Qualitative analysis for an elliptic system in the punctured space

Hui Yang; Wenming Zou

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arXiv · 1811.07079

Qualitative analysis for an elliptic system in the punctured space

Hui Yang·Wenming Zou

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Abstract

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$ \begin{cases} -\Delta u =\mu_1 u^{2q+1} + \beta u^q v^{q+1} \\ -\Delta v =\mu_2 v^{2q+1} + \beta v^q u^{q+1} \end{cases} \textmd{in} ~\mathbb{R}^n \backslash \{0\}, $$ where $\mu_1, \mu_2$ and $\beta$ are all positive constants, $n\geq 3$. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.

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Hui Yang, Wenming Zou. 2018-11-17. Qualitative analysis for an elliptic system in the punctured space. https://arxiv.org/abs/1811.07079

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