Skip to main content

New periodic solutions of singular Hamiltonian systems with fixed energies

Abstract

By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second-order Hamiltonian systems with a singular potential V C 2 ( R n O,R) and V C 1 ( R 2 O,R), which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as complements of the well-known theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.

MSC:35A15, 47J30.

1 Introduction

Seifert [1] in 1948 and Rabinowitz [2, 3] in 1978 and 1979 studied classical second-order Hamiltonian systems without singularity, based on their work, Benci [4, 5] and Gluck and Ziller [6] and Hayashi [7] used a Jacobi metric and very complicated geodesic methods and algebraic topology to study the periodic solutions with a fixed energy of the following system:

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

They proved a very general theorem.

Theorem 1.1 Suppose V C 2 ( R n ,R), if

{ x R n | V ( x ) h }

is bounded and non-empty, then (1.1)-(1.2) has a periodic solution with energy h.

Furthermore, if

V (x)O,x { x R n | V ( x ) = h } ,

then (1.1)-(1.2) has a nonconstant periodic solution with energy h.

For the existence of multiple periodic solutions for (1.1)-(1.2) with compact energy surfaces, we can refer to Groessen [8] and Long [9] and the references therein.

In the 1987 paper of Ambrosetti and Coti Zelati [10], Clark-Ekeland’s dual action principle, Ambrosetti-Rabinowitz’s mountain pass theorem etc. were used to study the existence of T-periodic solutions of the second-order equation

x ¨ =U(x),

where

U=V C 2 (Ω;R)

is such that

U(x),xΓ=Ω;

here Ω R n is a bounded and convex domain, and they got the following result.

Theorem 1.2 Suppose that

  1. (i)

    U(O)=0=minU;

  2. (ii)

    U(x)θ(x,U(x)) for some θ(0, 1 2 ) and for all x near Γ (superquadraticity near  Γ);

  3. (iii)

    ( U (x)y,y)k | y | 2 for some k>0 and for all (x,y)Ω× R N .

Let ω N be the greatest eigenvalue of U (0) and T 0 = ( 2 / ω N ) 1 / 2 . Then x ¨ =U(x) has for each T(0, T 0 ) a periodic solution with minimal period T.

For C r systems, a natural interesting problem is if

{ x R n | V ( x ) h }

is unbounded: can we get a nonconstant periodic solution for the system (1.1)-(1.2)?

In 1987, Offin [11] firstly generalized Theorem 1.1 to some non-compact cases under V C 3 ( R n ,R) and complicated geometrical assumptions on potential wells, but it seems to be difficult to verify this for concrete potentials under the geometrical conditions.

In 1988, Rabinowitz [12] studied multiple periodic solutions for classical Hamiltonian systems with potential V C 1 (R× R n ,R), where V( q 1 ,, q n ;t) is T i -periodic in positions q i R and is T-periodic in t.

In 1990, using Clark-Ekeland’s dual variational principle and Ambrosetti-Rabinowitz’s mountain pass lemma, Coti Zelati et al. [13] studied Hamiltonian systems with convex potential wells, they got the following result.

Theorem 1.3 Let Ω be a convex open subset of R n containing the origin O. Let V C 2 (Ω,R) be such that

(V1) V(q)V(O)=0, qΩ.

(V2) qO, V (q)>0.

(V3) ω>0, s.t. V(q) ω 2 q 2 , q<ϵ.

(V4) V ( q ) 1 0, q0, or

(V4)′ V ( q ) 1 0, qΩ.

Then, for every T< 2 π ω , (1.1) has a solution with minimal period T.

In Theorems 1.2 and 1.3, the authors assumed the convex conditions for potentials and potential wells so that they can apply Clark-Ekeland’s dual variational principle; we notice that Theorems 1.1-1.3 essentially made the following assumption:

V(x),xΓ=Ω.

So all the potential wells are bounded.

For singular Hamiltonian systems with a fixed energy hR, Ambrosetti and Coti Zelati in [14, 15] used Ljusternik-Schnirelmann theory on a C 1 manifold to get the following theorem.

Theorem 1.4 (Ambrosetti and Coti Zelati [14])

Suppose V C 2 ( R n {O},R) satisfies V(q), q0 and

  • (A1) 3 V (u)u+( V (u)u,u)0, u0;

  • (A2) V (u)u>0, u0;

  • (A3) α>2, s.t. V (u)uαV(u), u0;

  • (A4) β>2, r>0, s.t. V (u)uβV(u), 0<|u|<r;

  • (A5) V(u)+ 1 2 V (u)u0, u0.

