MULTIPLE SOLUTIONS OF SECOND ORDER HAMILTONIAN SYSTEMS

. The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via link- ing theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualiﬁed solutions.

For each x ∈ R n , the function V (t, x) is periodic in t with period T.
By assuming that the elements of the symmetric matrix B(t) are to be real-valued functions b jk (t) = b kj (t) and that (B1) each component of B(t) is an integrable function on I, i.e., for each j and k, b jk (t) ∈ L 1 (I), it was possible to exploit the property that there is an extension of the operator having a discrete, countable spectrum consisting of isolated eigenvalues of finite multiplicity with a finite lower bound −L −∞ < −L ≤ λ 0 < λ 1 < λ 2 < . . . < λ l < . . .
Here, inspired by the arguments adopted in [35], we consider the following problem where B is a symmetric matrix valued function satisfying an elliptic condition (see next assumption (B 3 )) and µ is a positive real parameter. In particular, first we simply require a suitable behaviour of the potential V (t, ·) near zero in order to establish the existence of positive interval of parameters for which problem (1.4) admits at least one qualified non trivial solution (see Theorem 3.1). Then, assuming in addition that V (t, ·) satisfies different conditions at infinity, a second non trivial solution is assured (see Theorems 3.2 -3.4). The multiplicity results are obtained combining a linking theorem for functionals depending on a parameter with a monotonicity trick.

Variational setting and preliminary results
In the sequel we will assume the following conditions on the matrix valued func- (B3) There exists a positive function γ ∈ L ∞ (I) such that for every x ∈ R n and a.e. t in I. Thus γ(t)|x| 2 ≤ B(t)x · x ≤ Λ(t)|x| 2 , for every t ∈ I and x ∈ R n , where Λ(t) ∈ L ∞ (I). Following the notation of [30], let H 1 T be the Sobolev space of functions u ∈ L 2 (I, R n ) having a weak derivativė u ∈ L 2 (I, R n ). It is well known that H 1 T , endowed with the norm , is a Hilbert space, compactly embedded in C 0 (I, R n ) and C ∞ T ⊂ H 1 T . Because of the previous conditions, it is possible to introduce on H 1 T the following inner product In fact, we have Remark 2.1. For an explicit estimate of the constant c 0 we refer to [12,21,30].
A solution of problem (1.4) is any function u 0 ∈ C 1 (I, R n ) such thatu 0 is absolutely continuous, and satisfies It follows that, if we put λ = 1/µ, a critical point of the functional is a solution of (1.4) where the system takes the form We introduced the parameter λ to make use of the monotonicity trick. This requires us to work in an interval of the parameter λ, and it allows us to obtain solutions under very weak hypotheses. However, we obtain solutions only for almost every value of the parameter. We can then obtain solutions for all values of the parameter by introducing appropriate mild assumptions.
In proving the theorems, we shall make use of the following results of linking. Let E be a reflexive Banach space with norm ∥ · ∥. The set Φ of mappings Γ(t) ∈ C(E × [0, 1], E) is to have following properties: map bounded sets to bounded sets and Λ is an open interval contained in (0, +∞). Assume one of the following alternatives holds. (

Statement of the theorems
where φ is an eigenfunction of D corresponding to the first eigenvalue λ 0 . Then the system (2.1) has a nontrivial solution u λ satisfying (1) and (2) of Theorem 3.1 are satisfied in addition to

Theorem 3.2. Assume that hypotheses
Then the system (2.1) has two nontrivial solutions u λ , v λ satisfying for almost all λ ∈ (K 0 , M ). (1) and (2) of Theorem 3.1 are satisfied. Moreover, There are a constant C and a function W (t) ∈ L 1 (I) such that

Then the system (2.1) has two nontrivial solutions
where

Proofs of the theorems
Before giving the proofs, we shall prove a few lemmas.

Lemma 4.1. If (3.4) holds, then
where the constant C does not depend on u, r.
Proof. This follows from (3.4) if we take u = x.

