O tochnykh resheniyakh mnogomernoy sistemy ellipticheskikh uravneniy so stepennymi nelineynostyami

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 multidimensionalexact solutions is studied. It is proposed to use a special type of ansatz that reduces the problemto solving systems of algebraic equations. A number of multiparameter families of new exactmultidimensional solutions (both radially symmetric and anisotropic) represented by explicitformulas are obtained. Examples are given to illustrate the exact solutions found.