Open Access

Mountain pass lemma and new periodic solutions of the singular second order Hamiltonian system

Boundary Value Problems20142014:49

https://doi.org/10.1186/1687-2770-2014-49

Received: 2 February 2014

Accepted: 18 February 2014

Published: 27 February 2014

Abstract

We generalize the classical Ambrosetti-Rabinowitz mountain pass lemma with the Palais-Smale condition for C 1 functional to some singular case with the Cerami-Palais-Smale condition and then we study the existence of new periodic solutions with a fixed period for the singular second-order Hamiltonian systems with a strong force potential.

MSC:34C15, 34C25, 58F.

Keywords

Ambrosetti-Rabinowitz’s mountain pass lemmasingular second-order Hamiltonian systemsperiodic solutionsCerami-Palais-Smale condition

1 Introduction

Many authors [119] studied the existence of periodic solutions t x ( t ) Ω , with a prescribed period, of the following second-order differential equations:
x ¨ = V ( t , x ) ,
(1.1)

where Ω = R N { 0 } ( N N , N 2 ) and V C 1 ( R × Ω , R ) ; V ( t , ) denotes the gradient of the function V ( t , ) defined on Ω.

In 1975, Gordon [10] firstly used variational methods to study periodic solutions of planar 2-body type problems, he assumed the condition nowadays called Gordon’s strong force condition.

Condition (V1): There exists a neighborhood of 0 and a function U C 1 ( Ω , R ) such that:
  1. (i)

    lim x 0 U ( x ) = ;

     
  2. (ii)

    V ( t , x ) | U ( x ) | 2 for every x N { 0 } and t [ 0 , T ] .

     
Moreover,
  1. (iii)

    lim x 0 V ( t , x ) = .

     

In the 1980s and 1990s, Ambsosetti-Coti Zelati, Bahri-Rabinowitz, Greco etc. [19, 1119] further studied 2-body type problems in R N ( N 2 ).

Suppose that V ( t , x ) is T-periodic in t; as regards the behavior of V ( t , x ) at infinity, they suppose that one of the following conditions holds.

Condition (V2): lim | x | V ( t , x ) = 0 , lim | x | V ( t , x ) = 0 (uniformly for t) and V ( t , x ) < 0 for every t [ 0 , T ] , x Ω .

Condition (V3): There exist c 1 , M 1 , R 1 , ν > 0 such that, for every t [ 0 , T ] and x R N with | x | R 1 :
  1. (i)

    | V ( t , x ) | M 1 ;

     
  2. (ii)

    V ( t , x ) c 1 | x | ν .

     
Condition (V4): There exist c 1 , R 1 > 0 , θ > 1 2 , ν > 1 such that, for every t [ 0 , T ] , | x | R 1 :
  1. (i)

    θ V ( t , x ) x V ( t , x ) ;

     
  2. (ii)

    V ( t , x ) c 1 | x | ν .

     

Setting K = { x Ω | V ( t , x ) = 0  for every  t [ 0 , T ] } , they got the following results.

Theorem 1.1 (Greco [11])

If (V1) and one of (V2)-(V4) hold, and moreover K = , then there is at least one non-constant T-periodic C 2 solution.

Theorem 1.2 (Bahri-Rabinowitz [3], Greco [11])

Suppose that V / t 0 , so V ( t , x ) V ( x ) ; moreover suppose we have the following condition.

Condition (V5): K is compact (or empty).

Then, if (V1) and one of (V2)-(V4) hold, there exist infinitely many non-constant T-periodic C 2 solutions.

In this paper, we prove the following new theorem.

Theorem 1.3 Suppose V C 1 ( R × Ω , R ) satisfies the conditions:

  • (V1) For the given T > 0 , V ( t + T , x ) = V ( t , x ) .

  • (V2) ( t , x ) R × Ω , V ( t + T 2 , x ) = V ( t , x ) .

  • (V3) There is a > 0 , α 2 such that for any given ϵ > 0 and
    t [ 0 , T ] , | x | ( T 12 ) 1 2 [ ( b α ) 1 ( α + 2 ) + ϵ ] ,
    we have
    V ( t , x ) a | x | α ,
    where
    b = a ( 2 π ) α T 1 α 2 .
  • (V4) There exists M > 0 such that ( t , x ) R × Ω ,
    3 V ( t , x ) V ( t , x ) x M .
  • (V5) V ( t , x ) + as | x | + uniformly for 0 t T .

