The One-Phase Bifurcation For The p-Laplacian

TL;DR

This paper investigates a bifurcation phenomenon related to the p-Laplacian in a free boundary problem, demonstrating the existence of multiple solutions and analyzing their stability and convergence properties.

Contribution

It introduces a new bifurcation analysis for the p-Laplacian, proving the existence of a third solution via the Mountain Pass Lemma and studying solution stability.

Findings

Existence of a third solution when boundary data decreases below a threshold.

Convergence of an evolution to stable solutions.

Mountain Pass solution is shown to be unstable.

Abstract

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.