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Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded ( ϕ 1 , ϕ 2 ) -Laplacian

Advances in Difference Equations20142014:218

https://doi.org/10.1186/1687-1847-2014-218

Received: 23 May 2014

Accepted: 10 July 2014

Published: 5 August 2014

Abstract

In this paper, we consider the multiplicity of periodic solutions for a class of difference systems involving the ( ϕ 1 , ϕ 2 ) -Laplacian in the cases when the gradient of the nonlinearity has a sublinear growth. By using the variational method, some existence results are obtained. Our results generalize some recent results in (Mawhin in Discrete Contin. Dyn. Syst. 6:1065-1076, 2013).

Keywords

  • difference systems
  • periodic solutions
  • critical point theorem
  • variational method

1 Introduction and main results

Let denote the real numbers and the integers. Given a < b in . Let Z [ a , b ] = { a , a + 1 , , b } . Let T > 1 and N be fixed positive integers.

In this paper, we investigate the multiplicity of periodic solutions for the following nonlinear difference systems:
{ Δ ϕ 1 ( Δ u 1 ( t 1 ) ) = u 1 F ( t , u 1 ( t ) , u 2 ( t ) ) + h 1 ( t ) , Δ ϕ 2 ( Δ u 2 ( t 1 ) ) = u 2 F ( t , u 1 ( t ) , u 2 ( t ) ) + h 2 ( t ) ,
(1.1)

where F : Z × R N × R N R and ϕ m , m = 1 , 2 , satisfy the following condition:

( A 0 ) ϕ m is a homeomorphism from R N onto B a R N ( a ( 0 , + ] ), such that ϕ m ( 0 ) = 0 , ϕ m = Φ m , with Φ m C 1 ( R N , [ 0 , + ] ) strictly convex and Φ m ( 0 ) = 0 , m = 1 , 2 .

Remark 1.1 Assumption ( A 0 ) is given in [1], which is used to characterize the classical homeomorphism and the bounded homeomorphism. ϕ m is called classical when a = + and bounded when a < + . If furthermore Φ m : R N R is coercive (i.e. Φ m ( x ) + as | x | ), there exists δ m > 0 such that
Φ m ( x ) δ m ( | x | 1 ) , x R N ,
(1.2)

where δ m = min | x | = 1 Φ m ( x ) , m = 1 , 2 (see [1]).

It is well known that the variational method has been an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contributions have been obtained (for example, see [113]). However, to the best of our knowledge, few people investigated system (1.1). Recently, in [1] and [14], by using the variational approach, Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian:
Δ ϕ [ Δ u ( n 1 ) ] = u F [ n , u ( n ) ] + h ( n ) ( n Z ) ,
(1.3)

where ϕ = Φ , Φ strictly convex, is a homeomorphism of R N onto the ball B a R N or of B a onto R N . By using the variational approach, under different conditions, the author found that system (1.3) has at least one or N + 1 geometrically distinct T-periodic solutions. It is interesting that Mawhin considered three kinds of ϕ: (1) ϕ : R N R N is a classical homeomorphism, for example, ϕ ( x ) = | x | p 1 x for some p > 1 and all x R N ; (2)  ϕ : R N B a ( a < + ) is a bounded homeomorphism, for example, ϕ ( x ) = x 1 + | x | 2 B 1 for all x R N ; (3) ϕ : B a R N R N is a singular homeomorphism, for example, ϕ ( x ) = x 1 | x | 2 for all x B 1 .

For a classical and bounded homeomorphism, in [14], Mawhin obtained the following multiplicity results.

Theorem A (see [14], Theorem 4.1)

Assume that the following assumptions hold:

(HB) ϕ is a homeomorphism from R N onto R N , such that ϕ ( 0 ) = 0 , ϕ = Φ , with Φ C 1 ( R N , [ 0 , + ] ) strictly convex and Φ ( 0 ) = 0 .

(HF) F C ( Z × R N , R ) , F ( n , ) C 1 ( R N , R ) , and there exist an integer T > 0 and real numbers ω 1 > 0 , ω 2 > 0 , , ω N > 0 such that
F ( t + T , u 1 + ω 1 , u 2 + ω 2 , , u N + ω N ) = F ( t , u 1 , u 2 , , u N )

for all t R and u = ( u 1 , u 2 , , u N ) R N .

If there exist γ > 0 and p > 1 such that
| Φ ( u ) | γ | u | p ( u R N ) .

Then, for any h H T such that 1 T t = 1 T h ( t ) = 0 (the definition of H T can be seen in [14]), system (1.3) has at least N + 1 geometrically distinct T-periodic solutions.

Theorem B (see [14], Theorem 4.2)

Assume that assumption (HF) and the following condition hold:

(HB)′ ϕ is a homeomorphism from R N onto B a R N ( a ( 0 , + ) ), such that ϕ ( 0 ) = 0 , ϕ = Φ , with Φ C 1 ( R N , [ 0 , + ] ) strictly convex and Φ ( 0 ) = 0 .

If Φ : R N R is coercive, h H T such that 1 T t = 1 T h ( t ) = 0 and | H | < δ , system (1.3) has at least N + 1 geometrically distinct T-periodic solutions, where δ > 0 is given by (5) in [14]and H = ( H ( n ) ) n Z H T is such that Δ H ( n ) = h ( n ) , n Z .

Obviously, (HF) implies that F is periodic on all variables u 1 , , u N . Hence, a natural question is that what will occur if F is periodic on some of variables u 1 , , u N . For differential systems, in [15] and [16], the arguments on this question have been given. In [15], Tang and Wu considered the second order Hamiltonian system
{ u ¨ ( t ) + F ( t , u ( t ) ) = e ( t ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 ,
(1.4)
and in [16], Zhang and Tang generalized and improved the results in [15]. They considered the following ordinary p-Laplacian system:
{ ( | u ( t ) | p 2 u ( t ) ) + F ( t , u ( t ) ) = e ( t ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 .
(1.5)

Inspired by [1, 14, 15] and [16], in this paper, we investigate system (1.1), which is different from (1.3), and consider the case that F ( t , x 1 , x 2 ) is periodic on some of the variables x 1 ( 1 ) , , x N ( 1 ) and some of the variables x 1 ( 1 ) , , x N ( 2 ) , where x 1 = ( x 1 ( 1 ) , , x N ( 1 ) ) τ and x 2 = ( x 1 ( 1 ) , , x N ( 2 ) ) τ . We generalize Theorem A and Theorem B.

Next, in order to present our main results, we consider two decompositions R N = R 1 S 1 and R N = R 2 S 2 with
R 1 = span e i 1 , , e i r 1 , S 1 = span e i r 1 + 1 , , e i N , R 2 = span e j 1 , , e j r 2 , S 2 = span e j r 2 + 1 , , e j N ,

where e i k and e j s are the canonical basis of R N for 1 k N , 1 s N , 1 r 1 N , and 1 r 2 N .