Lemma 4.2.
If u satisfies G ′ λ (u) = 0 for some λ > 0, then there is a constant C independent of u, λ, r such that Proof. From G ′ λ (u) = 0 one has that Then we have Proof of Theorem 3.1. Fix λ ∈ (K 0 , M ), put r 2 = m 2 /c 0 and define Indeed, let δ > 0 be such that K 0 < K 0 + δ < λ < M , then for every u ∈ ∂B r one has for every |c| <σ. Hence, for c sufficiently small one has cφ ∈ B r , as well as Thus Also λd(u) ≤ lim inf λd(u k ) = µ(λ) + 2 ∫ I V (t, u) dt, namely G λ (u) ≤ µ(λ) < 0 and u / ∈ ∂B r . Hence, u is in the interior of B r and we have G ′ λ (u) = 0. Proof of Theorem 3.2. First observe that, if we define T one has that G λ = G λ . Hence, taking in mind that I(u) ≥ 0 for all u ∈ H 1 T , it is clear that (H 1 ) holds. Moreover, as in the proof of Theorem 3.1, take r 2 = m 2 /c 0 . Then By hypothesis, there are c 1 , c 2 such that c 1 φ ∈ B r and c 2 φ / ∈ B r with G λ (c i φ) < 0, i = 1, 2. The set A = (c 1 φ, c 2 φ) links B = ∂B r (cf.,e.g., [33]). Applying Theorem 2.1, for almost every λ we obtain a bounded sequence (y Since the sequence is bounded, there is a renamed subsequence such that y k ⇀ y ∈ H 1 T and y k → y ∈ L ∞ (I). Since G ′ λ (y k ) → 0, we have In the limit this gives G ′ λ (y) = 0. We also have λd(y k ) → ∫ I ∇V (t, y)y = λd(y).
The proof is completed taking u λ as already assured by Theorem 3.1 and v λ = y.

The remaining proofs
Proof of Theorem 3.3. Note that (3.3) implies (3.2). By Theorem 3.2, for a.e. λ ∈ . Then d(ũ n ) = 1 and there is a renamed subsequence such thatũ n →ũ weakly in H 1 T , strongly in L ∞ (I) and a.e. in I. Let Ω 0 ⊂ I be the set whereũ ̸ = 0. Then |u n (t)| → ∞ for t ∈ Ω 0 . If Ω 0 had positive measure, then, observing that (3.3) and the continuity of V assure the existence of β ∈ R such that At this point, we obtain a contradiction passing to the liminf and applying the Fatou lemma, since from (3.3) it is clear that for every t ∈ Ω 0 , 2V (t,un) |un| 2 |ũ n | 2 → +∞ as n → ∞. This shows thatũ = 0 a.e. in I. Hence,ũ n → 0 in L ∞ (I). For any s > 0 and h n = sũ n , we have Take r n = s/d 1/2 (u n ) → 0. By Lemma 4.2 Hence, 1). This implies G λn (sũ n ) → ∞ as s → ∞, contrary to (5.3). This contradiction shows that ∥u n ∥ H 1 T ≤ C. Then there is a renamed subsequence such that u n → u weakly in H 1 T , strongly in L ∞ (I) and a.e. in I. It now follows that for the bounded renamed subsequence, We can now follow the proof of Theorem 3.2 to obtain the desired solution.
Proof of Theorem 3.4. We follow the proof of Theorem 3.3 until we conclude that u n → 0 in L ∞ (I) as a consequence of the fact that we assume that ∥u n ∥ H 1 T → ∞. We define θ n ∈ [0, 1] by For any c > 0 and h n = cũ n , we have ∫ Thus, for every fixed c > 0, if n is large enough one has that 0 < c/d 1/2 (u n ) < 1 and namely, lim n→∞ G λn (θ n u n ) = ∞. If there is a renamed subsequence such that θ n = 1 for every n, then If 0 ≤ θ n < 1 for all n, then we have (G ′ λn (θ n u n ), θ n u n ) = 0. Indeed, defined h(θ) = G λn (θu n ) for every θ ∈ [0, 1], one has Hence, if θ n = 0 then (G ′ λn (θ n u n ), θ n u n By hypothesis, Thus, (5.4) holds in any case. But This contradiction shows that ∥u n ∥ H 1 T ≤ C. It now follows that for a renamed subsequence, We can now follow the proof of Theorem 3.2 to obtain the desired solution.

Some examples
Here we show that the assumptions required in the main theorems are naturally satisfied in many simple and meaningful cases.
For simplicity, in the following, we suppose that n = 1, I = [0, π] and B(t) ≡ 1 for all t ∈ I while α, β ∈ L 1 (I) are two positive functions. A direct computation shows that the eigenvalues of D, with periodic boundary conditions, are λ l = 4l 2 + 1.