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

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 $p⩾2$ 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 $p⩾2$.