Open Access

Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

Boundary Value Problems20132013:16

DOI: 10.1186/1687-2770-2013-16

Received: 8 November 2012

Accepted: 10 January 2013

Published: 6 February 2013

Abstract

In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

1 Introduction

Consider the second-order Hamiltonian systems
{ u ¨ ( t ) + u W ( t , u ) = 0 , t R , u ( 0 ) = u ( T ) , u ˙ ( 0 ) = u ˙ ( T ) , T > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ1_HTML.gif
(1.1)
where W ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq1_HTML.gif is also T-periodic and satisfies the following assumption (A):
  1. (A)
    W ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq1_HTML.gif is measurable in t for all u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq2_HTML.gif, continuously differentiable in u for a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq3_HTML.gif and there exist a C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq4_HTML.gif and b L 1 ( [ 0 , T ] , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq5_HTML.gif such that
    | W ( t , u ) | a ( | u | ) b ( t ) , | u W ( t , u ) | a ( | u | ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equa_HTML.gif
     

for all u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq2_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq3_HTML.gif.

Here and in the sequel, , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq6_HTML.gif and | | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq7_HTML.gif always denote the standard inner product and the norm in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq8_HTML.gif respectively.

There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [17] and the references therein). In [6], Zhang and Liu studied the asymptotically quadratic case of W ( t , u ) = 1 2 U ( t ) u , u + W 1 ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq9_HTML.gif under the following assumptions:

(AQ1) W 1 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq10_HTML.gif for all ( t , u ) [ 0 , T ] × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq11_HTML.gif, and there exist constants μ ( 0 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq12_HTML.gif and R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq13_HTML.gif such that
u W 1 ( t , u ) , u μ W 1 ( t , u ) , t [ 0 , T ]  and  | u | R 1 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equb_HTML.gif
(AQ2) lim | u | 0 W 1 ( t , u ) | u | 2 = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq14_HTML.gif uniformly for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq15_HTML.gif, and there exist constants c 2 , R 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq16_HTML.gif such that
W 1 ( t , u ) c 2 | u | , t [ 0 , T ]  and  | u | R 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equc_HTML.gif

(AQ3) lim inf | u | W 1 ( t , u ) | u | d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq17_HTML.gif uniformly for t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq3_HTML.gif.

They obtained the existence of infinitely many periodic solutions of (1.1) provided W 1 ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq18_HTML.gif is even in u (see Theorem 1.1 of [6]).

The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example, [4] of Jiang and Tang. This paper considers the case of U ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq19_HTML.gif, then W ( t , u ) = W 1 ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq20_HTML.gif. Motivated by [4] and [6], we replace (AQ1) with the following condition:

( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq21_HTML.gif) W ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq22_HTML.gif for all ( t , u ) [ 0 , T ] × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq11_HTML.gif, and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equd_HTML.gif
The condition ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq23_HTML.gif) implies that for some constant R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq24_HTML.gif,
u W ( t , u ) , u 2 W ( t , u ) , t [ 0 , T ]  and  | u | R 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ2_HTML.gif
(1.2)
By the assumption (A) and the condition ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq23_HTML.gif), for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq25_HTML.gif, there exists a δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq26_HTML.gif such that
W ( t , u ) ϵ | u | 2 + max s [ 0 , δ ] a ( s ) b ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ3_HTML.gif
(1.3)

for u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq27_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq3_HTML.gif.

Meanwhile, we weaken the condition (AQ3) to ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq28_HTML.gif) as follows:

( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq28_HTML.gif) There exists a constant ϱ ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq29_HTML.gif such that
lim inf | u | W ( t , u ) | u | ϱ d > 0 uniformly for  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Eque_HTML.gif

Then our main result is the following theorem.

Theorem 1.1 Assume that ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq21_HTML.gif), (AQ2), ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq28_HTML.gif) hold and W ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq1_HTML.gif is even in u. Then (1.1) possesses infinitely many solutions.

Remark The conditions (AQ1) and (AQ3) are stronger than ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq21_HTML.gif) and ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq28_HTML.gif). Then Theorem 1.1 above is different from Theorem 1.1 of [6].

