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A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems


Authors: Xianjin Chen and Jianxin Zhou
Journal: Math. Comp. 79 (2010), 2213-2236
MSC (2010): Primary 35A15, 58E05, 58E30
Published electronically: March 26, 2010
MathSciNet review: 2684362
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Abstract: A local min-max-orthogonal method together with its mathematical justification is developed in this paper to solve noncooperative elliptic systems for multiple solutions in an order. First it is discovered that a noncooperative system has the nature of a zero-sum game. A new local characterization for multiple unstable solutions is then established, under which a stable method for multiple solutions is developed. Numerical experiments for two types of noncooperative systems are carried out to illustrate the new characterization and method. Several important properties for the method are explored or verified. Multiple numerical solutions are found and presented with their profiles and contour plots. As a new bifurcation phenomenon, multiple asymmetric positive solutions to the second type of noncooperative systems are discovered numerically but are still open for mathematical verification.


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Additional Information

Xianjin Chen
Affiliation: Institute for Mathematics & Its Application, University of Minnesota, Minneapolis, Minnesota 55455
Email: xchen@ima.umn.edu

Jianxin Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jzhou@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-10-02336-7
Keywords: Cooperative/noncooperative systems, multiple solutions, local min-orthogonal method, saddle points, strongly indefinite
Received by editor(s): January 27, 2009
Received by editor(s) in revised form: August 1, 2009
Published electronically: March 26, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.