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

TL;DR

This paper investigates the existence, asymptotic behavior, and blow-up solutions of nonlinear elliptic equations involving the Finsler-Laplacian, focusing on solutions that diverge at the boundary and their behavior as a parameter approaches zero.

Contribution

It provides new existence results and asymptotic analysis for solutions of Finsler-Laplacian equations with boundary blow-up, including the study of the associated ergodic problem.

Findings

Existence of solutions with boundary blow-up behavior.

Asymptotic analysis as the parameter approaches zero.

Gradient estimates critical for ergodic problem analysis.

Abstract

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\textrm{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \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 $L^{\infty}_{\textrm{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, 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}.