2 Preliminaries

In this section, we establish the variational setting for our problem and give the variant fountain theorem. Let E = H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq30_HTML.gif be the usual Sobolev space with the inner product
u , v E = 0 T u ( t ) , v ( t ) d t + 0 T u ˙ ( t ) , v ˙ ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equf_HTML.gif
We define the functional on E by
Φ ( u ) = 1 2 0 T | u ˙ | 2 d t Ψ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equg_HTML.gif
where Ψ ( u ) = 0 T W ( t , u ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq31_HTML.gif. Then Φ and Ψ are continuously differentiable and
Φ ( u ) , v = 0 T u ˙ , v ˙ d t 0 T u W ( t , u ) , v d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equh_HTML.gif
Define a self-adjoint linear operator B : L 2 ( [ 0 , T ] ; R N ) L 2 ( [ 0 , T ] ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq32_HTML.gif by
0 T B u , v d t = 0 T u ˙ ( t ) , v ˙ ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equi_HTML.gif
with the domain D ( B ) = E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq33_HTML.gif. Then has a sequence of eigenvalues σ k = 4 k 2 π 2 T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq34_HTML.gif ( k = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq35_HTML.gif). Let { e j } j = 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq36_HTML.gif be the system of eigenfunctions corresponding to { σ j } j = 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq37_HTML.gif, it forms an orthogonal basis in L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq38_HTML.gif. Denote by E + = { u E | 0 T u ( t ) d t = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq39_HTML.gif, E 0 = R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq40_HTML.gif, it is well known that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equj_HTML.gif
and E possesses orthogonal decomposition E = E 0 E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq41_HTML.gif. For u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq42_HTML.gif, we have
u = u 0 + u + E 0 E + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equk_HTML.gif
We can define on E a new inner product and the associated norm by
u , v 0 = B u + , v + L 2 + u 0 , v 0 L 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equl_HTML.gif
and
u = u , u 0 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equm_HTML.gif
Therefore, Φ can be written as
Φ ( u ) = 1 2 u + 2 Ψ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ4_HTML.gif
(2.1)
Direct computation shows that
Ψ ( u ) , v = 0 T u W ( t , u ) , v d t , Φ ( u ) , v = u + , v + 0 Ψ ( u ) , v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ5_HTML.gif
(2.2)

for all u , v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq43_HTML.gif with u = u 0 + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq44_HTML.gif and v = v 0 + v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq45_HTML.gif respectively. It is known that Ψ : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq46_HTML.gif is compact.

Denote by | | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq47_HTML.gif the usual norm of L P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq48_HTML.gif, then there exists a τ p > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq49_HTML.gif such that
| u | p τ p u , u E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ6_HTML.gif
(2.3)
We state an abstract critical point theorem founded in [8]. Let E be a Banach space with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq50_HTML.gif and E = j N X j ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq51_HTML.gif with dim X j < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq52_HTML.gif for any j N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq53_HTML.gif. Set Y k = j = 1 k X j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq54_HTML.gif and Z k = j = k X j ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq55_HTML.gif . Consider the following C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq56_HTML.gif-functional Φ λ : E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq57_HTML.gif defined by
Φ λ ( u ) : = A ( u ) λ B ( u ) , λ [ 1 , 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equn_HTML.gif

Theorem 2.1 [[8], Theorem 2.2]

Assume that the functional Φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq58_HTML.gif defined above satisfies the following:

(T1) Φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq58_HTML.gif maps bounded sets to bounded sets uniformly for λ [ 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq59_HTML.gif, and Φ λ ( u ) = Φ λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq60_HTML.gif for all ( λ , u ) [ 1 , 2 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq61_HTML.gif;

(T2) B ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq62_HTML.gif for all u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq42_HTML.gif, and B ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq63_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq64_HTML.gif on any finite-dimensional subspace of E;

(T3) There exist ρ k > r k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq65_HTML.gif such that
α k ( λ ) : = inf u Z k , u = ρ k Φ λ ( u ) 0 > β k ( λ ) : = max u Y k , u = r k Φ λ ( u ) , λ [ 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equo_HTML.gif
and
ξ k ( λ ) : = inf u Z k , u ρ k Φ λ ( u ) 0 as k uniformly for λ [ 1 , 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equp_HTML.gif
Then there exist λ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq66_HTML.gif, u λ n Y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq67_HTML.gif such that
Φ λ n Y n ( u λ n ) = 0 , Φ λ n ( u λ n ) η k [ ξ k ( 2 ) , β k ( 1 ) ] as n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equq_HTML.gif

Particularly, if { u λ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq68_HTML.gif has a convergent subsequence for every k, then Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq69_HTML.gif has infinitely many nontrivial critical points { u k } E { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq70_HTML.gif satisfying Φ 1 ( u k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq71_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq72_HTML.gif.

