Generalized Monge–Ampère type equation and its multidimensional exact solutions

Several variants of the multidimensional generalized Monge–Ampère equation are considered containing, in addition to the determinant of the Hessian matrix, also additional terms depending on the Laplace operator and the gradient of the desired function. It is proposed to construct exact solutions in the form of superposition of quadratic form and solutions of ordinary differential equations generated by the initial partial differential equation. We give a number of examples of exact solutions, both radially symmetric and anisotropic, expressed through combinations of elementary functions.