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

\[ (p-1)^{-1}(\phi_p(x'))'+[\al\phi_p(x^+)-\beta\phi_p(x^-)]+g(x) = h(t) \]

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