Two-sided game for competitive systems with nonlocal interactions

A system of two stationary semilinear elliptic partial differential equations (PDEs) with rather general competitive interactions is considered. The motivation is related to possible applications of the formalism to competitive/conflict situations and, in particular, to classical warfare for which systems of this type have been applied with rather promising results. The optimal strategies, mathematically defined in terms of minimizing or maximizing a certain function, are realized by varying the control functions. The authors show that a unique saddle point exists and is represented by the solution of the optimality system. The optimality system consists of two elliptic PDE state equations coupled with two adjoint equations