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New periodic solutions for a class of singular Hamiltonian systems

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Abstract

We use the variational minimizing method to study the existence of new nontrivial periodic solutions with a prescribed energy for second order Hamiltonian systems with singular potential V C 1 ( R n {0},R), which may have an unbounded potential well.

MSC:34C15, 34C25, 58F.

1 Introduction and main results

For singular Hamiltonian systems with a fixed energy hR,

q ¨ + V (q)=0,
(1.1)
1 2 | q ˙ | 2 +V(q)=h.
(1.2)

Ambrosetti-Coti Zelati [1, 2] used Ljusternik-Schnirelmann theory on an C 1 manifold to get the following theorem.

Theorem 1.1 (Ambrosetti-Coti Zelati [1])

Suppose V C 2 ( R n {0},R) satisfies

(A0)

V(u),u0,

(A1)

3 V (u)u+ ( V ( u ) u , u ) 0,

(A2)

V (u)u>0,u0,

(A3) α>2, s.t.

V (u)uαV(u),

(A4) β>2, r>0, s.t.

V (u)uβV(u),0<|u|<r,

(A5)

lim sup | u | + [ V ( u ) + 1 2 V ( u ) u ] 0.

Then (1.1)-(1.2) have at least one non-constant periodic solution.

After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems. Here we only mention a related recent paper of Carminati-Sere-Tanaka [3], in which they used complex variational and geometrical and topological methods to generalize Pisani’s results [5]. They got the following theorems.

Theorem 1.2 Suppose h>0, L 0 >0 and V C ( R n {0},R) satisfies (A0), (A4), and

(B1) V(q)0;

(B2) V(q)+ 1 2 V (q)qh, |q| e L 0 ;

(B3) V(q)+ 1 2 V (q)qh, |q| e L 0 .

Then (1.1)-(1.2) have at least one periodic solution with the given energy h and whose action is at most 2π r 0 with

r 0 =max { [ 2 ( h V ( q ) ) ] 1 2 ; | q | = 1 } .

Theorem 1.3 Suppose h>0, ρ 0 >0 and V C ( R n {0},R) satisfies (B1), (A4), and

(B2′) lim | q | + V (q)=0;

(B3′) V(q)+ 1 2 V (q)qh, |q| ρ 0 .

Then (1.1)-(1.2) have at least one periodic solution with the given energy h and whose action is at most 2π r 0 .

Using variational minimizing methods, we get the following theorem.

Theorem 1.4 Suppose V C 1 ( R n {0},R) satisfies

(V1) α>0, β>2, r>0, s.t.

V(q)α | q | β ,0<|q|<r;

(V2)

V(q)<0,q0;

(V3)

V(q)=V(q),q0.

Then for any h>0, (1.1)-(1.2) have at least one non-constant periodic solution with the given energy h.

2 A few lemmas

Let

H 1 = W 1 , 2 ( R / Z , R n ) = { u : R R n , u L 2 , u ˙ L 2 , u ( t + 1 ) = u ( t ) } .

Then the standard H 1 norm is equivalent to

u= u H 1 = ( 0 1 | u ˙ | 2 d t ) 1 / 2 +| 0 1 u(t)dt|.

Let

Λ= { u H 1 | u ( t ) 0 , t } .

By symmetry condition (V3), similar to Ambrosetti-Coti Zelati [1], let

Λ 0 = { u H 1 = W 1 , 2 ( R / Z , R n ) , u ( t + 1 / 2 ) = u ( t ) , u ( t ) 0 } .

We define the equivalent norm in E={u H 1 = W 1 , 2 (R/Z, R n ),u(t+ 1 2 )=u(t)}:

u= u E = ( 0 1 | u ˙ | 2 d t ) 1 / 2 .

Lemma 2.1 ([1, 4])

Let f(u)= 1 2 0 1 | u ˙ | 2 dt 0 1 (hV(u))dt and u ˜ Λ be such that f ( u ˜ )=0 and f( u ˜ )>0. Set

1 T 2 = 0 1 ( h V ( u ˜ ) ) d t 1 2 0 1 | u ˜ ˙ | 2 d t .
(2.1)

Then q ˜ (t)= u ˜ (t/T) is a non-constant T-periodic solution for (1.1)-(1.2). Furthermore, if V(x)<h, x0, then f(u)0 on Λ and f(u)=0, uΛ if and only if u is a nonzero constant.

