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

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.