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Zeitschrift für Analysis und ihre Anwendungen

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Volume 26, Issue 2, 2007, pp. 207–220
DOI: 10.4171/ZAA/1319

Published online: 2007-06-30

Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations

Xiaojing Yang[1] and KUEIMING LO[2]

(1) Tsinghua University, Beijing, China
(2) Tsinghua University, Beijing, China

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 $\phi_p(u)=|u|^{p-2}u, \,p>1$, \,$\al, \beta$ 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, $x^{\pm}=\max \{\pm x, 0\}, \,\pi_p=\frac{2\pi}{p\sin(\pi/p)}.$

Keywords: Aubry–Mather sets, p-Laplacian, resonance, quasi-periodic solutions

Yang Xiaojing, LO KUEIMING: Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations. Z. Anal. Anwend. 26 (2007), 207-220. doi: 10.4171/ZAA/1319