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: $\{\begin{array}{ll}-{\mathcal{Q}}_{p}u=[H(Du){]}^q+f(x)& \text{in }\Omega ,\\ u=0& \text{on }\partial \Omega .\end{array}$ 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 ${\mathcal{Q}}_{p}u=\mathrm{div}\left([H(Du){]}^{p-1}{H}_{\xi }(Du)\right)$, where $H$ is a suitable norm of ${\mathbb{R}}^N$. Moreover, $f$ belongs to an appropriate Marcinkiewicz space.