The Existence of Homoclinic Solutions for Second Order Hamiltonian System

The existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized from Poincaré[1]. In the past decade, this problem has been studied intensively by many mathematicians. Many authors have studied the existence of homoclinic solutions for Hamiltonian systems via critical point theory and variational methods, see[2]-[5] and references therein.

The purpose of this paper is to study the existence of homoclinic solutions for second order Hamiltonian system u - A( t) u + Vu (t, u) = 0.                                                                              (1.1)

Wher t e R , u e R ” , V_u ( t , u ) denotes the gradient of V(t , u ) with respect to u .

Definition 1.1 If the solution u(t) e C (R, R ) of system (1.1) satisfies boundary value condition u (±«) = u(±^) = 0.

* Corresponding author.

Then u ( t ) is called the homoclinic solution (to 0) of (1.1). If u ( t ) ^ 0, then u ( t ) is called the nontrivial homoclinic solution .

Definition 1.2 I ^e C ⁽ B ^{, R} ) is said to satisfiy the (PS) condition if any sequence { u j } j ^e N °^ for which { I ( U j )} j e N

Is bounded and I '( u _y) ^ 0 as j ^ +^ , possesses a convergent subquence in B .

To state the main result, we state the basic conditions on A ( t ) and V ( t , u ) :

• (I)    A e C ( R , R " ) is a symmetric and positive definite matrix for all t e R ; there exists a continuous function f : R ^ R such that

( A ( t ) u , u ) >  f ( t ) u |²,   V ( t , u ) e R x R " .

where f satisfies: f ( t ) >  0 for all t e R ,and f ( t ) ^ +» as | t | ^ +^.

From this condition, we see that there exists a constant b > 0 such that

( A ( t ) u , u ) >  b\u\ ², V ( t , u ) e R x R " .                                                                               (1.2)

• (II)    V ( t , u ) >  0 for all ( t , u ) e R x R" and there exists constants K >  0and M_x >  0 such that

V(t,u) < K|u|2, (t,u) e R x R", |u| < M 1, where 2K < b, with b is defined in (1.2).

• (III)    there exists function f : R ^ R ⁺ , constant c_x >  2 and M ₂ > 0 such that

V ( t , u ) >  f ( t )| u\c ¹, V ( t , u ) e R x R " , | u | > M ₂.

l - 2

• (IV)    there exists constants l >  2 and c ₂ with 0 <  c ₂<  ^  ,

such that

IV ( t , u ) - ( V ( t , u ), u ) <  c ₂ ( A ( t ) u , u ), V( t , u ) e R x R " .

• (V)    V u ( t , u ) = O (| u\ ) as | u | ^ 0 uniformly with respect to t e R .

• (VI)    there exists W e C ( R " , R ) such that

• 2.    Preliminary knowledges

I V ( t , u )| <  W ( u )|,   V ( t , u ) e R x R " .

We donote H ^{( R} , ^R ) and L ^{( R} , ^{R )(2}<  p <  ⁺”) as the Banach space of functions on ^R with value in ^{R n} under the norms

II " Li : = (I u 12 +1 u 2 - 2 ; HI , = (f, u ( ‘ ) ^pd- ) p

Respectively.

Let

E = { u e H 1 ( R , R n ): f [| u ( t )| 2 + ( A(t ) u ( t ), u ( t ))] dt <  +^ }, then E is a Hilbert space with the inner product

⁽ x ’ У ) = f„^{[( x(} t),^y( t)) ^{+ ( A (} t) ^{x (} t ), У ⁽ t ^{))] dt} ’

R and the corresponding norm is ||x||2 = (x, x). Note that for p e [2

, да), E c H * ( R , Rⁿ ) c L^p ( R , Rⁿ ) is continuous embedding.

In particular, for p = +да , there exists a constant C >  0 such that || u ||_m<  C||u ||, V u e E .

Lemma 2.1 [6] Suppose that A ( t ) satisfies condition (I), then E embedding into l ² ( R , r ⁿ ) is compact.

Lemma 2.2 [6] Suppose that conditions (I),(V),(VI) are satisfied . If u weakly convergence to u in E , then V ( t , u ) strongly convergence to V ( t , u )in L ² ( R , Rⁿ ).

Now we difine the functional I : E ^ R as follows:

r I .71

I ( u ) =[ [- «( t )   + -( A ( t ) u ( t ), u ( t )) - V ( t , u ( t ))] dt

R 2         2(2.1)

• = ^- H u | । ² — L V ( t , u ( t )) dt .

Lemma 2.3 [7] Suppose that conditions (I)-(VI) are satisfied, then for all u , v e E , we have

I'(u)v = f [(u-(t), V(t)) + (A(t)u(t),v(t)) - (V (t, u(t)), v(t))]dt.(2.2)

R

From this we obtain

I'(u)u = |u||2 - f (V (t, u(t)), u(t))dt.(2.3)

R

Moreover, I is a continuously Frechet-differentiable functional defined on E , namely, I e C * ( E , R ) and any critical point u of I on E is a classical solution of (1.1) with u ( ±» ) = 0 = z-( ±» ).

