A Note on Generalized Solutions of Singular Hamiltonian Systems

We consider T-periodic solutions of singular Hamiltonian systems with weak force q + VV(q,t) = 0, where V(q, t) ~ -l/\q\a near q = 0 with a e (0, 2). In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz. 0. Introduction In this paper we study properties of the generalized solutions of singular Hamiltonian systems which were introduced in [BR]. We consider a Hamiltonian system (HS.l) q + vV(q,t) = 0 in R, (HS.2) q(t + T) = q(t) inR, where q = (qx, q2,... , q^) £ RN (N > 3), F > 0, and V(q, t) is a function such that (VI) V £ C2((RJV\{0}) x R, R) is F-periodic in t; (VI) V(q, t) < 0 for all (q, t) and V(q, t), VV(q, i) 0 as \q\ — co uniformly in t ; (V3) V(q, t) is of the form V(q,t) = -l/\q\a + U(q,t), where a £ (0, 2) and U(q, t) £ C2((RN\{0}) x R, R) is F-periodic in t and satisfies \q\a-»U(q,t), \q\a-"+xVU(q,t), \q\a~P+2V2U(q, t), \q\a~"U,(q ,t)->0 as q -» 0 uniformly in t for some p £ (0, a). Received by the editors November 23, 1992. 1991 Mathematics Subject Classification. Primary 58F05; Secondary 58E05.