Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^2,2_\textrmloc(\Omega)$ of the nonlinear elliptic problem \beginequation \tagP \labelabstr \left\ \beginarrayll -\Delta_Hu+H(\nabla u)^q+\lambda u=f&\textin \Omega,\\ u\rightarrow +\infty &\texton \partial\Omega, \endarray \right. \endequation where $H$ is a suitable norm of $\mathbb R^n$, $\Omega$ is a bounded domain, $\Delta_H$ is the Finsler Laplacian, $1<q\le 2$, $\lambda>0$ and $f$ is a suitable function in $L^\infty_\textrmloc$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^+$, studying the so-called ergodic problem associated to \eqrefabstr. A key role in order to study the ergodic problem will be played by local gradient estimates for \eqrefabstr.