Abstract

Equations and systems of elliptic type with power-law nonlinearities are considered. Such equations are found in modeling distributed robotic formations, as well as in chemical kinetics, biology, astrophysics, and many other fields. The problem of constructing multidimensional exact solutions is studied. It is proposed to use a special type of ansatz that reduces the problem to solving systems of algebraic equations. A number of multiparameter families of new exact multidimensional solutions (both radially symmetric and anisotropic) represented by explicit formulas are obtained. Examples are given to illustrate the exact solutions found.

中文翻译：

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具有幂律非线性的多维椭圆方程组的精确解

摘要

考虑具有幂律非线性的椭圆型方程和系统。此类方程存在于分布式机器人编队建模以及化学动力学、生物学、天体物理学和许多其他领域中。研究了构造多维精确解的问题。建议使用一种特殊类型的 ansatz 将问题简化为求解代数方程组。获得了由显式公式表示的许多新的精确多维解（径向对称和各向异性）的多参数族。给出了示例来说明找到的确切解决方案。