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 $({\varphi }_p(x^{\prime }){)}^{\prime }+a{\varphi }_p(x^{+})-b{\varphi }_p(x^{-})+f(x)=e(t)$, where $x^{+}=\max (x,0)$, $x^{-}=\max (-x,0)$, ${\varphi }_p(s)=|s{|}^{p-2}s$, $p\ge 2$, $a$ and $b$ are positive constants $(a\ne b)$, and satisfy $\frac{1}{a^{\frac{1}{p}}}+\frac{1}{b^{\frac{1}{p}}}=2{\omega }^{-1}$,where $ω\in \RR^+ \backslash \QQ$, the perturbation $f$ is unbounded, $e(t)\in {\mathcal{C}}^6$ is is a smooth $2{\pi }_p$-periodic function on $t$, where ${\pi }_p=\frac{2\pi (p-1{)}^{\frac{1}{p}}}{p\sin \frac{\pi }{p}}$.

arXiv admin note: substantial text overlap with arXiv:1301.5388