Then (1.1)-(1.2) has at least one nonconstant periodic solution.

Besides Ambrosetti-Coti Zelati, many other mathematicians [1634] studied singular Hamiltonian systems, here we only mention a related recent paper of Carminati, Sere and Tanaka [16]. They used complex variational and topological methods to generalize Pisani’s results [17], and they got the following theorem.

Theorem 1.5 Suppose h>0, L 0 >0 and V C ( R n {O},R) satisfies V(q), q0 and

  • (B1) V(q)0, q0;

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

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

  • (A4) β>2, r>0, s.t. V (q)qβV(q), 0<|q|<r.

Then (1.1)-(1.2) has 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 1.6 Suppose h>0, ρ 0 >0, and V C ( R n {O},R) satisfies V(q), q0 and (B1), (A4) and

(B2)′ lim | q | + V (q)=O;

(B3)′ V(q)+ 1 2 V (q)qh, |q| ρ 0 .

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

By using the variational minimizing method with a special constraint, we obtain the following result.

Theorem 1.7 Suppose V C 2 ( R n {O},R) and V(q), q0 and satisfies (A1)-(A3) and

(A4)′ β>2, s.t. V (q)qβV(q), 0<|q|<+;

(A5)′ V(q)=V(q), qO.

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

Using the direct variational minimizing method, we get the following theorem.

Theorem 1.8 Suppose V C 1 ( R 2 {O},R) and V(q), q0 and satisfies

(B1)′ V(q)<h, qO;

(P1)′ V (u)O, u+;

(A3)′ α>2, μ 2 >0, s.t. V (u)uαV(u)+ μ 2 , u0;

(A4) β>2, r>0, s.t. V (u)uβV(u), 0<|u|<r.

Then for any h> μ 2 α , (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.

Corollary 1.9 Suppose α=β>2 and

V(x)= | x | α .

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

Remark In Theorem 1.8, the assumption on regularity for potential V is weaker than Theorems 1.1-1.6. Comparing Theorem 1.5 with Theorem 1.8, our (B1)′ is also weaker than (B1), and (A3)′ is also different from (B2)-(B3) and (B3)′.

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 + | u ( 0 ) | .

Let

Λ= { u H 1 | u ( t ) O , t } .

Lemma 2.1 ([14])

Let

F= { u H 1 | 0 1 ( V ( u ) + 1 2 V ( u ) u ) d t = h } .

If (A1) holds, then F is a C 1 manifold with codimension 1 in H 1 . Let

f(u)= 1 4 0 1 | u ˙ | 2 dt 0 1 V (u)udt

and let u ˜ F be such that f ( u ˜ )=O and f( u ˜ )>0. Set

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

If (A2) holds, then q ˜ (t)= u ˜ (t/T) is a nonconstant T-periodic solution for (1.1)-(1.2). Moreover, if (A2) holds, then f(u)0 on F and f(u)=0, uF if and only if u is constant.

Lemma 2.2 ([8, 14])

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

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

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

Lemma 2.3 (Sobolev-Rellich-Kondrachov [35, 36])

W 1 , 2 ( R / Z , R n ) C ( R / Z , R n )

and the imbedding is compact.

Lemma 2.4 ([35, 36])

Let q W 1 , 2 (R/TZ, R n ).

  1. (1)

    If q(0)=q(T)=O, then we have the Friedrics-Poincaré inequality:

    0 T | q ˙ ( t ) | 2 dt ( π T ) 2 0 T | q ( t ) | 2 dt.
  2. (2)

    If 0 T q(t)dt=0, then we have Wirtinger’s inequality:

    0 T | q ˙ ( t ) | 2 dt ( 2 π T ) 2 0 T | q ( t ) | 2 dt

    and Sobolev’s inequality:

    0 T | q ˙ ( t ) | 2 dt 12 T | q ( t ) | 2 .

Lemma 2.5 (Eberlein-Shmulyan [37])

A Banach space X is reflexive if and only if any bounded sequence in X has a weakly convergent subsequence.

Definition 2.6 (Tonelli [35])

Let X be a Banach space, f:XR.

  1. (i)

    If for any { x n }X strongly converges to x 0 : x n x 0 , we have

    lim inff( x n )f( x 0 ),

    then we call f(x) lower semi-continuous at x 0 .

