Local solvability for semilinear partial differential equations of constant strength

The main goal of the present paper is to study the local solvability of semilnear partial differential operators of the form F(u) = P(D)u + ∫(x, Q 1(D)u, ....... Q M(D)u). where P(D), Q 1(D), .... Q M(D) are linear partial differntial operators of constant coefficients and ∫(x, v) is a C ∞ function with respect to x and an entire function with respect to v. Under suitable assumptions on the nonlinear function ∫ and on P, Q 1, .... Q M, we will solve locally near every point x 0 ∈ ℝ n the next equation F(u) = g, g ∈ B p,k. where B p,k is a wieghted Sobolev space as in Hörmander [13].