Abstract:
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.
Keywords:system of equations, Monge–Ampère operator, exact multidimensional solutions.
Fulltext:
PDF file (284 kB)
First page:
PDF file