Exact solutions of the Liouville equation in multidimensional spaces

Exact solutions of the multidimensional Liouville equation, which are sought in the class of functional forms of the degree $n$ equal to the coordinate space dimension, are constructed based on a special representation of the Laplace and D'Alembert equations. The corresponding Liouville equation solutions are completely described in the coordinate space dimensions $d=3,4$. For $d>4$, the general solution form and the method for obtaining algebraic equations for the coefficients of functional $n$-forms is presented.