In this paper, we make the following assumptions:

( A 1 ) Let p > 1 , q > 1 , β 1 [ 0 , p ) , and β 2 [ 0 , q ) . Assume that there exist positive constants γ 1 , γ 2 , γ 3 , γ 4 such that
Φ 1 ( x ) γ 1 | x | p γ 2 | x | β 1 , Φ 2 ( y ) γ 3 | y | q γ 4 | y | β 2 , x , y R N .
( A 2 ) There exist positive constants d 1 , d 2 , d 3 , d 4 with d 1 > 1 p and d 3 > 1 q , β 3 [ 0 , p ) , and β 4 [ 0 , q ) such that
( ϕ 1 ( x ) , x ) d 1 | x | p d 2 | x | β 3 , ( ϕ 2 ( x ) , x ) d 3 | x | q d 4 | x | β 4 , x R N .
( A 3 ) There exist constants c m 0 > 0 , k m 1 > 0 , k m 2 > 0 , α 1 [ 0 , p 1 ) , α 2 [ 0 , q 1 ) , and two nonnegative functions w m C ( [ 0 , + ) , [ 0 , + ) ) , where m = 1 , 2 , with the properties:
  1. (i)

    w m ( s ) w m ( t ) s t , s , t [ 0 , + ) ,

     
  2. (ii)

    w m ( s + t ) c m 0 ( w m ( s ) + w m ( t ) ) s , t [ 0 , + ) ,

     
  3. (iii)

    0 w 1 ( t ) k 11 t α 1 + k 12 , 0 w 2 ( t ) k 21 t α 2 + k 22 , t [ 0 , + ) ,

     
  4. (iv)

    w m ( t ) + , as t + .

     

( F 1 ) F : Z × R N × R N R N , ( t , x 1 , x 2 ) F ( t , x 1 , x 2 ) is T-periodic in t for all ( x 1 , x 2 ) R N × R N and continuously differentiable in ( x 1 , x 2 ) for every t Z [ 1 , T ] , where x 1 = ( x 1 ( 1 ) , , x N ( 1 ) ) τ , x 2 = ( x 1 ( 2 ) , , x N ( 2 ) ) τ .

( F 2 ) F ( t , x 1 , x 2 ) is T i k ( 1 ) -periodic in x i k ( 1 ) , where x i k ( 1 ) is a component of vector x 1 and T i k ( 1 ) > 0 , 1 k r 1 , and T j s ( 2 ) -periodic in x j s ( 2 ) , where x j s ( 2 ) is a component of vector x 2 and T j s ( 2 ) > 0 , 1 s r 2 .

( F 3 ) There exist f m , g m : Z [ 1 , T ] R , m = 1 , 2 , such that
| x 1 F ( t , x 1 , x 2 ) | f 1 ( t ) w 1 ( | x 1 | ) + g 1 ( t ) , | x 2 F ( t , x 1 , x 2 ) | f 2 ( t ) w 2 ( | x 2 | ) + g 2 ( t )
for all ( x 1 , x 2 ) R N × R N and t Z [ 1 , T ] .()
t = 1 T h 1 ( t ) = t = 1 T h 2 ( t ) = 0 .
Remark 1.2 A condition similar to ( A 3 ) and ( F 3 ) was given first in [17] for the second order Hamiltonian systems
{ u ¨ ( t ) = F ( t , u ( t ) ) , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) .
(1.6)
The condition presented some advantages over the following subquadratic condition: there exist α [ 0 , 1 ) and f , g L 1 ( [ 0 , T ] ; R N ) such that
| F ( t , x ) | f ( t ) | x | α + g ( t ) .

We refer readers to [17] for more details.

Moreover, assume that p > 1 and q > 1 satisfying 1 / p + 1 / p = 1 and 1 / q + 1 / q = 1 . Let
C ( p ) = min { ( T 1 ) ( p + 1 ) / p T , ( ( T + 1 ) p + 1 2 T p ( p + 1 ) ) 1 / p } ,
(1.7)
C ( q ) = min { ( T 1 ) ( q + 1 ) / q T , ( ( T + 1 ) q + 1 2 T q ( q + 1 ) ) 1 / q } ,
(1.8)
C ( p , p ) = min { ( T 1 ) 2 p 1 T p 1 , T p 1 Θ ( p , p ) ( p + 1 ) p / p } ,
(1.9)
C ( q , q ) = min { ( T 1 ) 2 q 1 T q 1 , T q 1 Θ ( q , q ) ( q + 1 ) q / q } , Θ ( p , p ) = t = 1 T [ ( t T ) p + 1 + ( 1 t T + 1 T ) p + 1 2 T p + 1 ] p / p , Θ ( q , q ) = t = 1 T [ ( t T ) q + 1 + ( 1 t T + 1 T ) q + 1 2 T q + 1 ] q / q .
(1.10)

Next, we present our main results.

(I) For classical homeomorphism

Theorem 1.1 Assume that ( A 0 ) with a = + , ( A 1 ), ( A 3 ), ( F 1 )-( F 3 ), and () hold. Assume that F satisfies the following condition:

( F 4 ) For ( x 1 , x 2 ) S 1 × S 2 ,
lim | x 1 | + | x 2 | + t = 1 T F ( t , x 1 , x 2 ) w 1 p ( | x 1 | ) + w 2 q ( | x 2 | ) > max { [ c 10 C ( p ) ] p [ p γ 1 ] p 1 p ( t = 1 T f 1 ( t ) ) p , [ c 20 C ( q ) ] q [ q γ 3 ] q 1 q ( t = 1 T f 2 ( t ) ) q } .

Then system (1.1) has at least r 1 + r 2 + 1 geometrically distinct solutions in , where the definition of is given in Section  2 below.

Theorem 1.2 Assume that ( A 0 ) with a = + , ( A 1 ), ( A 2 ), ( A 3 ), ( F 1 )-( F 3 ), and () hold. Assume that F satisfies the following condition:

( A 1 )′ Let θ 1 [ 0 , p ) and θ 2 [ 0 , q ) . Assume that there exist positive constants ζ 1 , ζ 2 , ζ 3 , ζ 4 such that
Φ 1 ( x ) ζ 1 | x | p + ζ 2 | x | θ 1 , Φ 2 ( y ) ζ 3 | y | q + ζ 4 | y | θ 2 , x , y R N ;
( F 4 )′ For ( x 1 , x 2 ) S 1 × S 2 ,
lim | x 1 | + | x 2 | + t = 1 T F ( t , x 1 , x 2 ) w 1 p ( | x 1 | ) + w 2 q ( | x 2 | ) < max { [ C ( p ) c 10 ] p p [ 1 + p ζ 1 d 1 p 1 + 1 + q ζ 3 d 3 q 1 + 1 ] ( t = 1 T f 1 ( t ) ) p , [ C ( q ) c 20 ] q q [ 1 + p ζ 1 d 1 p 1 + 1 + q ζ 3 d 3 q 1 + 1 ] ( t = 1 T f 2 ( t ) ) q } .

