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New periodic solutions for a class of singular Hamiltonian systems

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. □

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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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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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Wang, Xr., Shen, S. New periodic solutions for a class of singular Hamiltonian systems. Bound Value Probl 2014, 42 (2014). https://doi.org/10.1186/1687-2770-2014-42

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