Local solvability for nonlinear partial differential equations

In the introduction we give a short survey on known results concerning loca l solvability for nonlinear partial differential equations; the next sectio ns will be then devoted to the proof of a new result in the same direction. Specifically we study the semilinear 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 oper ators with constant coefficients and f (x, v), x is an element of R-n, v is an element of C-M, is a smooth function with respect to x and entire with respect to v. Let g be in the Hbrmander space Bp,k we want to solve locally near a point x(0) is an element of R-n the equation F(u) = g.