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

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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.

1 Introduction

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

x ¨ = V (t,x),
(1.1)

where Ω= R N {0} (NN, N2) 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 xN{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 (N2).

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], xΩ.

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)xV(t,x);

  2. (ii)

    V(t,x) c 1 | x | ν .

Setting K={xΩ| 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/t0, 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, α2 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,

    3V(t,x) V (t,x)xM.
  • (V5) V(t,x)+ as |x|+ uniformly for 0tT.

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

Corollary 1.1 Suppose α2, β3, 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/ZT, R N ) and 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.
  2. (ii)

    Let q W 1 , 2 (R/ZT, R N ) and 0 T q(t)dt=0, then we have Sobolev’s inequality:

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

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, ΛX, 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, ΛX, 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, ΛX, 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 ΛX, 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 dt 0 T V(t,q)dt,qΛ,

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 dt 0 T V( x n )dtc+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 =2f( x n )+ 0 T [ 2 V ( t , x n ) V ( t , x n ) x n ] dt0.
(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 dtd.
(3.5)

So 0 T | x ˙ n | 2 dt 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

xΛ, 0 T x(t)dt=0,

so by (V3) and Wirtinger’s inequality we have

f(x)= 1 2 0 T | x ˙ | 2 dt 0 T V(t,x)dt
(3.6)
1 2 0 T | x ˙ | 2 dt+a 0 T | x | α dt
(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 dt) 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)=+. □

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Correspondence to Fengying Li.

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Keywords

  • Ambrosetti-Rabinowitz’s mountain pass lemma
  • singular second-order Hamiltonian systems
  • periodic solutions
  • Cerami-Palais-Smale condition