  2. (ii)

    If for any { x n }X weakly converges to x 0 : x n x 0 , we have

    lim inff( x n )f( x 0 ),

    then we call f(x) weakly lower semi-continuous at x 0 .

Using the famous Ekeland variational principle, Ekeland proved the following.

Lemma 2.7 (Ekeland [38])

Let X be a Banach space, FX be a closed (weakly closed) subset, let δ( x 1 , x 2 ) be the geodesic distance between two points x 1 and x 2 in X, δ(x,F) be the geodesic distance between x and the set F. Suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous (or weakly lower semi-continuous) and assume Φ | F restricted on F is bounded from below. Then there is a sequence { x n }F such that

δ ( x n , F ) 0 , Φ ( x n ) inf F Φ , ( 1 + x n ) Φ | F ( x n ) 0 .

Definition 2.8 ([38, 39])

Let X be a Banach space, FX be a closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence { x n }F is such that

δ ( x n , F ) 0 , Φ ( x n ) c , ( 1 + x n ) Φ | F ( x n ) 0 ,

then { x n } has a strongly convergent subsequence.

Then we say that f satisfies the ( C P S ) c , F condition at the level c for the closed subset FX.

We notice that if F=X, then the above condition is the classical Cerami-Palais-Smale condition [40].

We can give a weaker condition than the ( C P S ) c , F condition.

Definition 2.9 Let X be a Banach space, FX be a weakly closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence { x n }F such that

δ ( x n , F ) 0 , Φ ( x n ) c , Φ | F ( x n ) 0 ,

then { x n } has a weakly convergent subsequence.

Then we say that f satisfies the ( W C P S ) c , F condition.

Lemma 2.10 (Gordon [18])

Let V satisfy the so-called Gordon strong force condition:

There exists a neighborhood N of O 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{O}.

Let

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

Then we have

0 1 V(u)dt, u n uΛ.

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.

By Lemmas 2.7 and 2.10, it is easy to prove the following.

Lemma 2.11 Let X be a Banach space, let FX be a weakly closed subset. Suppose that Φ defined on F is Gateaux-differentiable and weakly lower semi-continuous and bounded from below on F. If Φ satisfies the ( C P S ) inf Φ , F condition or the ( W C P S ) inf Φ , F condition, and suppose that

Φ( u n )+, u n uΛ,

then Φ attains its infimum on F.

The next lemma is a variant on the classical Tonelli’s theorem, whose proof is easy, so we omit its proof.

Lemma 2.12 Let X be a Banach space, FX be a weakly closed subset. Suppose that ϕ(u) is defined on an open subset ΛX and is Gateaux-differentiable on Λ and weakly lower semi-continuous and bounded from below on ΛF, if ϕ is coercive, that is, ϕ(x)+ as x+, and suppose that

ϕ( u n )+, u n uΛ,

then ϕ attains its infimum on ΛF.

3 The proof of Theorem 1.7

By the symmetrical condition (A5)′, it is easy to prove that the critical point of the functional f on Λ 0 is also the critical point of the functional f on Λ.

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 } .

Lemma 3.1 Assume (A4)′ holds, then for any weakly convergent sequence u n u Λ 0 , we have

f( u n )+.

Proof Similar to the proof of Zhang [19]. □

Lemma 3.2 FΛ is a weakly closed subset in H 1 .

Proof Let { u n }FΛ be a weakly convergent sequence, we use the embedding theorem to find which uniformly converges to u H 1 .

Now we claim uΛ, and then it is obvious that uF. In fact, if uΛ, by V(q), q0 and the condition (A4)′ we have

V(u) C 1 | u | β ,0<|u|< r <r.

So V(u) satisfies Gordon’s strong force condition, and by his lemma, we have

0 1 V( u n )dt+, u n uΛ.

The condition (A4)′ implies

V( u n )+ 1 2 V ( u n ) , u n ( 1 β 2 ) V( u n ).

Hence

h= 0 1 [ V ( u n ) + 1 2 V ( u n ) , u n ] dt+.

This is a contradiction. □

Lemma 3.3 f(u) is weakly lower semi-continuous on F Λ 0

Proof For any { u n }F: u n u, then by Sobolev’s embedding theorem and functional analysis, 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.

