Using an operator approach, we discuss stationary solutions to Fokker-Planck equations and systems with nonlinear reaction terms. The existence of solutions is obtained by using Banach, Schauder and Schaefer fixed point theorems, and for systems by means of Perov’s fixed point theorem. Using the Ekeland variational principle, it is proved that the unique solution of the problem minimizes the energy functional, and in case of a system that it is the Nash equilibrium of the energy functionals associated to the component equations.

中文翻译：

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有界域中带有非线性反应项的Fokker-Planck方程的平稳解

使用算子方法，我们讨论了带有非线性反应项的Fokker-Planck方程和系统的平稳解。解的存在是通过使用Banach，Schauder和Schaefer不动点定理来获得的，对于系统而言，是通过Perov不动点定理来实现的。使用Ekeland变分原理，证明了该问题的唯一解决方案使能量泛函最小化，并且在系统的情况下，这是与组件方程式相关联的能量泛函的Nash平衡。