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

Abstract: 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.