Abstract

For p > 1, and ?p(s) = |s|p-2s, we consider the equation (?p(x?))? + ??p(x+) - ??p(x-) = f(t, x), where x+ = max{x, 0}; x- = max{-x, 0}, in a situation of resonance or near resonance for the period T, i.e. when ?, ? satisfy exactly or approximately the equation ?p/?1/p + ?p/?1/p = T/n, for some integer n. We assume that f is continuous, locally Lipschitzian in x, T-periodic in t, bounded on R2, and having limits f±(t) for x ? ±?, the limits being uniform in t. Denoting by v a solution of the homogeneous equation (?p(x?))? + ??p(x+) - ??p(x-) = 0, we study the existence of T-periodic solutions by means of the function Z(?) = ?{t?I|v?(t)>0} f+(t)v(t + ?) dt + ?{t?I|v?(t)<0} f-(t)v(t + ?) dt, where I def= [0, T]. In particular, we prove the existence of T-periodic solutions at resonance when Z has 2z zeros in the interval [0, T/n), all zeros being simple, and z being different from 1.