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.