2 Breakdown of the Palais-Smale condition In the sequel, we assume that V is autonomous and that the period T—1. First, we show that if q € E and I(q) oo , then q 6 An where An = {q e E/qi(t) ^ q^t) for all i ^ j and t e [0,1]} (2.1) Proposition 2.1 Suppose V satisfies (Vi), (V^), (V^), (V6). Then for any c 0, there exists 6 = 6(c) such that if q € E and I(q) c then inf \qi(t) -qj(t)\6 (2.2) t G [0, 1] Proof: The proof was given by A. Bahri and P. Rabinowitz in [4]. It is based on the earlier basic work of W.B. Gordon [6] and C. Greco [7]. We recall it here. For any two distinct indices z, j G {1, 2,..., n}, (]/2) implies that °-JQ Vij(*(t) ~ Qj(t)) dt c. Since c oo, (V4) implies that there exists 61 = 61(c) 0 and r G [0,1] such that \qi(T) - qj(T)\ ^i(c). We can assume that \qi(r) - qj(r)\ = 6X. (V2) also implies that ||ft-%||L2 V§C. 5

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