Then system (1.1) has at least r 1 + r 2 + 1 geometrically distinct solutions in .

(II) For bounded homeomorphism

Theorem 1.3 Assume that ( A 0 ) with a < + , Φ m : R N R are coercive, m = 1 , 2 , ( F 1 ), ( F 2 ), and () hold. Assume that F satisfies the following conditions:

( F 5 ) There exists a nonnegative b m : Z [ 1 , T ] R + , m = 1 , 2 , such that
| x 1 F ( t , x 1 , x 2 ) | b 1 ( t ) , | x 2 F ( t , x 1 , x 2 ) | b 2 ( t )

for all ( x 1 , x 2 ) R N × R N and t Z [ 1 , T ] ;

( F 6 ) For ( x 1 , x 2 ) S 1 × S 2 ,
lim | x 1 | + | x 2 | + t = 1 T F ( t , x 1 , x 2 ) = + ;
( F 7 )
t = 1 T b 1 ( t ) + t = 1 T | h 1 ( t ) | < δ 1 C ( p ) , t = 1 T b 2 ( t ) + t = 1 T | h 2 ( t ) | < δ 2 C ( q ) ,

where δ m , m = 1 , 2 are given in (1.2). Then system (1.1) has at least r 1 + r 2 + 1 geometrically distinct solutions in .

Theorem 1.4 Assume that ( A 0 ) with a < + , Φ m : R N R are coercive, m = 1 , 2 , ( F 1 ), ( F 2 ), ( F 5 ), and () hold. If F satisfies the following conditions:

( F 6 )′ For ( x 1 , x 2 ) S 1 × S 2 ,
lim | x 1 | + | x 2 | + t = 1 T F ( t , x 1 , x 2 ) = ;
( F 7 )′
C ( p ) t = 1 T b 1 ( t ) + C ( p ) t = 1 T | h 1 ( t ) | + ( C ( p , p ) + 1 ) 1 / p < δ 1 , C ( q ) t = 1 T b 2 ( t ) + C ( q ) t = 1 T | h 2 ( t ) | + ( C ( q , q ) + 1 ) 1 / q < δ 2 ,

where δ m , m = 1 , 2 are given in (1.2), then system (1.1) has at least r 1 + r 2 + 1 geometrically distinct solutions in .

2 Preliminaries

First, we present some basic notations. We use | | to denote the usual Euclidean norm in  R N . Define
V = { u = ( u 1 , u 2 ) τ = { u ( t ) } | u ( t ) = ( u 1 ( t ) , u 2 ( t ) ) τ R 2 N , u m = { u m ( t ) } , u m ( t ) R N , m = 1 , 2 , t Z } .
is defined as a subspace of V by
H = { u = { u ( t ) } V | u ( t + T ) = u ( t ) , t Z } .
Define
H m = { u m = { u m ( t ) } | u m ( t + T ) = u m ( t ) , u m ( t ) R N , t Z } , m = 1 , 2 .
Then H = H 1 × H 2 . For u m H m , set
u m r = ( t = 1 T | u m ( t ) | r ) 1 / r and u m = max t Z [ 1 , T ] | u m ( t ) | , m = 1 , 2 , r > 1 .
Obviously, we have
u m u m 2 , m = 1 , 2 .
(2.1)
For 1 < p , q < + , on H 1 , we define
u 1 p = ( t = 1 T | Δ u 1 ( t ) | p + t = 1 T | u 1 ( t ) | p ) 1 / p
and, on H 2 , we define
u 2 q = ( t = 1 T | Δ u 2 ( t ) | q + t = 1 T | u 2 ( t ) | q ) 1 / q .
For u = ( u 1 , u 2 ) τ H , we define
u = u 1 p + u 2 q .
Let
W = { u = ( u 1 , u 2 ) τ H | u m ( 1 ) = = u m ( T ) = 1 T t = 1 T u m ( t ) , m = 1 , 2 }
and
H ˜ = { u = ( u 1 , u 2 ) τ H | t = 1 T u m ( t ) = 0 , m = 1 , 2 } .

Then can be decomposed into the direct sum H = W H ˜ . So, for any u H , u can be expressed in the form u = u ˜ + u ¯ , where u ˜ = ( u ˜ 1 , u ˜ 2 ) τ V and u ¯ = ( u ¯ 1 , u ¯ 2 ) τ W . Obviously, u m = u ˜ m + u ¯ m , m = 1 , 2 .

For u = ( u 1 , u 2 ) τ H ˜ , let
Δ u m r = ( t = 1 T | Δ u m ( t ) | r ) 1 / r ,
where m = 1 , 2 , r > 1 . It is easy to verify that
Δ u = Δ u 1 p + Δ u 2 q

is also a norm on H ˜ . Since H ˜ is finite-dimensional, the norm Δ u is equivalent to the norm u in if u H ˜ .

Lemma 2.1 (see [12])

Let u = ( u 1 , u 2 ) H ˜ . Then
max t Z [ 1 , T ] | u m ( t ) | C ( p ) ( s = 1 T | Δ u m ( s ) | p ) 1 / p , m = 1 , 2 ,
(2.2)
max t Z [ 1 , T ] | u m ( t ) | C ( q ) ( s = 1 T | Δ u m ( s ) | q ) 1 / q , m = 1 , 2 ,
(2.3)
and
t = 1 T | u m ( t ) | p C ( p , p ) s = 1 T | Δ u m ( s ) | p , m = 1 , 2 ,
(2.4)
t = 1 T | u m ( t ) | q C ( q , q ) s = 1 T | Δ u m ( s ) | q , m = 1 , 2 ,
(2.5)

where C ( p ) , C ( q ) , C ( p , p ) , and C ( q , q ) are defined by (1.7)-(1.10).

Lemma 2.2 (see [16])

Let a > 0 , b , c 0 , ε > 0 .
  1. (i)

    If α ( 0 , 1 ] , then ( a + b + c ) α a α + b α + c α ;

     
  2. (ii)
    if α ( 1 , + ) , then there exists B ( ε ) > 1 such that
    ( a + b + c ) α ( 1 + ε ) a α + B ( ε ) b α + B ( ε ) c α .
     