    Then the system (1.1) has at least a non-constant T-periodic solution.

Corollary 1.1 Suppose α 2 , β 3 , a > 0 , a > 0 , V C 1 ( Ω , R ) and
V ( x ) = a | x | α , 0 < | x | r 1 = ( T 12 ) 1 2 [ ( b α ) 1 ( α + 2 ) + ϵ ] ; V ( x ) = a | x | β , | x | r 2 > r 1 ;

then T > 0 , (1.1) has at least a T-periodic solution.

2 A few lemmas

Lemma 2.1 (Sobolev-Rellich-Kondrachov [20])

We have
H 1 = W 1 , 2 ( R / T Z , R N ) C ( R / T Z , R N )

and the embedding is compact.

Lemma 2.2 (Eberlein-Shmulyan [20])

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

Lemma 2.3 ([21])
  1. (i)
    Let q W 1 , 2 ( R / Z T , R N ) and 0 T q ( t ) d t = 0 , then we have Wirtinger’s inequality:
    0 T | q ˙ ( t ) | 2 d t ( 2 π T ) 2 0 T | q ( t ) | 2 d t .
     
  2. (ii)
    Let q W 1 , 2 ( R / Z T , R N ) and 0 T q ( t ) d t = 0 , then we have Sobolev’s inequality:
    q 2 T 12 0 T | q ˙ ( t ) | 2 d t .
     
  3. (iii)
    Let ϕ be a convex function on the real line; f : [ a , b ] R is a non-negative real-valued function which is Lebesgue-integrable, then
    ϕ ( a b f ( x ) d x ) 1 b a a b ϕ ( ( b a ) f ( x ) ) d x .
     

Lemma 2.4 (Ekeland [8])

Let X be a Banach space; suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous and bounded from below. Then there is a sequence { x n } such that
Φ ( x n ) inf Φ , ( 1 + x n ) Φ ( x n ) 0 .

Definition 2.1 (Palais and Smale [22])

Let X be a Banach space; f C 1 ( X , R ) , if { x n } X s.t.
f ( x n ) c , f ( x n ) 0 ,

and { x n } has a strongly convergent subsequence; then we say that f satisfies the ( PS ) c condition.

Cerami [23] presented a weaker compact condition than the above classical ( PS ) c condition.

Definition 2.2 ([8])

Let X be a Banach space, Λ X , and suppose that Φ is defined on Λ is Gateaux-differentiable, if the sequence { x n } is such that
Φ ( x n ) c , ( 1 + x n ) Φ ( x n ) 0 ,

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

Then we say that f satisfies the ( CPS ) c condition.

We can give a weaker condition than the ( CPS ) c condition.

Definition 2.3 Let X be a Banach space, Λ X , and suppose that Φ defined on Λ is Gateaux-differentiable; if the sequence { x n } is such that
Φ ( x n ) c , ( 1 + x n ) Φ ( x n ) 0 ,

and { x n } has a weakly convergent subsequence in Λ, then we say that f satisfies the ( WCPS ) c condition.

Lemma 2.5 (Ambrosetti-Rabinowitz [24], mountain pass lemma)

Let X be a Banach space, Λ X , f C 1 ( Λ , R ) . We have
B ρ = { x Λ | x ρ } , S ρ = B ρ X , ρ > 0 .
If there are two points e 1 B ρ S ρ , e 2 Λ B ρ such that
f | S ρ α > 0
and
f ( e 1 ) , f ( e 2 ) 0 ,

then C = inf ϕ Γ sup t [ 0 , 1 ] f ( ϕ ( t ) ) α , where Γ = { h ( t ) C 1 ( [ 0 , 1 ] , Λ ) , h ( 0 ) = e 1 , h ( 1 ) = e 2 } . If f satisfies the ( CPS ) C condition on Λ X , furthermore, if f ( x n ) + as x n Λ , then C is a critical value for f.

