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Regularity and radial symmetry of positive solutions for a higher order elliptic system

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Abstract

We discuss the properties of solutions for the following elliptic partial differential equations system in Rn,

$$\begin{cases}(-\triangle)^{\frac{\alpha}{2}} u&= u^{p_{1}} v^{p_{2}}\\(-\triangle)^{\frac{\alpha}{2}} v &= u^{q_{1}} v^{q_{2}}\end{cases}$$

where 0 < α < n, p i and q i (i = 1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1 + p2 = q1 + q2 = \(\frac{n+\alpha}{n-\alpha}\), we classify the solutions of the PDEs system.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable suggestions.

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Authors and Affiliations

  1. School of Statistics, Xi’an University of Finance and Economics, Xi’an, 710100, China

    Huai-yu Zhou & Su-fang Tang

Authors
  1. Huai-yu Zhou
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  2. Su-fang Tang
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Correspondence to Huai-yu Zhou.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11571268), the foundation of Xi’an University of Finance and Economics (Grant No. 12XCK07).

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Zhou, Hy., Tang, Sf. Regularity and radial symmetry of positive solutions for a higher order elliptic system. Acta Math. Appl. Sin. Engl. Ser. 33, 551–560 (2017). https://doi.org/10.1007/s10255-017-0662-5

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  • DOI: https://doi.org/10.1007/s10255-017-0662-5

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