Lemma 2.3 For any u = ( u 1 , u 2 ) , v = ( v 1 , v 2 ) H , the following two equalities hold:
t = 1 T ( Δ ϕ 1 ( Δ u 1 ( t 1 ) ) , v 1 ( t ) ) = t = 1 T ( Δ ϕ 1 ( Δ u 1 ( t ) ) , Δ v 1 ( t ) ) ,
(2.6)
t = 1 T ( Δ ϕ 2 ( Δ u 2 ( t 1 ) ) , v 2 ( t ) ) = t = 1 T ( Δ ϕ 2 ( Δ u 2 ( t ) ) , Δ v 2 ( t ) ) .
(2.7)
Proof In fact, since u 1 ( t ) = u 1 ( t + T ) and v 1 ( t ) = v 1 ( t + T ) for all t Z , we have
t = 1 T ( Δ ϕ 1 ( Δ u 1 ( t 1 ) ) , v 1 ( t ) ) = t = 1 T ( ϕ 1 ( Δ u 1 ( t ) ) , v 1 ( t ) ) + t = 1 T ( ϕ 1 ( Δ u 1 ( t 1 ) ) , v 1 ( t ) ) = t = 1 T ( ϕ 1 ( Δ u 1 ( t ) ) , v 1 ( t ) ) + t = 1 T 1 ( ϕ 1 ( Δ u 1 ( t ) ) , v 1 ( t + 1 ) ) + ( ϕ 1 ( Δ u 1 ( 0 ) ) , v 1 ( 1 ) ) = t = 1 T ( ϕ 1 ( Δ u 1 ( t ) ) , Δ v 1 ( t ) ) + ( ϕ 1 ( Δ u 1 ( 0 ) ) , v 1 ( 1 ) ) ( ϕ 1 ( Δ u 1 ( T ) ) , v 1 ( T + 1 ) ) = t = 1 T ( ϕ 1 ( Δ u 1 ( t ) ) , Δ v 1 ( t ) ) .

Hence, (2.6) holds. Similarly, it is easy to obtain (2.7). The proof is complete. □

Lemma 2.4 Let L : Z [ 1 , T ] × R N × R N × R N × R N R , ( t , x 1 , x 2 , y 1 , y 2 ) L ( t , x 1 , x 2 , y 1 , y 2 ) and assume that L is continuously differential in ( x 1 , x 2 , y 1 , y 2 ) for all t Z [ 1 , T ] . Then the function φ : H R defined by
φ ( u ) = φ ( u 1 , u 2 ) = t = 1 T L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) )
is continuously differentiable on and
φ ( u ) , v = φ ( u 1 , u 2 ) , ( v 1 , v 2 ) = t = 1 T [ ( D x 1 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , v 1 ( t ) ) + ( D y 1 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , Δ v 1 ( t ) ) + ( D x 2 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , v 2 ( t ) ) + ( D y 2 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , Δ v 2 ( t ) ) ] ,

where u , v H .

Proof Define G : [ 1 , 1 ] × Z [ 1 , T ] R , ( λ , t ) G ( λ , t ) by
G ( λ , t ) = L ( t , u 1 ( t ) + λ v 1 ( t ) , u 2 ( t ) + λ v 2 ( t ) , Δ u 1 ( t ) + λ Δ v 1 ( t ) , Δ u 2 ( t ) + λ Δ v 2 ( t ) ) .
Since L is continuously differential in ( x 1 , x 2 , y 1 , y 2 ) for all t Z [ 1 , T ] , G ( λ , t ) is differential in λ and
G ( λ , t ) = ( D x 1 L ( t , u 1 ( t ) + λ v 1 ( t ) , u 2 ( t ) + λ v 2 ( t ) , Δ u 1 ( t ) + λ Δ v 1 ( t ) , Δ u 2 ( t ) + λ Δ v 2 ( t ) ) , v 1 ( t ) ) + ( D x 2 L ( t , u 1 ( t ) + λ v 1 ( t ) , u 2 ( t ) + λ v 2 ( t ) , Δ u 1 ( t ) + λ Δ v 1 ( t ) , Δ u 2 ( t ) + λ Δ v 2 ( t ) ) , v 2 ( t ) ) + ( D y 1 L ( t , u 1 ( t ) + λ v 1 ( t ) , u 2 ( t ) + λ v 2 ( t ) , Δ u 1 ( t ) + λ Δ v 1 ( t ) , Δ u 2 ( t ) + λ Δ v 2 ( t ) ) , Δ v 1 ( t ) ) + ( D y 2 L ( t , u 1 ( t ) + λ v 1 ( t ) , u 2 ( t ) + λ v 2 ( t ) , Δ u 1 ( t ) + λ Δ v 1 ( t ) , Δ u 2 ( t ) + λ Δ v 2 ( t ) ) , Δ v 2 ( t ) ) .
Hence,
φ ( u ) , v = lim λ 0 φ ( u + λ v ) φ ( u ) λ = lim λ 0 t = 1 T G ( λ , t ) t = 1 T G ( 0 , t ) λ = t = 1 T lim λ 0 G ( λ , t ) G ( 0 , t ) λ = t = 1 T G ( 0 , t ) = t = 1 T [ ( D x 1 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , v 1 ( t ) ) + ( D y 1 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , Δ v 1 ( t ) ) + ( D x 2 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , v 2 ( t ) ) + ( D y 2 L ( t , u 1 ( t ) , u 2 ( t ) , Δ u 1 ( t ) , Δ u 2 ( t ) ) , Δ v 2 ( t ) ) ] .

The proof is complete. □

Let
L ( t , x 1 , x 2 , y 1 , y 2 ) = Φ 1 ( y 1 ) + Φ 2 ( y 2 ) + F ( t , x 1 , x 2 ) + ( h 1 ( t ) , x 1 ) + ( h 2 ( t ) , x 2 ) .
Then
φ ( u ) = φ ( u 1 , u 2 ) = t = 1 T [ Φ 1 ( Δ u 1 ( t ) ) + Φ 2 ( Δ u 2 ( t ) ) + F ( t , u 1 ( t ) , u 2 ( t ) ) + ( h 1 ( t ) , u 1 ( t ) ) + ( h 2 ( t ) , u 2 ( t ) ) ] .
(2.8)
It follows from ( A 0 ), ( F 1 ), and Lemma 2.4 that
φ ( u ) , v = φ ( u 1 , u 2 ) , ( v 1 , v 2 ) = t = 1 T [ ( ϕ 1 ( Δ u 1 ( t ) ) , Δ v 1 ( t ) ) + ( ϕ 2 ( Δ u 2 ( t ) ) , Δ v 2 ( t ) ) + ( u 1 F ( t , u 1 ( t ) , u 2 ( t ) ) , v 1 ( t ) ) + ( u 2 F ( t , u 1 ( t ) , u 2 ( t ) ) , v 2 ( t ) ) + ( h 1 ( t ) , v 1 ( t ) ) + ( h 2 ( t ) , v 2 ( t ) ) ] , u , v H .
(2.9)

By Lemma 2.3, it is easy to see that the critical points of φ in are periodic solutions of system (1.1).

