Boundedness for Second Order Differential Equations with Jumping p-Laplacian and an Oscillating Term

TL;DR

This paper investigates the boundedness of solutions for a class of second order differential equations involving p-Laplacian, oscillating terms, and perturbations, establishing conditions under which solutions remain bounded.

Contribution

It provides new boundedness criteria for solutions of second order p-Laplacian equations with oscillating and perturbation terms, extending previous results to more complex oscillating functions.

Findings

Solutions are bounded under specified conditions.

Boundedness holds for perturbations with high regularity.

The results apply to equations with oscillating terms satisfying certain derivative bounds.

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 and an oscillating term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$\phi_p(s)=|s|^{p-2}s$,$p\geq2$, $a $ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2\pi_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2\pi_p \ZZ)$,where $\pi_p=\frac{2\pi(p-1)^{\frac{1}{p}}}{p\sin\frac{\pi}{p}},$ and $G(x,t)$ satisfies $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21,$ and $\label{hatG} |D_t^j\hat{G}|\le C,\quad 0\le j\le 21$ for some $C>0$, where $\hat{G}$ is some function satisfying $\frac{\pa \hat{G}}{\pa x}=G$.