Qualitative properties of a nonlinear system involving the $p$-Laplacian operator

TL;DR

This paper investigates the qualitative properties of a nonlinear system involving the p-Laplacian operator, focusing on symmetry, asymptotic behavior, and non-degeneracy, to inform higher-dimensional analyses.

Contribution

It establishes fundamental qualitative properties of a specific nonlinear p-Laplacian system, aiding understanding of higher-dimensional cases under certain conditions.

Findings

Proves symmetry of solutions

Analyzes asymptotic behavior at infinity

Establishes non-degeneracy properties

Abstract

In this article we consider the nonlinear system involving the $p$-Laplacian $$\left\{\begin{array}{lc} |u^\prime |^{p-2} u^{\prime \prime} = u^{p-1} v^p& |v^\prime |^{p-2} v^{\prime \prime} = v^{p-1} u^p&\ {\rm on} \ \R, u\geq 0, v\geq 0& \end{array}\right.$$ for which we prove symmetry, asymptotic behavior and non degeneracy properties. This can help to a better understanding to what happens in the $N$ dimensional case, for which several authors prove a De Giorgi Type result under some additional growth and monotonicity assumptions.