MATHEMATICS

On multidimensional exact solutions of equations with the Laplace–Monge–Ampère operator

A. A. Kosov, È. I. Semenov

Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences (ISDCT SB RAS), ul. Lermontova, 134, Irkutsk, 664033, Russia

Abstract: The article considers a nonlinear equation with the Laplace–Monge–Ampère operator in cases of power-law or exponential nonlinearity. To construct exact solutions, it is proposed to apply the method of reduction to ordinary differential equations using a quadratic auxiliary function of spatial variables. Multidimensional anisotropic exact solutions are obtained, which are expressed explicitly in terms of elementary and special functions and/or solutions of ordinary differential equations. A number of examples are provided to illustrate the obtained results.

Keywords: Laplace–Monge–Ampère equation, power and exponential nonlinearities, exact solutions

UDC: 517.957

MSC: 35J96, 35C05

Received: 23.07.2025
Accepted: 10.09.2025

DOI: 10.35634/2226-3594-2025-66-05

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