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Liouville theorem and qualitative properties of solutions for an integral system
  • Ling Li,
  • Xiaoqian Liu
Ling Li
Nanjing Normal University

Corresponding Author:[email protected]

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Xiaoqian Liu
Nanjing Xiaozhuang University
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In this paper, we are concerned with an integral system $$ \left\{ \begin{aligned} &u(x)= W_{\beta,\gamma}(u^{p-1}v)(x), \ u>0 \ \text{in} \ R^{n},\\ &v(x)=I_{\alpha}(u^{p})(x), \ v>0 \ \text{in} \ R^{n}, \end{aligned} \right. $$ where $p>0,$ $0<\alpha, \beta\gamma1$. 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.
28 Jun 2022Submitted to Mathematical Methods in the Applied Sciences
29 Jun 2022Submission Checks Completed
29 Jun 2022Assigned to Editor
02 Sep 2022Reviewer(s) Assigned
22 Dec 2022Review(s) Completed, Editorial Evaluation Pending
09 Jan 2023Editorial Decision: Revise Minor
13 Jan 20231st Revision Received
14 Jan 2023Submission Checks Completed
14 Jan 2023Assigned to Editor
14 Jan 2023Review(s) Completed, Editorial Evaluation Pending
06 Feb 2023Reviewer(s) Assigned
06 Mar 2023Editorial Decision: Accept