Next, we recall a definition. Let G be a discrete subgroup of a Banach space X and let π : X X / G be the canonical surjection. A subset A of X is G-invariant if π 1 ( π ( A ) ) = A . A function f defined on X is G-invariant if f ( u + g ) = f ( u ) for every u X and every g G (see [18]).

Definition 2.1 (see [18], Definition 4.2)

A G-invariant differentiable functional φ : X R satisfies the (PS) G condition, if for every sequence { u k } in X such that φ ( u k ) is bounded and φ ( u k ) 0 , the sequence { π ( u k ) } contains a convergent subsequence.

We will use the following two lemmas to obtain the critical points of φ.

Lemma 2.5 (see [18], Theorem 4.12)

Let φ C 1 ( x , R ) be a G-invariant functional satisfying the (PS) G condition. If φ is bounded from below and if the dimension N of the space generated by G is finite, then φ has at least N + 1 critical orbits.

Lemma 2.6 (see [19])

Let X be a Banach space and have a decomposition: X = Y + Z where Y and Z are two subspaces of X with dim Y < + . Let V be a finite-dimensional, compact C 2 -manifold without boundary. Let f : X × V R be a C 1 -function and satisfy the (PS) condition. Suppose that f satisfies
inf u Z × V f ( u ) a , sup u S × V f ( u ) b < a ,

where S = D , D = { u Y | u R } , R, a, and b are constants. Then the function f has at least cuplength ( V ) + 1 critical points.

Let
u ˆ m ( t ) = P m u ¯ m + Q m u ¯ m + u ˜ m , m = 1 , 2 ,
where
P 1 u ¯ 1 = k = r 1 + 1 N ( u ¯ 1 , e i k ) e i k , Q 1 u ¯ 1 = k = 1 r 1 [ ( u ¯ 1 , e i k ) m i k T i k ] e i k , P 2 u ¯ 2 = s = r 2 + 1 N ( u ¯ 2 , e j s ) e j s , Q 2 u ¯ 2 = s = 1 r 2 [ ( u ¯ 2 , e j s ) m j s T j s ] e j s ,
and m i k , m j s are the unique integers such that
0 ( u ¯ 1 , e i k ) m i k T i k < T i k , 1 k r 1 , 0 ( u ¯ 2 , e j s ) m j s T j s < T j s , 1 s r 2 .
Let
G = { g = ( g 1 g 2 ) R N × R N | ( g 1 g 2 ) = ( k = 1 r 1 m i k T i k e i k s = 1 r 2 m j s T j s e j s ) , ( g 1 g 2 ) m i k  and  m j s  are integers , 1 k r 1 , 1 s r 2 } .
(2.10)
Let Z = H ˜ , Y = S 1 × S 2 , X = Y + Z , and V is the quotient space ( R 1 × R 2 ) / G which is isomorphic to the torus T r 1 + r 2 . Now define Ψ : X × T r 1 + r 2 R by
Ψ ( ( y + z ( t ) , v ) ) = φ ( y + v + z ( t ) ) , ( y , z , v ) Y × Z × T r 1 + r 2 ,
(2.11)
that is,
Ψ ( ( y + z ( t ) , v ) ) = φ ( y 1 + v 1 + z 1 ( t ) , y 2 + v 2 + z 2 ( t ) ) ,
(2.12)
where
y = ( y 1 , y 2 ) τ Y , v = ( v 1 , v 2 ) τ V , z = z ( t ) = ( z 1 ( t ) , z 2 ( t ) ) τ Z , y + v + z ( t ) = ( y 1 + v 1 + z 1 ( t ) , y 2 + v 2 + z 2 ( t ) ) τ .
It is easy to verify that Ψ is continuously differentiable and that
Ψ ( ( y [ 1 ] + z [ 1 ] ( t ) , v [ 1 ] ) ) , ( y [ 2 ] + z [ 2 ] ( t ) , v [ 2 ] ) = φ ( y [ 1 ] + v [ 1 ] + z [ 1 ] ( t ) ) , y [ 2 ] + v [ 2 ] + z [ 2 ] ( t ) = φ ( y 1 [ 1 ] + v 1 [ 1 ] + z 1 [ 1 ] ( t ) , y 2 [ 1 ] + v 2 [ 1 ] + z 2 [ 1 ] ( t ) ) , ( y 1 [ 2 ] + v 1 [ 2 ] + z 1 [ 2 ] ( t ) , y 2 [ 2 ] + v 2 [ 2 ] + z 2 [ 2 ] ( t ) ) , ( y [ m ] , z [ m ] , v [ m ] ) Y × Z × T r 1 + r 2 , m = 1 , 2 .
(2.13)
Then ( F 2 ) implies that
F ( t + T , x 1 + g 1 , x 2 + g 2 ) = F ( t , x 1 , x 2 ) , t Z  and  g G .
Hence, we have
F ( t , u 1 ( t ) , u 2 ( t ) ) = F ( t , u ˆ 1 ( t ) + k = 1 r 1 m i k T i k e i k , u ˆ 2 ( t ) + s = 1 r 2 m j s T j s e j s ) = F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) ,
(2.14)
F ( t , u 1 ( t ) , u 2 ( t ) ) = F ( t , u ˆ 1 ( t ) + k = 1 r 1 m i k T i k e i k , u ˆ 2 ( t ) + s = 1 r 2 m j s T j s e j s ) = F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) )
(2.15)
and, by (), we have
t = 1 T ( h 1 ( t ) , u 1 ( t ) ) = t = 1 T ( h 1 ( t ) , u ˆ 1 ( t ) + k = 1 r 1 m i k T i k e i k ) = t = 1 T ( h 1 ( t ) , u ˆ 1 ( t ) ) ,
(2.16)
t = 1 T ( h 2 ( t ) , u 2 ( t ) ) = t = 1 T ( h 2 ( t ) , u ˆ 2 ( t ) + k = 1 r 1 m i k T i k e i k ) = t = 1 T ( h 2 ( t ) , u ˆ 2 ( t ) ) .
(2.17)

Hence φ ( u ) = φ ( u ˆ ) and φ ( u ) = φ ( u ˆ ) .

