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

Abstract

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\operatorname{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation}\label{abstr} \begin{cases} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \end{cases}\tag{P} \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $\Omega\subset\mathbb R^{n}$ is a bounded domain, $\Delta_{H}$ is the Finsler Laplacian, $1\lt q\le 2$, $\lambda>0$, and $f$ is a suitable function in $L^{\infty}_{\operatorname{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).