the existence of solutions to certain quasilinear elliptic equations

let lu = −σⁿ_i,j=1 a_ij(x, u)d_iju + c(x, u)u. consider the quasilinear elliptic equation lu = f (x, u,∇u) on a bounded smooth domain ω in rⁿ , where c(x, r) ≥ α > 0, f (x, r, ξ ) = o[|r| + h(|r|)|ξ |²]. it is shown that if the oscillation of a_ij(x, r) with respect to r is sufficiently small, then there exists a solution u ∈ w^2,p(ω) ∩w^1,p₀ (ω) to the equation lu = f (x, u,∇u).