Existence of weak solutions for elliptic systems with p,q-growth

We consider a non-linear system of m equations in divergence form and a boundary condition: {Sigma(n)(i=1) partial derivative/partial derivative x(i) (A(i)(alpha)(x, Du(x))) = 0, 1 <= alpha <= m, in Omega u = (u) over tilde on partial derivative Omega. The functions A(i)(alpha)(x, z) are Holder continuous with respect to x and vertical bar z vertical bar(p) - c(1) <= Sigma(m)(alpha=1) Sigma(n)(i=1) A(i)(alpha)(x, z)z(i)(alpha) <= c(2)(1 + vertical bar z vertical bar)(q), 2 <= p <= q. We prove the existence of a weak solution u in ((u) over tilde + W-0(1,p)(Omega; R-m)) boolean AND W-loc(1,q)(Omega; R-m), provided p and q are close enough and under suitable sununability assumptions on the boundary datum (u) over tilde.