Nonlinear Analysis 66 (2007) 571–581 www.elsevier.com/locate/na Subharmonics for not uniformly coercive Hamiltonian systems Adel Daouas a,∗ , Mohsen Timoumi b a Preparatory Institute for Engineering Studies of Monastir, 5019, Monastir, Tunisia b Faculty of Sciences ofMonastir, 5019, Monastir, Tunisia Received 11 May 2005; accepted 1 December 2005 Abstract In this paper, we prove the existence of subharmonic solutions with prescribed minimal period for the Hamiltonian system ˙ u(t ) = J ∇ H (t , u(t )) when the Hamiltonian H is subquadratic with unbounded gradient and not uniformly coercive. c  2005 Elsevier Ltd. All rights reserved. Keywords: Subharmonic solutions; Hamiltonian system; The (SCe) condition; Minimal period 1. Introduction and statement of results In this paper we consider the Hamiltonian system of ordinary differential equations ˙ u (t ) = J ∇ H (t , u (t )) (H) where J is the standard symplectic (2n × 2n)-matrix J =  0 -I n I n 0  , and H : R × R 2n → R,(t , x ) → H (t , x ) is a continuous function, T -periodic in t (T > 0), differentiable with respect to the second variable and ∇ H (t , x ) is continuous. We are interested in the existence of subharmonic solutions of (H), i.e. of distinct kT -periodic solutions of (H), when H is subquadratic satisfying some conditions of Landesman–Lazer type. ∗ Corresponding address: Department of Mathematics, 5019 Monastir, Tunisia. E-mail addresses: daouas adel@yahoo.fr (A. Daouas), m Timoumi@yahoo.com (M. Timoumi). 0362-546X/$ - see front matter c  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.12.002