If u ˜ Λ 0 such that f ( u ˜ )=0 and f( u ˜ )>0, then we find that q ˜ (t)= u ˜ (t/T) is a non-constant T-periodic solution for (1.1)-(1.2).

Lemma 2.2 (Gordon [6])

Let V satisfy the so-called Gordon Strong Force condition: There exist a neighborhood of 0 and a function U C 1 (Ω,R) such that:

  1. (i)

    lim s 0 U(x)=;

  2. (ii)

    V(x) | U ( x ) | 2 for every xN{0}.

Let

Λ= { u H 1 = W 1 , 2 ( R / Z , R n ) , t 0 , u ( t 0 ) = 0 } .

Then we have

0 1 V(u)dt, u n uΛ.

Let

Λ 0 = { u H 1 = W 1 , 2 ( R / Z , R n ) , u ( t + 1 2 ) = u ( t ) , t 0 , u ( t 0 ) = 0 } .

Then we have

0 1 V(u)dt, u n u Λ 0 .

Lemma 2.3 Let X be a Banach space, and let EX be a weakly closed subset. Suppose that ϕ(u) is defined on an open subset ΛX and ϕ(u) for any uΛ. Let ϕ(u)=+ for uΛ. Assume ϕ(u)+ and is weakly lower semi-continuous on Λ ¯ E, and that it is coercive on ΛE:

ϕ(u)+,u+

and

ϕ( u n )+, u n uΛ.

Then ϕ attains its infimum in ΛE.

Proof We set

c= inf Λ E ϕ(u).

Then

<c<+,

in fact, by the assumptions, it is obvious that c<+. Now if c=, then there exists { u n }ΛE such that ϕ( u n ). Then we know that { u n } is bounded, since ϕ is coercive. By the Eberlein-Schmulyan theorem, { u n } has a weakly convergent subsequence. Finally, by the definition for c and the assumption for the weakly lower semi-continuity for ϕ(u), we know ϕ(u)=. This is a contradiction.

Now we know that there exists minimizing sequence { u n } such that ϕ( u n )c. Furthermore by the coercivity of ϕ we know that { u n } is bounded; then { u n } has a weakly convergent subsequence. We claim the weak limit uΛ, since otherwise ϕ(u)=+ by the assumption. On the other hand, by the definition of the infimum c and the assumption for the weak lower semi-continuity for ϕ(u) on Λ ¯ E, we know ϕ(u)=c<+. This is a contradiction. So the weak limit uΛE and ϕ(u)=c. □

3 The proof of Theorem 1.4

Lemma 3.1 Assume (V1) hold, then for any weakly convergent sequence u n u Λ 0 , we have

f( u n )+.

Proof Notice that (V1) imply Gordon’s strong force condition. By the weak limit uΛ and V satisfying Gordon’s strong force condition, we have

0 1 V( u n )dt+, u n uΛ.

By u n u in the Hilbert space H 1 , we know that u n is bounded.

  1. (1)

    If u0, then by Sobolev’s embedding theorem, we have the uniform convergence property:

    | u n | 0,n+.

By the symmetry of u(t+1/2)=u(t), we have 0 1 u(t)dt=0, then we have Sobolev’s inequality:

0 1 | u ˙ (t) | 2 dt12|u(t) | 2 .

Then we have

f( u n )6 | u n | 2 β +,n+.

So in this case we have

lim inff( u n )=+f(u).
  1. (2)

    If u0, then we have the following. By the weakly lower semi-continuity for the norm, we have

    lim inf u n u>0.

So, by Gordon’s lemma, we have

lim inf f ( u n ) = lim inf ( 1 2 0 1 | u ˙ n | 2 d t ) 0 1 ( h V ( u n ) ) d t = + 1 2 0 1 | u ˙ | 2 d t 0 1 ( h V ( u ) ) d t = f ( u ) .

