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

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• 作者：Shan Zhang ; Ling Zhou ; Zu Han Liu
• 关键词：Strongly coupled elliptic system ; uniqueness ; spatial segregation
• 刊名：Acta Mathematica Sinica, English Series
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：33
• 期：3
• 页码：419-438
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
• ISSN：1439-7617
• 卷排序：33

文摘

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.KeywordsStrongly coupled elliptic systemuniquenessspatial segregationSupported by PRC grant NSFC (Grant Nos. 11371310, 11401515)