3 Proofs

For the sake of convenience, we denote by C i j and D i j , i = 1 , 2 , j = 0 , 1 , , 9 below the various positive constants, by C i j ( ε ) and D i j ( ε ) , i = 1 , 2 , j = 0 , 1 , , 9 below the various positive constants depending on ε and
M 11 = t = 1 T f 1 ( t ) , M 12 = t = 1 T g 1 ( t ) , M 13 = ( k = 1 r 1 T i k 2 ) 1 / 2 , M 14 = t = 1 T | h 1 ( t ) | , M 15 = t = 1 T b 1 ( t ) , M 21 = t = 1 T f 2 ( t ) , M 22 = t = 1 T g 2 ( t ) , M 23 = ( s = 1 r 2 T j s 2 ) 1 / 2 , M 24 = t = 1 T | h 2 ( t ) | , M 25 = t = 1 T b 2 ( t ) .
Proof of Theorem 1.1 It follows from ( F 4 ) that there exist a 1 > C ( p ) p γ 1 and a 2 > C ( q ) q γ 3 such that
lim | x 1 | + | x 2 | F ( t , x 1 , x 2 ) w 1 p ( | x 1 | ) + w 2 q ( | x 2 | ) > max { c 10 p M 11 p a 1 p / p C ( p ) p , c 20 q M 21 q a 2 q / q C ( q ) q } ,
(3.1)
for ( x 1 , x 2 ) S 1 × S 2 . It follows from ( A 3 ), ( F 3 ), Lemma 2.1, and Lemma 2.2 that
t = 1 T | F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 ) | t = 1 T | F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) F ( t , P 1 u ¯ 1 , u ˆ 2 ( t ) ) | + t = 1 T | F ( t , P 1 u ¯ 1 , u ˆ 2 ( t ) ) F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 ) | t = 1 T | 0 1 ( x 1 F ( t , P 1 u ¯ 1 + s ( Q 1 u ¯ 1 + u ˜ 1 ( t ) ) , u ˆ 2 ( t ) ) , Q 1 u ¯ 1 + u ˜ 1 ( t ) ) d s | + t = 1 T | 0 1 ( x 2 F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 + s ( Q 2 u ¯ 2 + u ˜ 2 ( t ) ) ) , Q 2 u ¯ 2 + u ˜ 2 ( t ) ) d s | ( | Q 1 u ¯ 1 | + u ˜ 1 ) t = 1 T 0 1 | x 1 F ( t , P 1 u ¯ 1 + s ( Q 1 u ¯ 1 + u ˜ 1 ( t ) ) , u ˆ 2 ( t ) ) | d s + ( | Q 2 u ¯ 2 | + u ˜ 2 ) t = 1 T 0 1 | x 2 F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 + s ( Q 2 u ¯ 2 + u ˜ 2 ( t ) ) ) | d s ( M 13 + u ˜ 1 ) t = 1 T 0 1 [ f 1 ( t ) w 1 ( | P 1 u ¯ 1 + s ( Q 1 u ¯ 1 + u ˜ 1 ( t ) ) | ) + g 1 ( t ) ] d s + ( M 23 + u ˜ 2 ) t = 1 T 0 1 [ f 2 ( t ) w 2 ( | P 2 u ¯ 2 + s ( Q 2 u ¯ 2 + u ˜ 2 ( t ) ) | ) + g 2 ( t ) ] d s ( M 13 + u ˜ 1 ) t = 1 T 0 1 [ c 10 f 1 ( t ) w 1 ( | P 1 u ¯ 1 | ) + c 10 f 1 ( t ) w 1 ( s | Q 1 u ¯ 1 + u ˜ 1 ( t ) | ) ] d s + ( M 23 + u ˜ 2 ) t = 1 T 0 1 [ c 20 f 2 ( t ) w 2 ( | P 2 u ¯ 2 | ) + c 20 f 2 ( t ) w 2 ( s | Q 2 u ¯ 2 + u ˜ 2 ( t ) | ) ] d s + ( M 13 + u ˜ 1 ) t = 1 T 0 1 g 1 ( t ) d s + ( M 23 + u ˜ 2 ) t = 1 T 0 1 g 2 ( t ) d s ( M 13 + u ˜ 1 ) w 1 ( | P 1 u ¯ 1 | ) t = 1 T c 10 f 1 ( t ) + ( M 23 + u ˜ 2 ) w 2 ( | P 2 u ¯ 2 | ) t = 1 T c 20 f 2 ( t ) + ( M 13 + u ˜ 1 ) t = 1 T 0 1 [ c 10 f 1 ( t ) k 11 | s ( Q 1 u ¯ 1 + u ˜ 1 ( t ) ) | α 1 + f 1 ( t ) c 10 k 12 + g 1 ( t ) ] d s + ( M 23 + u ˜ 2 ) t = 1 T 0 1 [ c 20 f 2 ( t ) k 21 | s ( Q 2 u ¯ 2 + u ˜ 2 ( t ) ) | α 2 + f 2 ( t ) c 20 k 22 + g 2 ( t ) ] d s ( M 13 + u ˜ 1 ) w 1 ( | P 1 u ¯ 1 | ) t = 1 T c 10 f 1 ( t ) + 1 + ε 1 α 1 + 1 ( M 13 + u ˜ 1 ) t = 1 T f 1 ( t ) c 10 k 11 | Q 1 u ¯ 1 | α 1 + B ( ε 1 ) α 1 + 1 ( M 13 + u ˜ 1 ) t = 1 T f 1 ( t ) c 10 k 11 | u ˜ 1 ( t ) | α 1 + ( M 13 + u ˜ 1 ) t = 1 T [ f 1 ( t ) c 10 k 12 + g 1 ( t ) ] + ( M 23 + u ˜ 2 ) w 2 ( | P 2 u ¯ 2 | ) t = 1 T c 20 f 2 ( t ) + 1 + ε 2 α 2 + 1 ( M 23 + u ˜ 2 ) t = 1 T f 2 ( t ) c 20 k 21 | Q 2 u ¯ 2 | α 2 + B ( ε 2 ) α 2 + 1 ( M 23 + u ˜ 2 ) t = 1 T f 2 ( t ) c 20 k 21 | u ˜ 2 ( t ) | α 2 + ( M 23 + u ˜ 2 ) t = 1 T [ f 2 ( t ) c 20 k 22 + g 2 ( t ) ] M 11 c 10 w 1 ( | P 1 u ¯ 1 | ) u ˜ 1 + M 11 M 13 c 10 w 1 ( | P 1 u ¯ 1 | ) + ( 1 + ε 1 ) M 13 α 1 + 1 c 10 k 11 M 11 α 1 + 1 + ( 1 + ε 1 ) M 13 α 1 c 10 k 11 M 11 α 1 + 1 u ˜ 1 + B ( ε 1 ) M 13 c 10 k 11 M 11 α 1 + 1 u ˜ 1 α 1 + B ( ε 1 ) c 10 k 11 M 11 α 1 + 1 u ˜ 1 α 1 + 1 + M 13 t = 1 T [ f 1 ( t ) c 10 k 12 + g 1 ( t ) ] + u ˜ 1 t = 1 T [ f 1 ( t ) c 10 k 12 + g 1 ( t ) ] + M 21 c 20 w 2 ( | P 2 u ¯ 2 | ) u ˜ 2 + M 21 M 23 c 20 w 2 ( | P 2 u ¯ 2 | ) + ( 1 + ε 2 ) M 23 α 2 + 1 c 20 k 21 M 21 α 2 + 1 + ( 1 + ε 2 ) M 23 α 2 c 20 k 21 M 21 α 2 + 1 u ˜ 2 + B ( ε 2 ) M 23 c 20 k 21 M 21 α 2 + 1 u ˜ 2 α 2 + B ( ε 2 ) c 20 k 21 M 21 α 2 + 1 u ˜ 2 α 2 + 1 + M 23 t = 1 T [ c 20 f 2 ( t ) k 22 + g 2 ( t ) ] + u ˜ 2 t = 1 T [ c 20 f 2 ( t ) k 22 + g 2 ( t ) ] 1 p a 1 ( 1 C ( p ) ) p / p u ˜ 1 p + M 11 p c 10 p a 1 p / p C ( p ) p w 1 p ( | P 1 u ¯ 1 | ) + C 11 ( ε 1 ) u ˜ 1 α 1 + 1 + C 12 ( ε 1 ) u ˜ 1 α 1 + C 13 ( ε 1 ) u ˜ 1 + C 14 + 1 q a 2 ( 1 C ( q ) ) q / q u ˜ 2 q + M 21 q c 20 q a 2 q / q C ( q ) q w 2 q ( | P 2 u ¯ 2 | ) + C 21 ( ε 2 ) u ˜ 2 α 2 + 1 + C 22 ( ε 2 ) u ˜ 2 α 2 + C 23 ( ε 2 ) u ˜ 2 + C 24 + M 11 M 13 c 10 w 1 ( | P 1 u ¯ 1 | ) + M 21 M 23 c 20 w 2 ( | P 2 u ¯ 2 | ) C ( p ) p a 1 t = 1 T | Δ u 1 ( t ) | p + M 11 p c 10 p a 1 p / p C ( p ) p w 1 p ( | P 1 u ¯ 1 | ) + C 11 ( ε 1 ) [ C ( p ) ] α 1 + 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 + 1 p + C 14 + C 13 ( ε 1 ) C ( p ) ( t = 1 T | Δ u 1 ( t ) | p ) 1 p + C 12 ( ε 1 ) [ C ( p ) ] α 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 p + C ( q ) q a 2 t = 1 T | Δ u 2 ( t ) | q + M 21 q c 20 q a 2 q / q C ( q ) q w 2 q ( | P 2 u ¯ 2 | ) + C 21 ( ε 2 ) [ C ( q ) ] α 2 + 1 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 + 1 q + C 24 + C 23 ( ε 2 ) C ( q ) ( t = 1 T | Δ u 2 ( t ) | q ) 1 q + C 22 ( ε 2 ) [ C ( q ) ] α 2 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 q + M 11 M 13 c 10 w 1 ( | P 1 u ¯ 1 | ) + M 21 M 23 c 20 w 2 ( | P 2 u ¯ 2 | ) .
(3.2)
By ( A 1 ), (3.2), and Lemma 2.1, we have
φ ( u ) = φ ( u ˆ 1 , u ˆ 2 ) = t = 1 T [ Φ 1 ( Δ u 1 ( t ) ) + Φ 2 ( Δ u 2 ( t ) ) + F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) + ( h 1 ( t ) , u ˆ 1 ( t ) ) + ( h 2 ( t ) , u ˆ 2 ( t ) ) ] = t = 1 T [ Φ 1 ( Δ u 1 ( t ) ) + Φ 2 ( Δ u 2 ( t ) ) + F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 ) + F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 ) + ( h 1 ( t ) , u ˆ 1 ( t ) ) + ( h 2 ( t ) , u ˆ 2 ( t ) ) ] t = 1 T ( γ 1 | Δ u 1 ( t ) | p + γ 3 | Δ u 2 ( t ) | q γ 2 | Δ u 1 ( t ) | β 1 γ 4 | Δ u 2 ( t ) | β 2 ) C ( p ) p a 1 t = 1 T | Δ u 1 ( t ) | p M 11 p c 10 p a 1 p / p C ( p ) p w 1 p ( | P 1 u ¯ 1 | ) C 11 ( ε 1 ) [ C ( p ) ] α 1 + 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 + 1 p C 14 u ˜ 1 t = 1 T | h 1 ( t ) | C 13 ( ε 1 ) C ( p ) ( t = 1 T | Δ u 1 ( t ) | p ) 1 p C 12 ( ε 1 ) [ C ( p ) ] α 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 p C ( q ) q a 2 t = 1 T | Δ u 2 ( t ) | q M 21 q c 20 q a 2 q / q C ( q ) q w 2 q ( | P 2 u ¯ 2 | ) C 21 ( ε 2 ) [ C ( q ) ] α 2 + 1 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 + 1 q C 24 u ˜ 2 t = 1 T | h 2 ( t ) | C 23 ( ε 2 ) C ( q ) ( t = 1 T | Δ u 2 ( t ) | q ) 1 q C 22 ( ε 2 ) [ C ( q ) ] α 2 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 q M 11 M 13 c 10 w 1 ( | P 1 u ¯ 1 | ) M 21 M 23 c 20 w 2 ( | P 2 u ¯ 2 | ) ( γ 1 C ( p ) p a 1 ) t = 1 T | Δ u 1 ( t ) | p M 11 M 13 c 10 w 1 ( | P 1 u ¯ 1 | ) C 11 ( ε 1 ) [ C ( p ) ] α 1 + 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 + 1 p C 12 ( ε 1 ) [ C ( p ) ] α 1 ( t = 1 T | Δ u 1 ( t ) | p ) α 1 p C 13 ( ε 1 ) C ( p ) ( t = 1 T | Δ u 1 ( t ) | p ) 1 p γ 2 T 1 β 1 p ( t = 1 T | Δ u 1 ( t ) | p ) β 1 p C 14 + ( γ 3 C ( q ) q a 2 ) t = 1 T | Δ u 2 ( t ) | q M 21 M 23 c 20 w 2 ( | P 2 u ¯ 2 | ) C 21 ( ε 2 ) [ C ( q ) ] α 2 + 1 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 + 1 q C 22 ( ε 2 ) [ C ( q ) ] α 2 ( t = 1 T | Δ u 2 ( t ) | q ) α 2 q C 23 ( ε 2 ) C ( q ) ( t = 1 T | Δ u 2 ( t ) | q ) 1 q γ 4 T 1 β 2 q ( t = 1 T | Δ u 2 ( t ) | q ) β 2 q C 24 + [ w 1 p ( | P 1 u ¯ 1 | ) + w 2 q ( | P 2 u ¯ 2 | ) ] [ F ( t , P 1 u ¯ 1 , P 2 u ¯ 2 ) w 1 p ( | P 1 u ¯ 1 | ) + w 2 q ( | P 2 u ¯ 2 | ) max { M 11 p c 10 p a 1 p / p C ( p ) p , M 21 q c 20 q a 2 q / q C ( q ) q } ] C ( p ) ( t = 1 T | Δ u 1 ( t ) | p ) 1 p t = 1 T | h 1 ( t ) | C ( q ) ( t = 1 T | Δ u 2 ( t ) | q ) 1 q t = 1 T | h 2 ( t ) | .
(3.3)