3 The proof of Theorem 1.3

Let
H 1 = { q : R R n | q L 2 , q ˙ L 2 , q ( t + T ) = q ( t ) } , Λ = { q H 1 , q ( t + T 2 ) = q ( t ) , q ( t ) 0 , t } .

Lemma 3.1 ([2, 25])

If V C 1 ( R × Ω , R ) satisfies the conditions (V1)-(V2), let
f ( q ) = 1 2 0 T | q ˙ | 2 d t 0 T V ( t , q ) d t , q Λ ,

then the critical point of f ( q ) on Λ is a T-periodic solution of (1.1).

Lemma 3.2 If V satisfies (V3), (V4) in Theorem  1.1, then f satisfies the Cerami-Palais-Smale condition for any c > 0 , that is, for any { x n } Λ :
f ( x n ) c , ( 1 + x n ) f ( x n ) 0 ,
(3.1)

{ x n } has a strongly convergent subsequence and the limit is in Λ.

Proof By the condition (V3), it is well known [10] that f ( x n ) + as x n Λ . Since f ( x n ) c , we know that for any given ϵ > 0 , there exists N such that when n > N , we have
1 2 0 T | x ˙ n | 2 d t 0 T V ( x n ) d t c + ϵ .
(3.2)
By ( 1 + x n ) f ( x n ) 0 , we have
f ( x n ) x n 0 ,
(3.3)
f ( x n ) x n = 2 f ( x n ) + 0 T [ 2 V ( t , x n ) V ( t , x n ) x n ] d t 0 .
(3.4)
So by (V4) and (3.2) and (3.4), we have d > 0 such that when n large enough, we have
0 T | x ˙ n | 2 d t d .
(3.5)

So 0 T | x ˙ n | 2 d t is bounded. Then { x n } has a weakly convergence subsequence, and it is standard to further prove that this subsequence is strongly convergent in Λ.

Now we can prove our theorem.

In order to apply for Ambrosetti-Rabinowitz’s mountain pass lemma, we notice that
x Λ , 0 T x ( t ) d t = 0 ,
so by (V3) and Wirtinger’s inequality we have
f ( x ) = 1 2 0 T | x ˙ | 2 d t 0 T V ( t , x ) d t
(3.6)
1 2 0 T | x ˙ | 2 d t + a 0 T | x | α d t
(3.7)
1 2 0 T | x ˙ | 2 + a T 1 + α 2 ( 0 T | x | 2 d t ) α 2
(3.8)
1 2 0 T | x ˙ | 2 + a T 1 + α 2 ( T 2 π ) α ( 0 T | x ˙ | 2 d t ) α 2
(3.9)
= 1 2 s 2 + b s α = ϕ ( s ) ,
(3.10)
where
s = ( 0 T | x ˙ | 2 d t ) 1 / 2 , b = a T 1 + α 2 ( T 2 π ) α = a ( 2 π ) α T 1 α 2 .
(3.11)

It is easy to see that if s 0 = ( b α ) 1 α + 2 , ϕ attains its infimum which is a positive number. ϵ > 0 , we can take ρ = s 0 + ϵ , take e 1 ( t ) 0 , e 1 ˙ = s 0 < ρ . By Sobolev’s inequality, we know that ( 0 T | x ˙ | 2 d t ) 12 T x 2 , so if x ( t ) ( T 12 ) 1 2 [ ( b α ) 1 ( α + 2 ) + ϵ ] , then the above proofs hold.

Let us choose e 2 = constant value vector in R n , e 2 ˙ = 0 . Then by (V1) and (V5), we have
f ( e 2 ) = 0 T V ( t , e 2 ) T min 0 t T | V ( t , e 2 ) | as  | e 2 | = R + .
(3.12)
So if | e 2 | = R is large enough, we have
f | e 2 0 .

By Lemmas 2.5 and 3.2, f has a critical value C > 0 , and the corresponding critical point is a T-periodic solution of the system (1.1). Furthermore, we claim that the critical point is non-constant; in fact, if otherwise, by the anti- T / 2 periodic property, we know that the critical point must be constant zero, which is impossible since f ( 0 ) = + . □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University
(2)
School of Economic and Mathematics, Southwestern University of Finance and Economics

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© Li and Li; licensee Springer. 2014

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