Homoclinic Orbits for Second-Order Hamiltonian Systems with Some Twist Condition

and Applied Analysis 3 W0 ∇qW t, 0 ≡ 0 and B0 ∈ B, W∞ there exists some R0 > 0 and continuous symmetric matrix functions B1, B2 ∈ B with i B1 i B2 and ν B2 0 such that B1 t ≤ ∇qW t, z ≤ B2 t , ∀t ∈ R, |z| > R0. 1.6 Our first result reads as follows. Theorem 1.1. Assume L1 , W1 , W0 , and W∞ hold. If i B1 / ∈ i B0 , i B0 ν B0 , 1.7 then 1.1 has at least one nontrivial homoclinic orbit. Moreover, if ν B0 0 and |i B1 − i B0 | ≥ N, the problem possesses at least two nontrivial homoclinic orbits. Condition W∞ is a two-side pinching condition near the infinity, learning from the idea of 23, 24 , we can relax W∞ to condition W± ∞ as follows. W± ∞ There exist some R0 > 0 and a continuous symmetric matrix function B∞ ∈ B with ν B∞ 0 such that ±∇qW ( t, q ) ≥ ±B∞ t , ∀t ∈ R, ∣q ∣ > R0. 1.8 The uniform boundary of ∇qW t, q displayed in condition W1 can also be relaxed as W∗ 1 W ∈ C2 R × R,R , and there exists a constant c > 0 such that ∣∇qW ( t, q )∣ ≤ c∣q∣, ∀t, q ∈ R × R. 1.9 W∗∗ 1 For any M > 0, ∇qW t, q is bounded on R × −M,M . On the other hand, we need some sharply twisted conditions than the above theorem, and we have the following theorems. Theorem 1.2. Assume L1 , W∗ 1 , W ∗∗ 1 , W0 , W ∞ or W − ∞ , and ν B0 0 hold. If i B∞ ≥ i B0 2 (or i B∞ ≤ i B0 − 2), then 1.1 has at least one nontrivial homoclinic orbit. Theorem 1.3. Suppose that L1 , W∗ 1 , W ∗∗ 1 , W0 , W ∞ or W − ∞ , and ν B0 0 are satisfied. If, in addition, W is even in q and i B∞ ≥ i B0 2 (or i B∞ ≤ i B0 − 2), then 1.1 has at least |i B∞ − i B0 | − 1 pairs of nontrivial homoclinic orbits. Remark 1.4. Note that the assumption ν B∞ 0 in W± ∞ is not essential for our main results. For the case of W ∞ with ν B∞ / 0, let B̃∞ B∞ − εIn×n with ε > 0 small enough, where 4 Abstract and Applied Analysis In×n is the identity map on R , then i B̃∞ i B∞ and ν B̃∞ 0, and hence W ∞ holds for B̃∞. Therefore, Theorems 1.2 and 1.3 still hold in this case. While for the case of W− ∞ with ν B∞ / 0, if we replace i B∞ by i B∞ ν B∞ in Theorems 1.2 and 1.3, then similar results hold. Indeed, let B̃∞ B∞ εIn×n with ε > 0 small enough such that i B̃∞ i B∞ ν B∞ and ν B̃∞ 0, then this case is also reduced to the case of W− ∞ for B̃∞ with ν B̃∞ 0. Remark 1.5. Choose W t, q W t, q − W t, 0 instead of W in 1.1 , then conditions W∗ 1 , W∗∗ 1 , W0 , and W ∞ or W − ∞ still hold forW , so we can always assume W t, 0 ≡ 0. 2. Preliminaries Denote by Ã the self-adjoint extension of the operator −d2/dt2 L t with domain D Ã ⊂ L2 ≡ L2 R,R . Let {E λ : −∞ < λ < ∞} and |Ã| be the spectral resolution and the absolute value of Ã, respectively, and let |Ã|1/2 be the square root of |Ã| with domain D |Ã| . Set U I − E 0 − E −0 , where I is the identity map on L2. Then, U commutes with Ã, |Ã|, and |Ã|1/2, and Ã U|Ã| is the polar decomposition of Ã. Let E D |Ã|1/2 , and define on E the inner product and norm by u, v 0 (∣∣Ã ∣∣∣ 1/2 u, ∣∣∣Ã ∣∣∣ 1/2 v ) 2 u, v 2, ‖u‖0 u, u 1/2 0 , 2.1 where ·, · L2 denotes the usual inner product on L2 R,R . Then, E is a Hilbert space. It is easy to see that E is continuously embedded in W1 R,R . In fact, we further have the following lemmas. Lemma 2.1 see 16 , Lemma 2.2 . Suppose that L satisfies L1 . Then, E is compactly embedded in L R,R with the usual norm ‖ · ‖Lp for any 1 ≤ p ∈ 2/ 3 − α ,∞ . From 16 , under the assumption L1 on L and by Lemma 2.1, we know that Ã possesses a compact resolvent. Therefore, the spectrum σ Ã consists of only eigenvalues numbered in η1 ≤ η2 ≤ · · · → ∞ counted with multiplicity , and the corresponding system of eigenfunctions {en : n ∈ N} Ãen ηnen forms an orthogonal basis in L2. Let n− #{i | λi < 0}, n0 #{i | λi 0}, n n− n0, E− span{e1, . . . , en−}, E0 span{en− 1, . . . , en}, E span{en 1,...}, 2.2 where the closure is taken with respect to the norm ‖ · ‖0. Then, one has the orthogonal decomposition E E− ⊕ E0 ⊕ E with respect to the inner product ·, · 0. Now, we introduce on E the following inner product and norm: u, v E ( |Ã|u, ∣∣Ã ∣∣ 1/2 v )


