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 $u\in W_{\text{loc}}^{2,2}(\Omega )$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -Δ_{H}u+H(\nabla u)^{q}+λu=f&\text{in }Ω,\\ u\rightarrow +\infty &\text{on }\partialΩ, \end{array} \right. \end{equation} 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 $L_{\text{loc}}^{\infty }$. Furthermore, we are interested in the behavior of the solutions when $\lambda \to 0^{+}$, studying the so-called ergodic problem associated to \eqref{abstr}. A key role in order to study the ergodic problem will be played by local gradient estimates for \eqref{abstr}.