Nonlinear Equations of Infinite Order Defined by an Elliptic Symbol

The aim of this work is to show existence and regularity properties of equations of the form f(Δ)u=U(x,u(x)) on ℝn, in which f is a measurable function that satisfies some conditions of ellipticity and Δ stands for the Laplace operator on ℝn. Here, we define the class of functions to which f belongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition of f(Δ) explaining how to understand this operator