Introduction
Consider the following second-order non-autonomous Hamiltonian system q − L t q ∇ q W t, q 0, 1.1 where L t ∈ C R, R N×N is a symmetric matrix-valued function, W : R × R N → R and ∇ q W t, q denotes the gradient of W t, q with respect to q. As usual, we say that a nonzero solution q t of 1.1 is homoclinic to 0 if q t → 0 andq t → 0 as |t| → ∞.
As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics and nuclear physics. While it is well known that homoclinic solutions play an important role in analyzing the chaos of Hamiltonian systems, if a system has the transversely intersected homoclinic solutions, then it must be chaotic. If it has the smoothly connected homoclinic solutions, then it cannot stand the perturbation, its perturbed system probably produces chaotic phenomena. For the chaos theory, the readers can refer to 1-3 and the references therein for more details. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from 0.
In the past years, the existence and multiplicity of homoclinic orbits for 1.1 have been extensively investigated in many papers via the variational methods. Most of them see 4-13 treated the case where L t and W t, u are either independent of t or periodic in t. In this kind of problem, the function L t plays an important role. If L t is neither a constant nor periodic, the problem is quite different from the ones just described, because of the lack of compactness of the Sobolev embedding. After the work of Rabinowitz and Tanaka 13 , many results see, e.g., 9,[14][15][16][17][18][19][20][21][22] were obtained for the case where L t is neither a constant nor periodic. Among them, except for 13,16,18,[20][21][22] , all known results were obtained under the following assumption that L t is positive definite for all t ∈ R, that is, In the present paper, we will study the existence and multiplicity of homoclinic orbits for 1.1 under the condition that L t is coercive but unnecessarily positive definite for all t ∈ R. More precisely, L satisfies the following conditions: where l t is the smallest eigenvalue of L t , that is, Before presenting the conditions on the nonlinearity of 1.1 , we note that in the recent paper 23 , under a twisting of the nonlinearity between the origin and the infinity, the authors studied the existence and multiplicity of nontrivial solutions for nonlinear elliptic equations and also for nonlinear elliptic systems. Subsequently, this kind of twist conditions and the idea of the methods in 23 were also applied to first-order Hamiltonian systems in 24 .
Inspired by these works, we will present some similar twist condition on the nonlinearity of 1.1 to those in 23, 24 , which will be specified in what follows.
Here, we introduce some notations. Denote by B the set of all uniformly bounded symmetric N × N matric functions. That is to say, B ∈ B if and only if B T t B t for all t ∈ R and B t is uniformly bounded in t as the operator on R N . For any B ∈ B, in the next section, we will define an index pair i B , ν B , satisfying 0 ≤ i B , ν B < ∞.
With this index, we can present the conditions on W t, q and the nonlinearity ∇ q W t, q as follows. For notational simplicity, we set B 0 t ∇ 2 q W t, 0 , and in what follows the letter c will be repeatedly used to denote various positive constants whose exact value is irrelevant. Besides, for two N × N symmetric matrices M 1 and M 2 , M 1 ≤ M 2 means that M 2 − M 1 is semipositive definite.
Our first result reads as follows. Condition W ∞ is a two-side pinching condition near the infinity, learning from the idea of 23, 24 , we can relax W ∞ to condition W ± ∞ as follows.
W ± ∞ There exist some R 0 > 0 and a continuous symmetric matrix function B ∞ ∈ B with ν B ∞ 0 such that The uniform boundary of ∇ 2 q W t, q displayed in condition W 1 can also be relaxed as W * 1 W ∈ C 2 R × R N , R , and there exists a constant c > 0 such that On the other hand, we need some sharply twisted conditions than the above theorem, and we have the following theorems.
still hold for W, so we can always assume W t, 0 ≡ 0.

