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

TL;DR

This paper investigates the boundedness and existence of quasi-periodic solutions for a class of second-order p-Laplacian differential equations with jumping nonlinearities and unbounded forcing terms, extending understanding of such nonlinear systems.

Contribution

It establishes conditions for boundedness and quasi-periodic solutions in p-Laplacian equations with jumping nonlinearities and unbounded periodic forcing, a novel extension in nonlinear differential equations.

Findings

Solutions are bounded under specified parameter conditions.

Existence of quasi-periodic solutions is proven for the system.

The results extend previous work to unbounded forcing 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 $\pi_p=\frac{2\pi(p-1)^{\frac{1}{p}}}{p\sin\frac{\pi}{p}}$.