Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems

Author:

Luo Haijun1,Zhang Zhitao2

Affiliation:

1. College of Mathematics and Econometrics , Hunan University , Changsha 410082, Hunan , P. R. China

2. HLM , Academy of Mathematics and Systems Science , The Chinese Academy of Sciences , 100190; and School of Mathematical Sciences, Academy of Sciences, University of Chinese , Beijing 100049 , P. R. China

Abstract

Abstract We study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling β > 0 {\beta>0} , we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants μ β , ν β {\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as β {\beta\to\infty} , we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.

Funder

Fundamental Research Funds for the Central Universities

National Natural Science Foundation of China

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics,Statistical and Nonlinear Physics

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