Preliminaries
Denote by A the self-adjoint extension of the operator and | A| be the spectral resolution and the absolute value of A, respectively, and let | A| 1/2 be the square root of where I is the identity map on L 2 . Then, U commutes with A, | A|, and | A| 1/2 , and A U| A| is the polar decomposition of A. Let E D | A| 1/2 , and define on E the inner product and norm by where ·, · L 2 denotes the usual inner product on L 2 R, R N . Then, E is a Hilbert space. It is easy to see that E is continuously embedded in W 1 R, R N . In fact, we further have the following lemmas.
From 16 , under the assumption L 1 on L and by Lemma 2.1, we know that A possesses a compact resolvent. Therefore, the spectrum σ A consists of only eigenvalues numbered in η 1 ≤ η 2 ≤ · · · → ∞ counted with multiplicity , and the corresponding system of eigenfunctions {e n : n ∈ N} Ae n η n e n forms an orthogonal basis in L 2 . Let where the closure is taken with respect to the norm · 0 . Then, one has the orthogonal decomposition E E − ⊕ E 0 ⊕ E with respect to the inner product ·, · 0 . Now, we introduce on E the following inner product and norm: Abstract and Applied Analysis Clearly, norms · E and · 0 are equivalent cf. 16 . From now on, we take E with inner product ·, · E and norm · E as our working space.
Remark 2.2. Note that the decomposition E E − ⊕ E 0 ⊕ E with respect to the inner product ·, · 0 is also orthogonal with respect to both inner products ·, · E and ·, · L 2 . In what follows, we always denote by E E − ⊕ E 0 ⊕ E the orthogonal decomposition with respect to the inner products ·, · E unless specified otherwise.
In view of Lemma 2.1 and the equivalence of the norms · E and · 0 , there exists a constant c ∞ > 0 such that Define the quadratic form a on E by Then by definition, we have it is easy to check that K is a compact operator and a u, v Au, v E for all u, v in E. For any B ∈ B, it is easy to see B determines a bounded self-adjoint operator on L 2 , by z t → B t z t , for any z ∈ L 2 , we still denote this operator by B, then KB : E ⊂ L 2 → E is a self-adjoint compact operator on E and satisfies

2.10
Combining this with Lemma 2.1, we know that Ψ and Φ are both well defined. Furthermore, we have the following: First, we prove {u n } is bounded in E. For each ε ∈ 0, 1 , define C n ∈ B by

2.12
It is easy to verify that {C n } satisfies where c is the constant in condition W 1 and I is the identity map on R N . Since B 1 ≤ B 2 , i B 1 i B 2 , and ν B 1 ν B 2 0, we can choose ε small enough, such that for each n ∈ N , satisfying i C n i B 1 and ν C n 0. Thus A − KC n is reversible on E and there is a constant δ > 0, such that 2.14 Abstract and Applied Analysis 7 On the other hand, for b ∈ 0, 1 , there is a constant c > 0 depending on b, such that for each n ∈ N , ∇ q W t, u n t − C n u n t ≤ c|u n t | b , ∀t ∈ R.

2.16
As we claimed in the part of introduction, in 2. 15 for ε > 0 small enough; 3 there exist some c > 0 such that 4 for each m ∈ N, there exists some C m > 0 and a constant γ with γI n×n > B ∞ , ν γI n×n 0 such that where I n×n is the identity map on R N .

