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 ${\varphi }_p(u)=|u{|}^{p-2}u,p>1$, $\alpha ,\beta$ are positive constants satisfying ${\alpha }^{-\frac{1}{p}}+{\beta }^{-\frac{1}{p}}=\frac{2}{n}$ with $n\in \mathbb{N}$, $h$ is piece-wise two times differentiable and $2{\pi }_p$-periodic, $g\in C^1(R)$ is bounded, $x^{\pm }=\max \{\pm x,0\}$, ${\pi }_p=\frac{2\pi }{p\sin (\pi /p)}.$