Local solvability for nonlinear partial differential equations

In the introduction we give a short survey on known results concerning local solvability for nonlinear partial differential equations; the next sections will be then devoted to the proof of a new result in the same direction. Specifically we study the semi-linear operator F(u) = P(D)u + f(x, Q 1(D)u, .., Q M(D)u) where P, Q 1, .., Q M are linear partial differential operators with constant coefficients and f(x, v), x ∈ R n, v ∈ C M, is a smooth function with respect to x and entire with respect to v. Let g be in the Hörmander space B p,k we want to solve locally near a point x 0 ∈ R n the equation F(u) = g.