2.25
It is easy to see that η ∈ C 2 0, ∞ , R . Choose a sequence {R m } of positive numbers such that R 0 < R 1 < R 2 < · · · < R m < · · · → ∞ as m → ∞. For each m ∈ N, let η m t η t/R m and where ρ ∈ C 2 R, 0, 1 is a cut-off function with ρ s ≡ 1 for s ≤ 0 and ρ s ≡ 0 for s ≥ 1. If we choose T m large enough, then W m will satisfy 2.20 -2.24 .
Remark 2.6. Similar to Remark 1.4, we can choose ε > 0 small enough in 2.22 such that

Lemma 2.8. For each m ∈ N, Φ m satisfies the (PS) condition and the critical-point set
Proof. For any m ∈ N, assume {u n } ⊂ E with Φ m u n → 0 as n → ∞:

2.32
Since α < 2, ν γI n×n 0 and Ψ γ 0 has bounded inverse, from Lemma 2.1 and 2.30 , 2.32 , we have {u n } is bounded in E. From the similar argument in Lemma 2.4, {u n } has a convergent subsequence and the PS condition is verified. From the same reason, we can also prove that K m is compact set.
From Lemma II.5.1 in 25 , by standard argument, we have the following. Lemma 2.9. For any m ∈ N, there is an a m ∈ R with −a m large enough such that Note that θ is an isolated critical point of Φ m since ν B 0 0. For each m ∈ N, let K * m K m \ {θ}, then K * m is also compact since K m is compact. Then we have Lemma 2.10. For any , μ > 0 small enough, for each m ∈ N, there exists a functional Φ m such that Proof. We follow the idea of 26 , since K * m is a compact subset of E, for every μ > 0, there exists a C ∞ function l : E → 0, 1 , with all its derivatives bounded and l z 1, ∀z ∈ N μ K * m , l z 0, ∀z ∈ E \ N 2μ K * m .

2.34
Let We use the Sard-Smale Theorem to find y ∈ E, with y < min{ /C l 2 2M , δ/2 C l 1 2M }, and −y is a regular value for Φ m . For any z 0 ∈ N 2μ K * m , the functional is defined by By the fact y < /C l 2 2M and the definition of l z , it is easy to check that conclusions 1 , 2 , and 3 hold. Since y < δ/2C l 1 2M and −y is a regular value for Φ m , then all nontrivial critical points of Φ m are nondegenerate and lie in N μ K * m . In order to prove that Φ m satisfies the PS condition for each m ∈ N, assume there is when n is large enough. From the definition of Φ m and the proof in Lemma 2.8, we know that Φ m satisfies the PS condition and hence has a finite number of critical points.

Proof of the Main Results
From Lemma 2.4, Theorem 1.1 is a direct consequence of Theorem 5.1 and Corollary 5.2 in chapter II of 25 .
Step 2. We show that {z m } is bounded in L ∞ R, R N , so from the definition of Φ m , z m is a nontrivial critical point of Φ for m large enough.
We prove it indirectly, assume z m L ∞ → ∞, from 2.4 , we have z m E → ∞. Denote v m z m / z m E . Passing to a subsequence, we assume that for some Since z m satisfiesz m t − L t z m t ∇ q W m t, z m 0, we have  which implies m − z m ν z m ≤ i B ∞ . If 3.15 is false, then there exist m j → ∞ and u j ∈ E B ∞ with u j E 1 such that Φ m z m j u j , u j E ≤ 0, which can be rewritten as Passing to a subsequence if necessary, we assume that u j → u in L 2 and almost everywhere in If u 0, it is a contradiction, or we have which is also a contradiction. Thus, {z m } is bounded in L ∞ R, R N and Φ has a nontrivial critical point. By Proposition 2.3, 1.1 has a nontrivial homoclinic orbit. The proof is completed.
The proof of Theorem 1.3 is similar to the proof of Theorem 1.2. The difference is in Step 1, instead of Morse theory we make use of minimax arguments for multiplicity of critical points.
Let X be a Hilbert space and assume φ ∈ C 2 X, R is an even functional, satisfying the PS condition and φ 0 0. Denote S c {u ∈ X | u c}.
First, we consider the case of W ∞ , since W is even, we have W m is also even, and it satisfies Lemma 2.5. Let Y E − B ∞ , and Z E B 0 , and we have dim Y i B ∞ , codim Z i B 0 , dim Y > codim Z. Then, it is easy to prove that Φ m satisfies Lemma 3.1 for R and 1/r large enough. So, Φ m has l : i A B ∞ − i A B 0 pairs of nontrivial critical points: Then, we can complete the proof. In order to prove the case of W − ∞ , we need the following. The proof is similar to the case of W ∞ , we omit it here.