Sharp estimates and existence for anisotropic elliptic problems with general growth in the gradient

In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $\left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x) &\text{in }\Omega,\\ u=0&\text{on }\partial\Omega. \end{array} \right.$ Here $\Omega$ is a bounded open set of $\mathbb R^{N}$, $N\ge 2$, $0<p-1<q\le p<N$, and $\mathcal Q_{p}$ is the anisotropic operator $\mathcal Q_{p} u ={\rm div}\left( [H(Du)]^{p-1}H_{\xi}(Du) \right)$, where $H$ is a suitable norm of $\mathbb R^{N}$. Moreover, $f$ belongs to an appropriate Marcinkiewicz space.