Communications in Contemporary Mathematics Vol. 12, No. 3 (2010) 437–455 c  World Scientific Publishing Company DOI: 10.1142/S0219199710003877 ROTATION NUMBER, PERIODIC FU ˇ CIK SPECTRUM AND MULTIPLE PERIODIC SOLUTIONS PING YAN Department of Mathematical Sciences, Tsinghua University Beijing 100084, P. R. China pyan@math.tsinghua.edu.cn MEIRONG ZHANG Department of Mathematical Sciences, Tsinghua University Beijing 100084, P. R. China and Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University Beijing 100084, P. R. China mzhang@math.tsinghua.edu.cn Received 23 July 2008 Revised 6 January 2009 In this paper, we will introduce the rotation number for the one-dimensional asymmetric p-Laplacian with a pair of periodic potentials. Two applications of this notion will be given. One is a clear characterization of two unbounded sequences of Fuˇ cik curves of the periodic Fuˇ cik spectrum of the p-Laplacian with potentials. With the help of the Poincar´ e–Birkhoff fixed point theorem, the other application is some existence result of multiple periodic solutions of nonlinear ordinary differential equations concerning with the p-Laplacian. Keywords : Rotation number; p-Laplacian; Fuˇ cik spectrum; periodic solution. Mathematics Subject Classification 2010: 34L05, 37E45, 47E05 1. Introduction Rotation number is a fundamental notion introduced by Poincar´ e in describing the dynamics of homeomorphisms of the circle and differential equations of the 2-torus. The extension to systems of many degrees of freedom with certain recurrence is also very important in many problems such as spectrum of Schr¨ odinger operators. In this paper, we will develop some interesting applications of rotation numbers to the asymmetric p-Laplacian oscillators with potentials. More precisely, given two functions w, v ∈ L 1 (R/T Z) called potentials, we will consider the following scalar differential equation (φ p (x ′ )) ′ + w(t)φ p (x + )+ v(t)φ p (x - )=0. (1.1) 437