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

Jean Dolbeault, Marta García-Huidobro, Raúl Manásevich

TL;DR

This work extends the classical monotonicity results for the period function of one-dimensional oscillators from the linear ($p=2$) setting to the nonlinear $p$-Laplacian case with $p>2$. By deriving a first integral and an explicit derivative formula for the minimal period $T(E)$, the authors establish sufficient conditions for monotonicity: (i) a Chow–Wang-type criterion via the nonnegativity of $\mathcal{R}(w)=\mathcal{V}'(w)^2-p'\mathcal{V}(w)\mathcal{V}''(w)$, and (ii) a Chicone-type convexity criterion on $\mathcal{V}/(\mathcal{V}')^2$. They also analyze a Sobolev interpolation–type problem (ELS2) and prove that for $2<p<q$, the period $T(E)$ is increasing on $(0,E_*)$ with $T(E)\to0$ as $E\to0_+$ and $T(E)\to\infty$ as $E\to(E_*)_-$. A novel monotonicity criterion based on a carefully chosen change of variables and a kernel $K$ provides a unifying framework to prove monotonicity across different parameter regimes, including a detailed treatment for the case $\mathcal{V}$ arising from the interpolation problem. These results extend and sharpen prior $p=2$ outcomes and have implications for symmetry-breaking thresholds in related variational problems on Sobolev spaces.

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 $p\ge2$.

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

TL;DR

This work extends the classical monotonicity results for the period function of one-dimensional oscillators from the linear () setting to the nonlinear -Laplacian case with . By deriving a first integral and an explicit derivative formula for the minimal period , the authors establish sufficient conditions for monotonicity: (i) a Chow–Wang-type criterion via the nonnegativity of , and (ii) a Chicone-type convexity criterion on . They also analyze a Sobolev interpolation–type problem (ELS2) and prove that for , the period is increasing on with as and as . A novel monotonicity criterion based on a carefully chosen change of variables and a kernel provides a unifying framework to prove monotonicity across different parameter regimes, including a detailed treatment for the case arising from the interpolation problem. These results extend and sharpen prior outcomes and have implications for symmetry-breaking thresholds in related variational problems on Sobolev spaces.

Abstract

We investigate the monotonicity of the minimal period of periodic solutions of quasilinear differential equations involving the -Laplace operator. First, the monotonicity of the period is obtained as a function of a Hamiltonian energy in two cases. We extend to classical results due to Chow-Wang and Chicone for . 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 .
Paper Structure (10 sections, 13 theorems, 77 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A p-Laplacian version of some classical results
  3. Further properties of the minimal period
  4. Asymptotic results
  5. The monotonicity of the minimal period
  6. Proof of the main result
  7. Notations and strategy
  8. The case m=2
  9. The case m>2
  10. The case 1<m<2

Key Result

Theorem 1.1

Let $p>2$. Assume that $\mathcal{V}$ satisfies H1 and If $w\mapsto|\mathcal{V}'(w)|^2-p'\,\mathcal{V}(w)\,\mathcal{V}"(w)$ is positive, then $E\mapsto T(E)$ is increasing on $(0,E_*)$.

Figures (1)

  • Figure 1: Two examples of $\mathcal{V}$ satisfying assumption \ref{['H1']}. Our monotonicity results require the stronger assumptions on $\mathcal{V}$ of Theorems \ref{['Thm:Main1']} and \ref{['Thm:Main2']} which, typically, hold in the left case but not in the right case.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof : of Theorem \ref{['Tchow']}
  • proof : of Theorem \ref{['Thm:Main2']}
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 14 more