Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms

In this Note we prove an a priori estimate and the existence of a solution for a class of nonlinear elliptic problems whose model is -div A(x)Du + alpha(0)u = gamma vertical bar Du vertical bar(q) + f(x), when 1 < q < 2 and f is an element of L-m (Omega) for some suitable in. The main interest of the result lies in the a priori estimate, the complete proof of which is given in the Note