Rathore, Mohan Mallick, Ram Baran Verma

TL;DR

This work extends Lane–Emden theory to a fully nonlinear, non-divergence elliptic framework by studying a singular two-component system $F(D^2u,Du,u,x)=u^{-p}v^{-q}$, $F(D^2v,Dv,v,x)=u^{-r}v^{-s}$ in a bounded domain with Dirichlet boundary conditions. The authors develop barrier constructions based on eigenfunctions and distance-to-boundary estimates, and apply a Schauder fixed-point argument in a carefully defined cone to obtain existence results under precise relations among the exponents $(p,q,r,s)$. They establish sharp nonexistence criteria via boundary behavior and integral tests, and prove regularity and partial uniqueness results in various parameter regimes. The findings significantly generalize Lane–Emden-type analyses to fully nonlinear operators, clarifying how singular nonlinearities interact with boundary geometry to govern solvability and regularity of positive viscosity solutions.

Abstract

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

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

TL;DR

This work extends Lane–Emden theory to a fully nonlinear, non-divergence elliptic framework by studying a singular two-component system , in a bounded domain with Dirichlet boundary conditions. The authors develop barrier constructions based on eigenfunctions and distance-to-boundary estimates, and apply a Schauder fixed-point argument in a carefully defined cone to obtain existence results under precise relations among the exponents . They establish sharp nonexistence criteria via boundary behavior and integral tests, and prove regularity and partial uniqueness results in various parameter regimes. The findings significantly generalize Lane–Emden-type analyses to fully nonlinear operators, clarifying how singular nonlinearities interact with boundary geometry to govern solvability and regularity of positive viscosity solutions.

Abstract

Consider where is an open connected subset of and are two non-negative and are positive real numbers. This article discuses the conditions in terms of the relations among and 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}.
Paper Structure (6 sections, 16 theorems, 280 equations)

This paper contains 6 sections, 16 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalue
  4. Boundary Behavior and Singular Estimates
  5. Main Results
  6. Acknowledgement

Key Result

Theorem 2.2

Let $F$ satisfies the above assumptions. Then there exist $\varphi^{+}_{1},\varphi^{-}_{1} \in C^{1,\alpha}(\Omega)$ such that and $\varphi^{+}_{1}> 0\quad\text{and}~\quad \varphi^{-}_{1} < 0\quad \text{in}~\Omega.$ Moreover, the eigenvalue $\mu^{+}_{1}(F,\Omega)$ (resp. $\mu^{-}_{1}(F,\Omega)$) is unique in the sense that if $\mu$ is an eigenvalue of $F$ in $\Omega$ with a corresponding nonnegat

Theorems & Definitions (30)

  • Definition 2.1: see MHPishii1991viscosity
  • Theorem 2.2: Theorem 2.2, armstrong2009principal
  • Remark 2.3
  • Proposition 2.4: Lemma 2.3 in armstrong2009principal
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 20 more