Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866908792280580096 |
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| author | Rathore, Karan Mallick, Mohan Verma, Ram Baran |
| author_facet | Rathore, Karan Mallick, Mohan Verma, Ram Baran |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19874 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator Rathore, Karan Mallick, Mohan Verma, Ram Baran Analysis of PDEs 35J60, 35D40 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}. |
| title | Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator |
| topic | Analysis of PDEs 35J60, 35D40 |
| url | https://arxiv.org/abs/2601.19874 |