Existence of strong solutions to some quasilinear elliptic equations

Let Lu= - Σ_i,j=1^N a_ij(x, u)D _ijU. Consider the quasilinear elliptic equation Lu + f(x,u,Δu) = 0 on a bounded smooth domain Ω, in ℝ^N. It is shown that if the oscillation of a_ij(x,r) with respect to r is sufficiently small and f(x,r, ξ) has a sub-linear growth in r and ξ, then there exists a solution u ∈ W^2,p(Ω) ∩ W₀ ^1,p(Ω). The existence of W^2,p(Ω) ∩ W ₀^1,p(Ω) solutions to the equation Lu + c(x, u)u + f(x, u, ∇u) = 0, where β ≥ c(x, r) ≥ α > 0, remains valid if f has a sub-quadratic growth in ξ.