In order to apply this theorem to prove our main result, we define the functionals A, B and Φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq58_HTML.gif on our working space E by
A ( u ) = 1 2 u + 2 , B ( u ) = 0 T W ( t , u ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ7_HTML.gif
(2.4)
and
Φ λ ( u ) = A ( u ) λ B ( u ) = 1 2 u + 2 λ 0 T W ( t , u ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ8_HTML.gif
(2.5)

for all u = u 0 + u + E = E 0 + E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq73_HTML.gif and λ [ 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq59_HTML.gif. Then Φ λ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq74_HTML.gif for all λ [ 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq59_HTML.gif. Let X j = span { e j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq75_HTML.gif, j = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq76_HTML.gif . Note that Φ 1 = Φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq77_HTML.gif, where Φ is the functional defined in (2.1).

3 Proof of Theorem 1.1

We firstly establish the following lemmas.

Lemma 3.1 Assume that ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq23_HTML.gif) and ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq78_HTML.gif) hold. Then B ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq62_HTML.gif for all u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq42_HTML.gif and B ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq79_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq64_HTML.gif on any finite-dimensional subspace of E.

Proof Since W ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq22_HTML.gif, by (2.4), it is obvious that B ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq62_HTML.gif for all u E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq42_HTML.gif.

By the proof of Lemma 2.6 of [6], for any finite-dimensional subspace Y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq80_HTML.gif, there exists a constant ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq25_HTML.gif such that
m ( { t [ 0 , T ] : | u | ϵ u } ) ϵ , u Y { 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ9_HTML.gif
(3.1)

where m ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq81_HTML.gif is the Lebesgue measure.

For the ϵ given in (3.1), let
Λ u = { t [ 0 , T ] : | u | ϵ u } , u Y { 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equr_HTML.gif
Then m ( Λ u ) ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq82_HTML.gif. By ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq78_HTML.gif), there exists a constant R 3 > R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq83_HTML.gif such that
W ( t , u ) d | u | ϱ / 2 , t [ 0 , T ]  and  | u | R 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ10_HTML.gif
(3.2)
where R 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq84_HTML.gif is the constant given in (1.2). Note that
| u ( t ) | R 3 , t Λ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ11_HTML.gif
(3.3)
for any u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq85_HTML.gif with u R 3 / ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq86_HTML.gif. Thus,
B ( u ) = 0 T W ( t , u ) d t Λ u W ( t , u ) d t Λ u d | u | ϱ / 2 d t d ϵ ϱ u ϱ m ( Λ u ) / 2 d ϵ ϱ + 1 u ϱ / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equs_HTML.gif

for any u Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq85_HTML.gif with u R 3 / ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq86_HTML.gif. This implies B ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq79_HTML.gif as u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq64_HTML.gif on Y. □

Lemma 3.2 Assume that ( AQ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq23_HTML.gif), (AQ2) and ( AQ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq78_HTML.gif) hold. Then there exist a positive integer k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq87_HTML.gif and two sequences 0 < r k < ρ k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq88_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq72_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ12_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ13_HTML.gif
(3.5)
and
β k ( λ ) : = max u Y k , u = r k Φ λ ( u ) < 0 , k N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ14_HTML.gif
(3.6)

where Y k = j = 0 k X j = span { e 0 , e 1 , , e k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq89_HTML.gif and Z k = j = k X j ¯ = span { e k , e k + 1 , } ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq90_HTML.gif for all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq91_HTML.gif.

Proof Comparing this lemma with Lemma 2.7 of [6], we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of [6]. We can prove our lemma by using the same method of [6], so the details are omitted. □

Now it is the time to prove our main result Theorem 1.1.

Proof of Theorem 1.1 By virtue of (1.3), (2.3) and (2.5), Φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq58_HTML.gif maps bounded sets to bounded sets uniformly for λ [ 1 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq59_HTML.gif. Obviously, Φ λ ( u ) = Φ λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq60_HTML.gif for all ( λ , u ) [ 1 , 2 ] × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq92_HTML.gif since W ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq1_HTML.gif is even in u. Consequently, the condition (T1) of Theorem 2.1 holds. Lemma 3.1 shows that the condition (T2) holds, whereas Lemma 3.2 implies that the condition (T3) holds for all k k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq93_HTML.gif, where k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq87_HTML.gif is given there. Therefore, by Theorem 2.1, for each k k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq93_HTML.gif, there exist λ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq66_HTML.gif and u λ n Y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq67_HTML.gif such that
Φ λ n Y n ( u λ n ) = 0 , Φ λ n ( u λ n ) η k [ ξ k ( 2 ) , β k ( 1 ) ] as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ15_HTML.gif
(3.7)

For the sake of notational simplicity, in the following we always set u n = u λ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq94_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq95_HTML.gif.