    It’s well known that the norm is weakly lower semi-continuous, 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 ) .
  2. (ii)

    If u Λ 0 , then by our assumption on V which satisfies Gordon’s strong force condition, we have

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

      (1) If u0, then

      | u n | 0,n+.

      Then similar to the proof in [19], we have

      f( u n )6 | u n | 2 β +,n+.

      So in this case we have

      lim inff( u n )=+f(u).
      lim inf u n u>0.
    2. (2)

      (2) If u0, then by the weakly lower semi-continuity for norm, we have

      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 3.4 The functional f(u) has a positive lower bound on F.

Proof By the definitions of f(u) and F and the assumption (A2), we have

f(u)= 1 4 0 1 | u ˙ | 2 dt 0 1 ( V ( u ) u ) dt0,uF.

 □

By the definitions of the functional f(u) and its domain Λ 0 , and the conditions on the energy h>0 and the potential V(u)<0, it is easy to prove the following lemma.

Lemma 3.5 The functional f(u) is coercive.

Furthermore, we claim that

c= inf F 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)o, which contradicts the definition of Λ 0 . Now by Lemmas 3.1-3.4 and Lemmas 2.11 and 2.12, we know f(u) attains the infimum on F, furthermore we know that the minimizer is nonconstant.

4 The proof of Theorem 1.8

In order to prove the Cerami-Palais-Smale type condition and get a nonconstant periodic solution in non-symmetrical case, we need to add a topological condition, we know that there are winding numbers (degrees) in the planar case, so we define

Λ 1 = { u Λ , deg ( u ) 0 } .

Lemma 4.1 If u n u Λ 1 , then f( u n )+.

Proof By V satisfying Gordon’s strong force condition, we have

0 1 V( u n )dt+, u n u Λ 1 .
  1. (1)

    If u0, then by Sobolev’s embedding theorem, we have

    | u n | 0,n+.

    Then by deg( u n )0, we have c>0 such that

    c | u n | u ˙ n L 2

    and u ˙ n L 2 is an equivalent norm of W 1 , 2 and

    f( u n )c | u n | 2 β +,n+.

    So in this case, we have

    lim inff( u n )=+f(u).
  2. (2)

    If u0, then 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 4.2 Under the assumptions of Theorem  1.8,

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

satisfies the ( C P S ) + condition on Λ 1 , that is, if { u n } Λ 1 satisfies

f( u n )c>0, ( 1 + u n ) f ( u n )O,
(4.1)

then { u n } has a strongly convergent subsequence in Λ 1 .

Proof Since f ( u n ) makes sense, we know

{ u n } Λ 1 .

We claim 0 1 | u ˙ n | 2 dt is bounded. In fact, by f( u n )c, we have

1 2 u ˙ n L 2 2 0 1 V( u n )dtc h 2 u ˙ n L 2 2 .
(4.2)

By (A3)′ we have

f ( u n ) , u n = u ˙ n L 2 2 0 1 ( h V ( u n ) 1 2 V ( u n ) , u n ) d t u ˙ n L 2 2 0 1 [ h μ 2 2 ( 1 α 2 ) V ( u n ) ] d t .
(4.3)

By (4.2) and (4.3) we have

f ( u n ) , u n ( h μ 2 2 ) u ˙ n L 2 2 + ( 1 α 2 ) ( 2 c h u ˙ n L 2 2 ) = ( α 2 h μ 2 2 ) u ˙ n L 2 2 + C 1 ,
(4.4)

where C 1 =2(1 α 2 )c, α>2, h> μ 2 α . So u ˙ n 2 C 2 .

Then we claim | u n (0)| is bounded.

We notice that

f ( u n ) ( u n u n ( 0 ) ) = 0 1 | u ˙ n | 2 d t 0 1 ( h V ( u n ) ) d t 1 2 0 1 | u ˙ n | 2 d t 0 1 V ( u n ) , u n u n ( 0 ) d t = 2 f ( u n ) 1 2 0 1 | u ˙ n | 2 0 1 V ( u n ) , u n u n ( 0 ) d t .
(4.5)

If | u n (0)| is unbounded, then there is a subsequence, still denoted by u n s.t. | u n (0)|+. Since

u ˙ n M 1 ,

we have

min 0 t 1 | u n ( t ) | | u n ( 0 ) | u ˙ n 2 +,as n+.
(4.6)

By Friedrics-Poincaré’s inequality and the condition (P1), we have

0 1 | u ˙ n ( t ) | 2 dt π 2 0 1 | u n ( t ) u n ( 0 ) | 2 dt,
(4.7)
0 1 V ( u n ) ( u n u n ( 0 ) ) dt0,
(4.8)
f ( u n ) ( u n u n ( 0 ) ) 0.
(4.9)

So f( u n )0, which contradicts f( u n )c>0, hence u n (0) is bounded, and u n = u ˙ n L 2 +| u n (0)| is bounded. Furthermore, similar to the proof of Ambrosetti and Coti Zelati [15], u n strongly converges to uΛ. □

It is easy to prove the following.

