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Uniqueness and least energy property for solutions to a strongly coupled elliptic system

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Abstract

For the strongly coupled system of M ≥ 3 competing species:

$$ - \Delta \left[ {\left( {{d_i} + \sum\limits_{j = 1}^M {{\beta _{ij}}{u_j}} } \right){u_j}} \right] = \left( {{a_i} - {b_i}} \right){u_i} - k{u_i}\sum\limits_{j \ne i} {{u_j}} ,\;i = 1, \ldots ,M,$$

we prove the uniqueness of the limiting configuration as k →∞ under suitable conditions. Moreover, we prove that the limiting configuration minimizes a variational problem associated to the strongly coupled system among the segregated states with the same boundary conditions.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, 210003, P. R. China

    Shan Zhang

  2. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, P. R. China

    Ling Zhou & Zu Han Liu

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  1. Shan Zhang
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  2. Ling Zhou
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  3. Zu Han Liu
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Correspondence to Shan Zhang.

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Supported by PRC grant NSFC (Grant Nos. 11371310, 11401515)

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Zhang, S., Zhou, L. & Liu, Z.H. Uniqueness and least energy property for solutions to a strongly coupled elliptic system. Acta. Math. Sin.-English Ser. 33, 419–438 (2017). https://doi.org/10.1007/s10114-016-5686-x

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  • DOI: https://doi.org/10.1007/s10114-016-5686-x

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