Morse theory and multiple periodic solutions of some quasilinear difference systems with periodic nonlinearities

Abstract

AbstractWe consider the system of difference equations$\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$with${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$,${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$,${V_{n+T}=V_{n}}$,${h_{n+T}=h_{n}}$for all${n\in\mathbb{Z}}$and some positive integerT,${V_{n}(x)}$is${\omega_{i}}$-periodic (${\omega_{i}>0}$) with respect to each${x_{i}}$(${i=1,\ldots,N}$) and${\sum_{j=1}^{T}h_{j}=0}$. Applying a modification argument to the corresponding problem with a left-hand member ofp-Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least${2^{N}}$geometrically distinctT-periodic solutions.