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\begin{document}

\title{Liouville theorem and qualitative properties of solutions for an
integral system}



\author[1]{Ling Li}%
\author[2]{Xiaoqian Liu}%
\affil[1]{Nanjing Normal University}%
\affil[2]{Nanjing Xiaozhuang University}%


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  \date{June 18, 2024}


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\begin{abstract}
In this paper, we are concerned with an integral system \$\$
\textbackslash{}left\textbackslash{}\{ \textbackslash{}begin\{aligned\}
\&u(x)=
W\_\{\textbackslash{}beta,\textbackslash{}gamma\}(u\^{}\{p-1\}v)(x),
\textbackslash{} u\textgreater{}0 \textbackslash{}
\textbackslash{}text\{in\} \textbackslash{}
R\^{}\{n\},\textbackslash{}\textbackslash{}
\&v(x)=I\_\{\textbackslash{}alpha\}(u\^{}\{p\})(x), \textbackslash{}
v\textgreater{}0 \textbackslash{} \textbackslash{}text\{in\}
\textbackslash{} R\^{}\{n\}, \textbackslash{}end\{aligned\}
\textbackslash{}right. \$\$ where \$p\textgreater{}0,\$
\$0\textless{}\textbackslash{}alpha,
\textbackslash{}beta\textbackslash{}gamma1\$. Base on the integrability
of positive solutions, we obtain some Liouville theorems and the decay
rates of positive solutions at infinity. In addition, we use the
properties of the contraction map and the shrinking map to prove that
\$u\$ is Lipschitz continuous. In particular, the Serrin type condition
is established, which plays an important role to classify the positive
solutions.%
\end{abstract}%



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