GROUND STATE HOMOCLINIC SOLUTIONS FOR A SECOND-ORDER HAMILTONIAN SYSTEM

. Consider the second-order Hamiltonian system ¨ u − L ( t ) u + ∇ W ( t,u ) = 0 , where t ∈ R ,u ∈ R N , L : R → R N × N and W : R × R N → R . We mainly study the case when both L and W are periodic in t and 0 belongs to a spectral gap of σ (cid:16) − d 2 dt 2 + L (cid:17) . We prove that the above system possesses a ground state homoclinic solution under assumptions which are weaker than the ones known in the literature.


1.
Introduction. Consider the second-order Hamiltonian system u − L(t)u + ∇W (t, u) = 0, (1.1) where t ∈ R, u ∈ R N , L : R → R N ×N and W : R × R N → R satisfy the following basic assumptions: (L0) L ∈ C R, R N ×N is T -periodic (T > 0) and L(t) is an N × N symmetric matrix; (W1) W ∈ C(R × R N , R), W (t, x) is continuously differentiable with respect to x ∈ R N for each t ∈ R, W (t, 0) ≡ 0, W (t, x) is T -periodic in t, and W (t, x) ≥ 0 for all (t, x) ∈ R × R N ; (W2) ∇W (t, x) = o(|x|) as x → 0 uniformly for t ∈ R.
We say that a solution u(t) of (1.1) is homoclinic (to 0) if u(t) → 0 as t → ±∞. In addition, if u(t) ≡ 0 then u(t) is called a nontrivial homoclinic solution. A ground state homoclinic solution is a nontrivial homoclinic solution that minimizes the energy among all nontrivial homoclinic solutions.
We are now in a position to state the first result of this paper. To state the second result, we define a functional Φ on H 1 (R, R N ) by where E = E + ⊕ E − corresponds to the spectral decomposition of − d 2 dt 2 + L with respect to the positive and negative part of the spectrum, and u = u Otherwise, E − is infinitedimensional, for more detail, see Section 2. Pankov [18] introduced the following set, (1.4) By definition, M contains all nontrivial critical points of Φ. If u 0 ∈ E \ {0} satisfies Φ (u 0 ) = 0 and Φ(u 0 ) = inf M Φ, then it is a ground state solution of (1.1). There are many works on the existence of a ground state solution which minimizes the energy among all functions of the Nehari-Pankov mainfold for Schrödinger equations, see for example [12,18,[22][23][24]. However, to the best of our knowledge, there are no similar results on the existence of a ground state homoclinic solution for Hamiltonian system (1.1). The main reason is that for system (1.1), there is no analog of the Nehari-type monotone condition used for Schrödinger equation.
Being motivated by [22][23][24], in the present paper we generalize the Nehari-type monotone condition to high-dimensional case in such a way that (1.1) possesses a ground state homoclinic solution Before presenting our second theorem, we define a set N D as follows: We make the following assumptions: Now, we are ready to state the second result of this paper.
Remark 1.6. (W4) and (W5) (or (W6)) are complementary, but (W4) is satisfied by more functions. The ground state homoclinic solutions u 0 andū provided by Theorems 1.1 and 1.2, respectively, have the same characterization -they minimize the energy among all nontrivial homoclinic solutions. However,ū has another (minimax) characterization given by Similarly, in (W 1) one can assume that W and its derivative with respect to the R N -variable are Caratheodory functions with the obvious L ∞ -type boundedness with respect to the R-variable.

2.
Variational setting and preliminaries. Let X be a real Hilbert space with X = X − ⊕ X + and X − ⊥ X + . A functional ϕ ∈ C 1 (X, R) is said to be weakly sequentially lower semi-continuous if for any u n u in X, one has ϕ(u) ≤ lim inf n→∞ ϕ(u n ), and ϕ is said to be weakly sequentially continuous if Lemma 2.1. ( [12,13]) Let (X, · ) be a real Hilbert space with X = X − ⊕ X + and X − ⊥ X + , and let ϕ ∈ C 1 (X, R) be of the form Assume that the following conditions are satisfied: (KS1) ψ ∈ C 1 (X, R) is bounded from below and weakly sequentially lower semicontinuous; (KS2) ψ is weakly sequentially continuous; (KS3) there exist r > ρ > 0 and e ∈ X + with e = 1 such that Then there exist a constant c ∈ [κ, sup ϕ(Q)] and a sequence {u n } ⊂ X satisfying [11,Theorem 4.26]). Let {E(λ) : −∞ ≤ λ ≤ +∞} and |A| be the spectral family and the absolute value of A, respectively, and let |A| 1/2 be the square root of |A|. Set U = id − E(0) − E(0−). Then U commutes with A, |A| and |A| 1/2 , and A = U|A| is the polar decomposition of A (see [10,Theorem IV 3.3]).
For any u ∈ E, it is easy to see that u = u − + u + , where and Define an inner product and the corresponding norm where (·, ·) L 2 denotes the usual inner product in L 2 (R, R N ) and · s stands for the usual norm in L s (R, R N ). Since E = H 1 (R, R N ) with equivalent norms under (L), E is continuously embedded in L s (R, R N ) for all 2 ≤ s ≤ ∞, i.e., for all 2 ≤ s ≤ ∞, there exists γ s > 0 such that In addition, one has the decomposition E = E − ⊕ E + orthogonal with respect to both (·, ·) L 2 and (·, ·).
In view of (2.3) and (2.5), we have From (1.3) and (2.7), one has Then we can prove the following lemma by a standard argument.

3.
Proof of Theorem 1.1. The following lemmas will be useful in the proof of Theorem 1.1.
Proof. Since Φ(w) ≤ 0 for all w ∈ E − , we only need to show that Φ(w + se) → −∞ as w + se → ∞. Arguing indirectly, assume that there exists a sequence {w n + s n e} ⊂ E − ⊕ Re with w n + s n e → ∞ such that Φ(w n + s n e) ≥ 0 for all n ∈ N. Set v n = (w n + s n e)/ w n + s n e = v − n + τ n e, then v − n + τ n e = 1. Passing to a subsequence, we may assume that is bounded in E.
This shows that (4.1) holds. From Lemma 4.1, we have the following two corollaries.
Combining Lemma 3.1 and Corollary 4.2 with the argument used in [23] to prove Lemma 3.6, we can establish the following statement.
Following the argument used in the proof of Lemma 3.8 from [23], it is easy to show the following lemma.
is bounded in E.