Periodic Solutions of Singular Lagrangian Systems by Antonio Ambrosetti, Vittorio Coti Zelati (auth.)
By Antonio Ambrosetti, Vittorio Coti Zelati (auth.)
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Example text
Our next result deals with the case when (V2) is replaced by (V2') V(t,x) ---t 0 as Ixl ---t +00, uniformly in t; and ::J r2 > 0 such that V'(t,x)· x < 0 for alllxl2: r2. 12 Suppose that V satisfies (VO), (V3), (V2') and the (SF) condition. 8 holds. PROOF. 6 readily show that f satisfies (£7) provided (V2') holds. 12) is a trivial, T-periodic solution of q + V'(q) for all T > sup{f(O : ~ o. = 0 (CS) However, if V is bounded on Zv then b := E Zv} is finite. Since f possesses a sequence of 51 9.
4 below. 5 Let The Strong Force assumption n= jRn \ {O}, V satisfy (VO) and define g(u) = l T V(t,u)dt. In order to control the behavior of 9 on 8A, we give the following definition. 1 We say that V satisfies condition (SF) (Strong Force) if there exist a, r > 0 such that V(t, x) ~ a -l x l2 v 0 < Ixl < r, V t E [0, T]. 3 Suppose V satisfies (VO) and (SF) and let be a sequence such that Urn ~ U E 8A weakly. Then g(u rn ) Urn E A ~ -00. Without relabeling, we can assume that Urn ~ U uniformly in [0, T].
The function {3(t) = t ala{e - (~r}t achieves its minimum at to = ~{1 - (ala)2}-t Let a and {3(t o) = ~{1 - (ala)2}t. Then R{1-(ala)2}t a ~ lu(t)1 ~ 1~I+ada21~12-R2}t Vu E Sa,R. CHAPTER III. THE STRONGLY ATTRACTIVE CASE 54 Let R > rl, (rl given by (V2)) be such that ~{I-(ala)2}t ~ rl. The left hand side of the above inequality shows that min{lu(t)1 : t E ~} ~ rl, Vu E Sa,R and hence, by (V2), VuE Sa,R, V t E JR. 38) It follows that, for all u E Sa,R one has If ~(a2IeI2 - R 2 ) ~ 1, then one deduces that f(u) ~ 1.