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 ) D u + α 0 u = γ | D u | q + f ( x ) , when 1 < q < 2 and f ∈ L m ( Ω ) for some suitable m . The main interest of the result lies in the a priori estimate, the complete proof of which is given in the Note.