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Classification of Solutions to Mixed Order Conformally Invariant Systems in \({\mathbb {R}}^2\)

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Abstract

In this paper, we are concerned with the following mixed-order conformally invariant system with coupled nonlinearity in \({\mathbb {R}}^{2}\):

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{\frac{1}{2}}u(x)=u^{p_1}(x)e^{q_1 v(x)}, \qquad &{}x\in {\mathbb {R}}^{2}, \\ \\ (-\Delta ) v(x)=u^{p_2}(x)e^{q_2 v(x)}, \qquad &{}x\in {\mathbb {R}}^{2}, \end{array}\right. } \end{aligned}$$
(0.1)

where \(0\le p_1 < \frac{1}{1+K}\), \( p_2 >0\), \(q_1 >0\), \(q_2 \ge 0\), \(u>0\) and satisfies \(\int _{{\mathbb {R}}^{2}}u^{p_2}(x)e^{q_2 v(x)}dx<+\infty \). Under the assumptions, \(u(x)=O(|x|^{K})\) at \(\infty \) for some \(K\ge 1\) arbitrarily large and \(v^{+}(x)=O(\ln |x|)\) if \(q_2>0\) at \(\infty .\) We firstly derived the equivalent integral representation formula for (0.1). Then we discuss the exact asymptotic behavior of the solutions to system (0.1) as \(|x|\rightarrow \infty \). At last, by using the method of moving spheres in integral form, we give the classification of the classical solutions to (0.1).

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Acknowledgements

The authors are grateful to the anonymous referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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  1. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Yuxia Guo & Shaolong Peng

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  1. Yuxia Guo
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Correspondence to Shaolong Peng.

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Yuxia Guo was supported by NSFC (No. 12031015). Shaolong Peng is supported by the NSFC (No. 11971049)

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Guo, Y., Peng, S. Classification of Solutions to Mixed Order Conformally Invariant Systems in \({\mathbb {R}}^2\). J Geom Anal 32, 178 (2022). https://doi.org/10.1007/s12220-022-00916-0

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