Monotonicity of the period and positive periodic solutions of a quasilinear equation

TL;DR

This paper studies how the period of solutions to certain quasilinear differential equations involving the p-Laplace operator changes with energy levels, extending classical results to cases where p is greater than or equal to 2.

Contribution

It generalizes existing monotonicity results of the period for quasilinear equations to the case p ≥ 2, including equations related to Sobolev space inequalities.

Findings

Monotonicity of the period as a function of Hamiltonian energy established for p ≥ 2.

Extension of classical results from p=2 to p ≥ 2 for specific differential equations.

Generalization of monotonicity results related to Sobolev inequalities for p ≥ 2.

Abstract

We investigate the monotonicity of the minimal period of periodic solutions of quasilinear differential equations involving the $p$-Laplace operator. First, the monotonicity of the period is obtained as a function of a Hamiltonian energy in two cases. We extend to $p\ge2$ classical results due to Chow-Wang and Chicone for $p=2$. Then we consider a differential equation associated with a fundamental interpolation inequality in Sobolev spaces. In that case, we generalize monotonicity results by Miyamoto-Yagasaki and Benguria-Depassier-Loss to $p\ge2$.