Lemma 4.3 Under the assumption (B1)′, f(u)0 on Λ, that is, f has a lower bound.

Lemma 4.4 Under the assumptions of Theorem  1.8, f(u) is weakly lower semi-continuous on the closure Λ ¯ of Λ.

Now we can prove our Theorem 1.8, in fact, by Lemma 4.1, we know that the infimum of f on Λ 1 is equal to the infimum of f on the closure of Λ 1 . Furthermore, we can prove the infimum of f on Λ 1 is greater than zero, otherwise if it is zero, the corresponding minimizer must be constant, then the winding number is zero, which is a contradiction. Now by the above lemmas, especially Lemma 2.11, we know that f attains the positive infimum on Λ 1 and the corresponding minimizer must be nonconstant.

References

  1. Seifert H: Periodische bewegungen mechanischer systeme. Math. Z. 1948, 51: 197-216. 10.1007/BF01291002

    Article  MathSciNet  MATH  Google Scholar 

  2. Rabinowitz PH: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 1978, 31: 157-184. 10.1002/cpa.3160310203

    Article  MathSciNet  Google Scholar 

  3. Rabinowitz PH: Periodic solutions of a Hamiltonian systems on a prescribed energy surface. J. Differ. Equ. 1979, 33: 336-352. 10.1016/0022-0396(79)90069-X

    Article  MathSciNet  MATH  Google Scholar 

  4. Benci V: Normal modes of a Lagrangian system constrained in a potential well. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 379-400.

    MathSciNet  MATH  Google Scholar 

  5. Benci V: Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1984, 1: 401-412.

    MathSciNet  MATH  Google Scholar 

  6. Gluck H, Ziller W: Existence of periodic motions of conservative systems. In Seminar on Minimal Submanifolds. Edited by: Bombieri E. Princeton University Press, Princeton; 1983.

    Google Scholar 

  7. Hayashi K: Periodic solutions of classical Hamiltonian systems. Tokyo J. Math. 1983, 6: 473-486. 10.3836/tjm/1270213886

    Article  MathSciNet  MATH  Google Scholar 

  8. Van Groesen EWC: Analytical mini-max methods for Hamiltonian break orbits with a prescribed energy. J. Math. Anal. Appl. 1988, 132: 1-12. 10.1016/0022-247X(88)90039-X

    Article  MathSciNet  MATH  Google Scholar 

  9. Long Y: Index Theory for Symplectic Paths with Applications. Birkhäuser, Basel; 2002.

    Book  MATH  Google Scholar 

  10. Ambrosetti A, Coti Zelati V: Solutions with minimal period for Hamiltonian systems in a potential well. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1987, 4: 275-296.

    MathSciNet  MATH  Google Scholar 

  11. Offin D: A class of periodic orbits in classical mechanics. J. Differ. Equ. 1987, 66: 90-117. 10.1016/0022-0396(87)90042-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Rabinowitz PH:On a class of functionals invariant under a Z n action. Trans. Am. Math. Soc. 1988, 310: 303-311.

    MathSciNet  MATH  Google Scholar 

  13. Coti Zelati V, Ekeland I, Lions PL: Index estimates and critical points of functionals not satisfying Palais-Smale. Ann. Sc. Norm. Super. Pisa 1990, 17: 569-581.

    MathSciNet  MATH  Google Scholar 

  14. 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  16. 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

    Article  MathSciNet  MATH  Google Scholar 

  17. 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang SQ: Multiple geometrically distinct closed noncollision orbits of fixed energy for N -body type problems with strong force potentials. Proc. Am. Math. Soc. 1996, 124: 3039-3046. 10.1090/S0002-9939-96-03751-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Benci V, Giannoni G: Periodic solutions of prescribed energy for a class of Hamiltonian system with singular potentials. J. Differ. Equ. 1989, 82: 60-70. 10.1016/0022-0396(89)90167-8

    Article  MathSciNet  MATH  Google Scholar 

  21. Chang KC: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Basel; 1993.

    Book  MATH  Google Scholar 

  22. Degiovanni M, Giannoni F: Dynamical systems with Newtonian type potentials. Ann. Sc. Norm. Super. Pisa 1988, 15: 467-494.