Lemma 2.4 Suppose that conditions (I),(IV),(V),(VI) are satisfied, then I satisfies the (PS) condition.

Proof Assume that  {uy} eN c E is a sequence such that {I(u7)} еУ is bounded and I'(u,) ^ 0 as j ^ +да. Then there has a constant M > 0 such thatt

• 1 1 ( u j )| <  M ,  || I ' ( u j )||_£* <  M ,   j e ^                                                                     (2.4)

By (2.1),(2.3) and condition (IV), we have

( ^l - 1) u_j ² = lI ( u_j ) - I ′ ( u_j ) u_j

+   (lV(t,uj(t))-(Vu(t,uj(t)),uj(t)))dt

R

• ≤ lI ( u ) - I ′ ( u ) u + c ∫ ( A ( t ) u ( t ), u ( t )) dt                             (2.5)

Define

2 q(u) = £ l(2 )u(t) + ( 2 - c2)(A(t)u(t),u(t))]dt, then we have l1 u 2 ≤ q(u) ≤ l2 u 2,(2.6)

l-2

where l =      - c ,l =      . Thus, by (2.4),(2.5) and (2.6), we obtain that l1 uj 2 ≤q(uj)≤lI(uj)-I′(uj)uj ≤lM+M uj .(2.7)

Since l > 0, then (2.7) shows that {u } is bounded in E. By lemma 2.1, the sequence {u } has a subsequence, again denoted by {u }   , and there exists u ∈ E such that u weakly convergence to u in E , u strongly convergence to u in L2 (R,R n ).

Thus

(I′(u ) -I′(u))(u -u) → 0.

By lemma 2.2 and Hölder inequality ,we have

• ∫ ( V_u ( t , u_j ( t )) - V_u ( t , u ( t )), u_j ( t ) - u ( t )) dt → 0.

if j → +∞. On the other hand, an easy computation shows that

(I′(u ) -I′(u),u -u)

• = uj -u 2 -∫ (Vu(t, uj(t)) -Vu(t, u(t)), uj(t) -u(t))dt.

Hence, u_j - u → 0 as j → +∞. Namely, I satisfies the (PS) condition.

Lemma 2.5 [8] Let B is a real Banach space and I ∈ C ¹( B , R ) satisfying the (PS) condition, suppose that I (0) = 0 and

• (H 1 ) there exists constants ρ , a > 0 such that I ≥ a ;

• (H 2 ) there exists e ∈ B \ B ρ such that I ( e ) ≤ 0.

Then I possesses a critical value c ≥ a >  0 given by

Where

c = inf max I(g(s)).

g ∈Γ s ∈ [0,1]

Γ = { g ∈ C ([0,1], B ): g (0) = 0, g (1) = e }.

3. Main Result

Theorem 3.1 Suppose that conditions (I)-(VI) are satisfied, then the Hamiltonian system (1.1) possesses at least one nontrivial homoclinic solution.

Proof By condition (II), we know I (0) = 0. On the other hand, by lemma 2.3 and 2.4, we know I ∈ C ¹ ( E , R ) satisfies the (PS) condition.

Assume that u ∈ E , 0 < II u II ≤ M , then by (1.2)and condition (II), we have

∫ V(t, u(t))dt ≤ K∫ Iu(t)I2dt≤KIIuI

RR

II²≤ ^K ■II ^u II 2 _.

Combining (2.1), we have

I(u)≥1■IIu II 2-K ■IIu II  = 2 (1 - b ) IIu II 2.

(2.8)

Condition (II) implies 1 -    >  0. Let

M         ( b - 2 K ) M ²

ρ =    ¹ >  0; a =            ¹ >  0.

C           2 bC ²

(2.9)

Then there exists ρ > 0 and a >  0 such that I I ≥ a (by (2.8). Namely, I satisfies the condition (H 1 ) of lemma 2.5.

By (2.1), for every m ∈ R \ {0} and u ∈ E \ {0}, we have

I ( mu ) = ^{m 2} u II² - V ( t , mu ( t )) dt . 2 R

Take some U ∈ E such that I U II = 1. Then there exists a subset Ω of positive measure of R shuch that U ( t ) ≠ 0 for t ∈ Ω . . Let m >  0 such that mU ( t ) I ≥ M for t ∈ Ω . .Then by conditions (II) and (III), we have

m ²

I ( mU ) ≤     - m^{c 1} ∫ f ₀( t ) U ( t )I ^{c 1} dt .

(2.10)

Since f (t) > 0, c > 2, then (2.10) implies that I(mU) < 0 for some m > 0 such that mU(t) I ≥ M for t ∈ Ω and IImU II > ρ , where ρ is defined in (2.9). Namely, I satisfies the condition (H2) of lemma 2.5. Then by lemma 2.5, I possesses a critical value c ≥ a > 0given by c = inf max I(g(s)). g∈Γ s∈[0,1]

Where

Г = { g е C ([0,1], E ): g (0) = 0, g (1) = mU }.

Hence, there exists u е E such that

I ( u ) = c , I '( u ) = 0.

By lemma 2.3, hamiltonian system (1.1) possesses at least one nontrivial homoclinic solution. So we finish the proof of this theorem.