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<p>We study systems of two equations with the Monge–Ampère operator, whose right-hand sides may depend on the Laplace operator and the gradients of the unknown functions.To construct exact multidimensional solutions in the case where the right-hand sides involve power or exponential functions of the unknowns, we propose a reduction method to a system of ordinary differential equations.We obtain exact multidimensional solutions expressed explicitly as a superposition of quadratic forms in&#xa0;spatial variables and elementary functions.A series of examples of explicit exact solutions is&#xa0;presented, including global positive solutions and solutions anisotropic in the spatial variables.</p>

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System of Equations with the Monge–Ampère Operator and Its Exact Multidimensional Solutions

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • A. A. Kosov,
  • E. I. Semenov

摘要

We study systems of two equations with the Monge–Ampère operator, whose right-hand sides may depend on the Laplace operator and the gradients of the unknown functions.To construct exact multidimensional solutions in the case where the right-hand sides involve power or exponential functions of the unknowns, we propose a reduction method to a system of ordinary differential equations.We obtain exact multidimensional solutions expressed explicitly as a superposition of quadratic forms in spatial variables and elementary functions.A series of examples of explicit exact solutions is presented, including global positive solutions and solutions anisotropic in the spatial variables.