 □

Lemma 3.2 f(u) is weakly lower semi-continuous on Λ ¯ 0 .

Proof For any { u n } Λ ¯ 0 : u n u, by Sobolev’s embedding theorem, we have uniform convergence:

| u n (t)u(t) | 0.
  1. (i)

    If u Λ 0 , then by V C 1 ( R n {0},R), we have

    |V ( u n ( t ) ) V ( u ( t ) ) | 0.

By the weakly lower semi-continuity for norm, we have

lim inf u n u.

Hence

lim inf f ( u n ) = lim inf ( 1 2 0 1 | u ˙ n | 2 d t ) 0 1 ( h V ( u n ) ) d t 1 2 0 1 | u ˙ | 2 d t 0 1 ( h V ( u ) ) d t = f ( u ) .
  1. (ii)

    If u Λ 0 , then by Λ satisfying Gordon’s strong force condition, we have

    0 1 V( u n )dt+, u n u Λ 0 .
  2. (1)

    If u0, then

    | u n | 0,n+.

Then we have

f( u n )6 | u n | 2 β +,n+.

So in this case we have

lim inff( u n )=+f(u).
  1. (2)

    If u0. By the weakly lower semi-continuity for norm, we have

    lim inf u n u>0.

So by Gordon’s lemma, we have

lim inf f ( u n ) = lim inf ( 1 2 0 1 | u ˙ n | 2 d t ) 0 1 ( h V ( u n ) ) d t = + 1 2 0 1 | u ˙ | 2 d t 0 1 ( h V ( u ) ) d t = f ( u ) .

 □

Lemma 3.3 Λ ¯ 0 is a weakly closed subset of H 1 .

Proof By Sobolev’s embedding theorems, the proof is obvious. □

Lemma 3.4 The functional f(u) is coercive on Λ 0 .

Proof By the definition of f(u) and the assumption (V2), we have

f(u)= 1 2 0 1 | u ˙ | 2 dt 0 1 ( h V ( u ) ) dt h 2 0 1 | u ˙ | 2 dt,u Λ 0 .

 □

Lemma 3.5 The functional f(u) attains the infimum on Λ 0 ; furthermore, the minimizer is non-constant.

Proof By Lemma 2.2 and Lemmas 3.1-3.3, we know that the functional f(u) attains the infimum in Λ 0 ; furthermore, we claim that

inf Λ 0 f(u)>0,

since otherwise, u 0 (t)=const attains the infimum 0, then by the symmetry of Λ 0 , we have u 0 (t)0, which contradicts the definition of Λ 0 . Now we know that the minimizer is non-constant. □

References

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    Ambrosetti A, Coti Zelati V: Closed orbits of fixed energy for singular Hamiltonian systems. Arch. Ration. Mech. Anal. 1990, 112: 339-362. 10.1007/BF02384078

  2. 2.

    Ambrosetti A, Coti Zelati V: Periodic Solutions for Singular Lagrangian Systems. Springer, Berlin; 1993.

  3. 3.

    Carminati C, Sere E, Tanaka K: The fixed energy problem for a class of nonconvex singular Hamiltonian systems. J. Differ. Equ. 2006, 230: 362-377. 10.1016/j.jde.2006.01.021

  4. 4.

    Tanaka K: Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2000, 17: 1-33. 10.1016/S0294-1449(99)00102-X

  5. 5.

    Pisani L: Periodic solutions with prescribed energy for singular conservative systems involving strong forces. Nonlinear Anal. TMA 1993, 21: 167-179. 10.1016/0362-546X(93)90107-4

  6. 6.

    Gordon WB: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 1975, 204: 113-135.

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Acknowledgements

This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA172) and the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010).

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Correspondence to Xiong-rui Wang.

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The authors declare that they have no competing interests.

Authors’ contributions

XW proved the main theorem, SS participated in the proof and helped to draft the manuscript. Both authors read and approved the final manuscript.

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Keywords

  • singular Hamiltonian systems
  • periodic solutions
  • variational methods