Step 1. We firstly prove that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq96_HTML.gif is bounded in E.

Since { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq96_HTML.gif satisfies (3.7), one has
lim n ( Φ λ n Y n ( u n ) , u n 2 Φ λ n ( u n ) ) = 2 η k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equt_HTML.gif
More precisely,
lim n 0 T ( u W ( t , u n ) , u n 2 W ( t , u n ) ) d t = 2 η k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ16_HTML.gif
(3.8)
Now, we prove that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq96_HTML.gif is bounded. Otherwise, without loss of generality, we may assume that
u n as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equu_HTML.gif
Put z n = u n u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq97_HTML.gif, we have z n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq98_HTML.gif. Going to a subsequence if necessary, we may assume that
z n z in  E , z n z in  L 2 and z n ( t ) z ( t ) for a.e.  t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equv_HTML.gif
By (1.3), we have
Φ λ n ( u n ) = 1 2 u n + 2 λ n 0 T W ( t , u n ) d t 1 2 u n 2 1 2 u n 0 2 λ n ( ϵ 0 T | u n | 2 d t + max s [ 0 , δ ] a ( s ) 0 T b ( t ) d t ) 1 2 u n 2 ( 1 2 + λ n ϵ ) 0 T | u n | 2 d t λ n c 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equw_HTML.gif
where c 1 = max s [ 0 , δ ] a ( s ) 0 T b ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq99_HTML.gif. Therefore, one obtains
Φ λ n ( u n ) u n 2 1 2 ( 1 2 + λ n ϵ ) 0 T ( | u n | u n ) 2 d t λ n c 1 u n 2 = 1 2 ( 1 2 + λ n ϵ ) z n 2 2 λ n c 1 u n 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equx_HTML.gif
Passing to the limit in the inequality, by using Φ λ n ( u n ) η k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq100_HTML.gif and λ n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq66_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq101_HTML.gif, we obtain
1 2 ( 1 2 + ϵ ) z 2 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equy_HTML.gif

Thus, z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq102_HTML.gif on a subset Ω of [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq103_HTML.gif with positive measure.

By (1.2), we have
u W ( t , u ) , u 2 W ( t , u ) 0 , t [ 0 , T ]  and  | u | R 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equz_HTML.gif
and by the assumption (A), we obtain
u W ( t , u ) , u 2 W ( t , u ) c 3 b ( t ) , for all  | u | R 1  and a.e.  t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equaa_HTML.gif
where c 3 = ( 2 + R 1 ) max [ 0 , R 1 ] a ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq104_HTML.gif. So, we get
u W ( t , u ) , u 2 W ( t , u ) c 3 b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equab_HTML.gif
for all u R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq2_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq3_HTML.gif. Hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equac_HTML.gif
An application of Fatou’s lemma yields
Ω ( u W ( t , u n ) , u n 2 W ( t , u n ) ) d t as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equad_HTML.gif

which is a contradiction to (3.8).

Step 2. We prove that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq96_HTML.gif has a convergent subsequence in E.

Since { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq96_HTML.gif is bounded in E, E is reflexible and dim E 0 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq105_HTML.gif, without loss of generality, we assume
u n 0 u 0 0 , u n + u 0 + and u n u 0 as  n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ17_HTML.gif
(3.9)

for some u 0 = u 0 0 + u 0 + E = E 0 E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq106_HTML.gif.

Note that
0 = Φ λ n Y n ( u n ) = u n + λ n P n Ψ ( u n ) , n N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equae_HTML.gif
where P n : E Y n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq107_HTML.gif is the orthogonal projection for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq108_HTML.gif, that is,
u n + = λ n P n Ψ ( u n ) , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_Equ18_HTML.gif
(3.10)

In view of the compactness of Ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq109_HTML.gif and (3.9), the right-hand side of (3.10) converges strongly in E and hence u n + u 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq110_HTML.gif in E. Together with (3.9), we have u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq111_HTML.gif in E.

Now, from the last assertion of Theorem 2.1, we know that Φ = Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-16/MediaObjects/13661_2012_Article_269_IEq112_HTML.gif has infinitely many nontrivial critical points. The proof is completed. □

Declarations

Acknowledgements

The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).

Authors’ Affiliations

(1)
College of Science, Hohai University

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Copyright

© Gu and An; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.