It follows from (3.1), (3.3), a 1 > C ( p ) p γ 1 , and a 2 > C ( q ) q γ 3 that φ is bounded from below. Let G be a discrete subgroup of defined by (2.10) and let π : H H / G be the canonical surjection. By (2.14)-(2.17), it is easy to verify that φ is G-invariant. In what follows, we show that the functional φ satisfies the (PS) G condition, that is, for every sequence { u m } in such that { φ ( u n ) } is bounded and φ ( u n ) 0 , the sequence { π ( u n ) } has a convergent subsequence. In fact, the boundedness of φ ( u n ) , (3.1), (3.3), and the facts that a 1 > C ( p ) p γ 1 and a 2 > C ( q ) q γ 3 imply that ( P u ¯ n ) and t = 1 T | Δ u n ( t ) | 2 are bounded. Furthermore, by Lemma 2.1, we know that ( u ˜ n ) is also bounded. Hence { u ˆ n } is bounded in . Since dim H < , we know that { u ˆ n } has a convergent subsequence. Since π ( u n ) = π ( u ˆ n ) , { π ( u n ) } also has a convergent subsequence. Thus, by Lemma 2.5, we know that φ has r 1 + r 2 + 1 critical orbits. Hence, system (1.1) has at least r 1 + r 2 + 1 geometrically distinct solutions in . The proof is complete. □

Proof of Theorem 1.2 First, we prove that Ψ defined by (2.11) satisfies the (PS) condition. Assume that { ( y [ n ] + z [ n ] , v [ n ] ) } n = 1 X × T r 1 + r 2 is (PS) sequence for Ψ, that is, { Ψ ( ( y [ n ] + z [ n ] , v [ n ] ) ) } is bounded and Ψ ( ( y [ n ] + z [ n ] , v [ n ] ) ) 0 , where y [ n ] = ( y 1 [ n ] , y 2 [ n ] ) τ Y , z [ n ] = z [ n ] ( t ) = ( z 1 [ n ] ( t ) , z 2 [ n ] ( t ) ) τ Z , v [ n ] = ( v 1 [ n ] , v 2 [ n ] ) τ T r 1 + r 2 for n = 1 , 2 ,  . Let
u [ n ] = y [ n ] + v [ n ] + z [ n ] = ( y 1 [ n ] + v 1 [ n ] + z 1 [ n ] , y 2 [ n ] + v 2 [ n ] + z 2 [ n ] ) τ , n = 1 , 2 .
Then it is easy to see that
y m [ n ] = P m u ¯ m [ n ] , v m [ n ] = Q m u ¯ m [ n ] , z m [ n ] ( t ) = u ˜ m [ n ] ( t ) , m = 1 , 2 , n = 1 , 2 .
By (2.12) and (2.13), we find that { φ ( u 1 [ n ] , u 2 [ n ] ) } is bounded and φ ( u 1 [ n ] , u 2 [ n ] ) 0 . Then there exists a positive constant D 0 such that
| φ ( u 1 [ n ] , u 2 [ n ] ) | D 0 , φ ( u 1 [ n ] , u 2 [ n ] ) D 0 , n R .
(3.4)
By ( F 4 )′, there exist a 3 > C ( p ) and a 4 > C ( q ) such that
lim | x 1 | + | x 2 | F ( t , x 1 , x 2 ) w 1 p ( | x 1 | ) + w 2 q ( | x 2 | ) < max { [ a 3 M 11 c 10 ] p p [ 1 + p ζ 1 d 1 p 1 + 1 + q ζ 3 d 3 q 1 + 1 ] , [ a 4 M 21 c 20 ] q q [ 1 + p ζ 1 d 1 p 1 + 1 + q ζ 3 d 3 q 1 + 1 ] } .
(3.5)
It follows from ( F 3 ), Lemma 2.1, and Young’s inequality that, for all ( u 1 , u 2 ) H ,
| t = 1 T ( x 1 F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) , u ˜ 1 ( t ) ) + t = 1 T ( x 2 F ( t , u ˆ 1 ( t ) , u ˆ 2 ( t ) ) , u ˜ 2 ( t ) ) | | t = 1 T ( x 1 F ( t , P 1 u ¯ 1 + Q 1 u ¯ 1 + u ˜ 1 ( t ) , u ˆ 2 ( t ) ) , u ˜ 1 ( t ) ) | + | t = 1 T ( x 2 F ( t , u ˆ 1 ( t ) , P 2 u ¯ 2 + Q 2 u ¯ 2 + u ˜ 2 ( t ) ) , u ˜ 2 ( t ) ) | t = 1 T f 1 ( t ) w 1 ( | P 1 u ¯ 1 + Q 1 u ¯ 1 + u ˜ 1 ( t ) | ) | u ˜ 1 ( t ) | + t = 1 T g 1 ( t ) | u ˜ 1 ( t ) | + t = 1 T f 2 ( t ) w 2 ( | P 2 u ¯ 2 + Q 2 u ¯ 2 + u ˜ 2 ( t ) | ) | u ˜ 2 ( t ) | + t = 1 T g 2 ( t ) | u ˜ 2 ( t ) | u ˜ 1 c 10 ( w 1 ( | P 1 u ¯ 1 | ) + w 1 ( | Q 1 u ¯ 1 + u ˜ 1 | ) ) t = 1 T f 1 ( t ) + u ˜ 1 t = 1 T g 1 ( t ) + u ˜ 2 c 20 ( w 2 ( | P 2 u ¯ 2 | ) + w 2 ( | Q 2 u ¯ 2 + u ˜ 2 | ) ) t =