Asymptotics of Large Solutions of p-Laplace Equations on Cylinders Becoming Unbounded

TL;DR

This paper investigates the asymptotic behavior of large solutions to p-Laplace equations on expanding cylindrical domains, establishing convergence to cross-sectional solutions and extending results to the case 1<p<2 with explicit rates.

Contribution

It extends existing convergence results for large solutions of p-Laplace equations to the range 1<p<2 and provides explicit convergence rates independent of boundary data.

Findings

Solutions converge locally to cross-sectional problem solutions.

Established explicit convergence rates for 1<p<2.

Unified framework for solutions with finite Dirichlet boundary data.

Abstract

In this article, we study the asymptotic behavior of large solutions for a quasi-linear equation involving the p-Laplacian, defined on a sequence of finite cylindrical domains converging to an infinite cylinder. We demonstrate that the sequence of solutions converges locally, in the Sobolev norm, to a solution of the corresponding cross-sectional problem. Moreover, we establish a convergence rate. As part of our analysis, we extend existing convergence results for the case $p\geq 2$, which previously lacked explicit convergence rates, to the range $1<p<2$. We additionally address solutions with finite Dirichlet boundary data within a unified framework and exhibit that this rate of convergence is independent of the boundary data.