Existence of strong solutions to some quasilinear elliptic problems on bounded smooth domains

We consider the following quasilinear elliptic problems in a bounded smooth domain Z of ℝ^N, N ≥ 3: (formula presented) where f(x) ∈ L^p(Z) and all the coefficients a_ij, b_i, c are Carathédory functions. Suppose that a_ij ∈ C^0,1 (Z̄ × ℝ), a_ij, ∂a_ij/∂x_i, ∂a_ij,/∂r, b_i, c ∈ L^∞(Z × ℝ), c ≤ 0 for i,j = 1, ...N and the oscillations of a_ij = a_ij(x,r) with respect to r are sufficiently small. A global estimate for a solution u ∈ W^2,p(Z) ∩ W^1,p₀(Z) is established and the existence of a strong solution u ∈ W^2,p(Z) ∩ W^1,p₀(Z) is proved for p > N. Furthermore, we replace f(x) by f(x,r,ξ) which is defined on Z × ℝ × ℝ^N and is a Carathédory function. Assume that |f(x,r,ξ)| ≤ C₀ + h(|r|)|ξ|^θ, 0 ≤ θ < 2, where C₀ is a nonnegative constant, h(|r|) is a locally bounded function, and -c ≥ α₀ > 0 for some constant α₀. We prove the existence of solution u ∈ W^2,p(Z) ∩ W^1,p₀(Z) for the equation Lu = f(x,u,∇u).