    MathSciNet  MATH  Google Scholar 

  23. Fadell E, Husseini S: A note on the category of free loop space. Proc. Am. Math. Soc. 1989, 102: 527-536.

    Article  MathSciNet  MATH  Google Scholar 

  24. Ambrosetti A, Coti Zelati V: Critical points with lack of compactness and applications to singular dynamical system. Ann. Mat. Pura Appl. 1987, 149: 237-259. 10.1007/BF01773936

    Article  MathSciNet  MATH  Google Scholar 

  25. Greco C: Periodic solutions of a class of singular Hamiltonian systems. Nonlinear Anal. TMA 1988, 12: 259-269. 10.1016/0362-546X(88)90112-5

    Article  MathSciNet  MATH  Google Scholar 

  26. Majer P: Ljusternik-Schnirelmann theory with local Palais-Smale conditions and singular dynamical systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1991, 8: 459-476.

    MathSciNet  MATH  Google Scholar 

  27. Rabinowitz PH: A note on periodic solutions of prescribed energy for singular Hamiltonian systems. J. Comput. Appl. Math. 1994, 52: 147-154. 10.1016/0377-0427(94)90354-9

    Article  MathSciNet  MATH  Google Scholar 

  28. Serra E, Terracini S: Noncollision solutions to some singular minimization problems with Keplerian-like potentials. Nonlinear Anal. TMA 1994, 22: 45-62. 10.1016/0362-546X(94)90004-3

    Article  MathSciNet  MATH  Google Scholar 

  29. Tanaka K: A prescribed energy problem for a singular Hamiltonian system with weak force. J. Funct. Anal. 1993, 113: 351-390. 10.1006/jfan.1993.1054

    Article  MathSciNet  MATH  Google Scholar 

  30. Tanaka K: A prescribed energy problem for conservative singular Hamiltonian system. Arch. Ration. Mech. Anal. 1994, 128: 127-164. 10.1007/BF00375024

    Article  MathSciNet  MATH  Google Scholar 

  31. 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

    Article  MathSciNet  MATH  Google Scholar 

  32. Terracini S: Multiplicity of periodic solutions of prescribed energy problem for singular dynamical system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1992, 9: 597-641.

    MathSciNet  MATH  Google Scholar 

  33. Ambrosetti A, Struwe M: Periodic motions of conservative systems with singular potentials. NoDEA Nonlinear Differ. Equ. Appl. 1998, 1: 179-202.

    Article  MathSciNet  MATH  Google Scholar 

  34. Bahri A, Rabinowitz PH: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 1989, 82: 412-428. 10.1016/0022-1236(89)90078-5

    Article  MathSciNet  MATH  Google Scholar 

  35. Mawhin J, Willem M: Critical Point Theory and Applications. Springer, Berlin; 1989.

    MATH  Google Scholar 

  36. Ziemer WP: Weakly Differentiable Functions. Springer, Berlin; 1989.

    Book  MATH  Google Scholar 

  37. Yosida K: Functional Analysis. Springer, Berlin; 1978.

    Book  MATH  Google Scholar 

  38. Ekeland I: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin; 1990.

    Book  MATH  Google Scholar 

  39. Ghoussoub N, Preiss D: A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1989, 6: 321-330.

    MathSciNet  MATH  Google Scholar 

  40. Cerami G: Un criterio di esistenza per i punti critici so variete illimitate. Rend. - Ist. Lomb., Accad. Sci. Lett., a Sci. Mat. Fis. Chim. Geol. 1978, 112: 332-336.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the referees for their many valuable comments. This paper was partially supported by NSF of China and the Grant for the Advisors of PhD students.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fengying Li.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The research and writing of this manuscript was a collaborative effort made by all the authors. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, F., Hua, Q. & Zhang, S. New periodic solutions of singular Hamiltonian systems with fixed energies. J Inequal Appl 2014, 400 (2014). https://doi.org/10.1186/1029-242X-2014-400

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-400

Keywords