Quasi-periodic solutions for p-Laplacian equations with jumping nonlinearity and unbounded potential terms

Abstract

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)+f(x)=e(t)$, where $x^+=\max (x,0)$, $x^- =\max(-x,0)$, $\phi_p(s)=|s|^{p-2}s$, $p\geq2$, $a$ and $b$ are positive constants $(a\not=b)$, and satisfy $\frac{1}{a^{\frac{1}{p}}}+\frac{1}{b^{\frac{1}{p}}}=2\omega^{-1} $,where $\omega \in \RR^+ \backslash \QQ$, the perturbation $f$ is unbounded, $e(t)\in {\cal C}^{6}$ is is a smooth $2\pi_p$-periodic function on $t$, where $\pi_p=\frac{2\pi(p-1)^{\frac{1}{p}}}{p\sin\frac{\pi}{p}}$.