Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations

Abstract

The existence of Aubry--Mather sets and infinitely many subharmonic solutions to the following $p$-Laplacian like nonlinear equation

is discussed, where $\phi_p(u)=|u|^{p-2}u, \,p>1$, \,$\al, \beta$ are \vspace{-0.05cm} positive constants satisfying \linebreak $\al^{-\frac{1}{p}}+\beta^{-\frac{1}{p}}=\frac2n$ with $n\in \N, \,h$ is piece-wise two times differentiable and $2\pi_p$-periodic, $g\in C^1(R)$ is bounded, $x^{\pm}=\max \{\pm x, 0\}, \,\pi_p=\frac{2\pi}{p\sin(\pi/p)}.$