AN EXISTENCE RESULT FOR A CLASS OF SEMILINEAR DEGENERATE ELLIPTIC EQUATIONS ∗

In this work, we study the semilinear degenerate Dirichlet prob-lem, ∑m j=1X jXju + cu + f(x, u,Xu) = 0, inΩ; u = ϕ, on ∂Ω, where X = {X1,..., Xm} is a system of real smooth vector fields which sat-isfies the Hörmander’s condition. Assume that X1,..., Xm satisfies some supplementary conditions on the boundary ∂Ω, f ∈ C∞(Ω × IR × IRm), ∂zf(x, z, ξ) ≥ 0, sign zf(x, z, 0) ≥ µ> −∞, c(x) ≥ c0> 0. With some growth hypothesis of f(x, z, ξ) in the variables ξ, we have proved the existence and the uniqueness of solution u ∈ C∞(Ω) of above semilinear Dirichlet problem, if ϕ ∈ C∞(∂Ω)