BOUNDEDNESS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH JUMPING $p$-LAPLACIAN AND AN OSCILLATING TERM

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 $x^+ = \max (x,0)$, $x^- = \max(-x,0)$, $\phi_p(s) = |s|^{p-2}s$, $p \geq 2$, $a$ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t) \in {\cal C}^{23}(\mathbb{R}/2\pi_p \mathbb{Z})$, the oscillating term $G\in {\cal C}^{21}(\mathbb{R} \times \mathbb{R}/2\pi_p \mathbb{Z})$ satisfying $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21$.