A priori estimates and existence for elliptic equations with gradient dependent terms

We consider, in a bounded domain $\Omega \subset \R^{N}$, a class of nonlinear elliptic equations in divergence form as $$\left\{\begin{array}{l} \alpha_0 u -\dive (a(x,u,Du))=H(x,u,Du)\mbox{ in }\Omega ,\\ u=0\mbox{ on }\partial\Omega\,,\end{array} \right. $$ where $\alpha_0\geq 0$, the second order part is a coercive, pseudomonotone operator of Leray-Lions type in the Sobolev space $W^{1,p}_0(\Omega)$, $p>1$, and the function $H$ grows at most like $|Du|^q+f(x)$, with $p-1<q<p$.Assuming $f(x)$ to belong to an (optimal) Lebesgue class $L^m$, with $m<\frac Np$, we prove a priori estimates and existence of solutions, discussing several ranges of the exponent $m,q$ and $p$ which include cases of singular data ($L^1$ data or measures). The obtention of a priori estimates is not straightforward because of the "super linear" character of the first order terms. To this purpose we use a new approach, generalizing the method introduced in our note [29]. We complete the results known in the previous literature where either $q\leq p-1$ or $m\geq \frac Np$.