Consider the second-order differential system with p(t)-Laplacian

{ − d d t ( | u ˙ ( t ) | p ( t ) − 2 u ˙ ( t ) ) + | u ( t ) | p ( t ) − 2 u ( t ) = ∇ F ( t , u ( t ) )     a . e .   t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ˙ ( 0 ) − u ˙ ( T ) = 0 ,

where T > 0, F: [0, T] × ℝ ^N → ℝ, and p(t) ∈ C([0, T], ℝ⁺) satisfies the following assumptions:

(A) p(0) = p(T) and p - : = min 0 ≤ t ≤ T p ( t ) > 1 , where q ⁺ > 1 which satisfies 1/p ^- + 1/q ⁺ = 1.

Moreover, we suppose that F: [0, T] × ℝ ^N → ℝ satisfies the following assumptions:

(A') F(t, x) is measurable in t for every x ∈ ℝ ^N and continuously differentiable in x for a.e. t ∈ [0, T], and there exist a ∈ C(ℝ⁺, ℝ⁺), b ∈ L ¹(0, T; ℝ⁺), such that

| F ( t , x ) | ≤ a ( | x | ) b ( t ) , | ∇ F ( t , x ) | ≤ a ( | x | ) b ( t )

for all x ∈ ℝ ^N and a.e. t ∈ [0, T].

The operator d d t ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) ) is said to be p(t)-Laplacian, and becomes p-Laplacian when p(t) ≡ p (a constant). The p(t)-Laplacian possesses more complicated nonlinearity than p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field. Another field of application of equations with variable exponent growth conditions is image processing (see 1 2 ). The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to 3 4 5 6 7 8 9 10 11 12 for an overview on this subject.

In 2003, Fan and Fan 13 studied the ordinary p(t)-Laplacian system and introduced a generalized Orlicz-Sobolev space W T 1 , p ( t ) , which is different from the usual space W T 1 , p , then Wang and Yuan 14 obtained the existence and multiplicity of periodic solutions for ordinary p(t)-Laplacian system under the generalized Ambrosetti-Rabinowitz conditions. Fountain and Dual Fountain theorems were established by Bartsch and Willem 15 16 , and both theorems are effective tools for studying the existence of infinitely many large energy solutions and small energy solutions. When we impose some suitable conditions on the growth of the potential function at origin or at infinity, we get three multiplicity results of infinitely many periodic solutions for system (1.1) using the Fountain theorem, the Dual Fountain theorem, and the Symmetric Mountain Pass theorem.

The rest of the article is divided as follows: Basic definitions and preliminary results are collected in Second 2. The main results and proofs are given in Section 3. The three examples are presented in Section 4 for illustrating our results.

In this article, we denote by p + : = max 0 ≤ t ≤ T p ( t ) > 1 throughout this article, and we use 〈·, ·〉 and |·| to denote the usual inner product and norm in ℝ ^N , respectively.

In this section, we recall some known results in nonsmooth critical point theory, and the properties of space W T 1 , p ( t ) are listed for the convenience of readers.

Definition 2.1 14 . Let p(t) satisfies the condition (A), define

L p ( t ) ( [ 0 , T ] , ℝ N ) = u ∈ L 1 ( [ 0 , T ] , ℝ N ) : ∫ 0 T | u | p ( t ) d t < ∞

with the norm

| u | p ( t ) : = inf λ > 0 : ∫ 0 T u λ p ( t ) d t ≤ 1 .

For u ∈ L l o c 1 ( [ 0 , T ] , ℝ N ) , let u' denote the weak derivative of u, if u ′ ∈ L loc 1 ( [ 0 , T ] , ℝ N ) and satisfies

∫ 0 T u ′ ϕ d t = - ∫ 0 T u ϕ ′ d t , ∀ ϕ ∈ C 0 ∞ ( [ 0 , T ] , ℝ N ) .

Define

W 1 , p ( t ) ( [ 0 , T ] , ℝ N ) = { u ∈ L p ( t ) ( [ 0 , T ] , ℝ N ) : u ′ ∈ L p ( t ) ( [ 0 , T ] , ℝ N ) } ⊳

with the norm u W 1 , p ( t ) : = | u | p ( t ) + | u ′ | p ( t ) .

In this article, we will use the following equivalent norm on W ^{1, p(t)} ([0, T], ℝ ^N ), i.e.,

u : = inf λ > 0 : ∫ 0 T u λ p ( t ) + u ˙ λ p ( t ) d t ≤ 1 ,

and some lemmas given in the following section have been proven under the norm of u W 1 , p ( t ) , and it is obvious that they also hold under the norm ||u||.

Remark 2.1. If p(t) = p, where p ∈ (1, ∞) is a constant, by the definition of |u| _p(t), it is easy to get | u | p = ( ∫ 0 T | u ( t ) | p d t ) 1 ∕ p , which is the same with the usual norm in space L ^p .

The space L ^p(t) is a generalized Lebesgue space, and the space W ^{1, p(t)} is a generalized Sobolev space. Because most of the following lemmas have appeared in 13 14 17 18 , we omit their proofs.

Lemma 2.1 13 . L ^p(t) and W ^{1, p(t)} are both Banach spaces with the norms defined above, when p ^- > 1, they are reflexive.

Lemma 2.2 14 . (i) The space L ^p(t) is a separable, uniform convex Banach space, its conjugate space is L ^q(t), for any u ∈ L ^p(t) and v ∈ L ^q(t), we have

∫ 0 T u v d t ≤ 2 | u | p ( t ) | v | q ( t ) ,

where 1 p ( t ) + 1 q ( t ) = 1 .

(ii) If p ₁(t) and p ₂(t) ∈ C([0, T], ℝ⁺) and p ₁(t) ≤ p ₂(t) for any t ∈ [0, T], then L p 2 ( t ) → L p 1 ( t ) , and the embedding is continuous.

Lemma 2.3 14 . If we denote ρ ( u ) = ∫ 0 T | u ( t ) | p ( t ) d t , ∀ u ∈ L ^p(t), then

(i) |u| _p(t) < 1 (= 1; > 1) ⇔ ρ(u) < 1 (= 1; > 1);

(ii) | u | p ( t ) > 1 ⇒ | u | p ( t ) p - ≤ ρ ( u ) ≤ | u | p ( t ) p + , | u | p ( t ) < 1 ⇒ | u | p ( t ) p + ≤ ρ ( u ) ≤ | u | p ( t ) p - ;

(iii) |u| _p(t) → 0 ⇔ ρ(u) → 0; |u| _p(t) → ∞ ⇔ ρ(u) → ∞.

(iv) For u ≠ 0, | u | p ( t ) = λ ⇔ ρ ( u λ ) = 1 .

Similar to Lemma 2.3, we have

Lemma 2.4. If we denote I ( u ) = ∫ 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t , ∀ u ∈ W ^1,p(t), then

(i) ||u|| < 1 (= 1; > 1) ⇔ I(u) < 1 (= 1; > 1);

(ii) u > 1 ⇒ u p - ≤ I ( u ) ≤ u p + , u < 1 ⇒ u p + ≤ I ( u ) ≤ u p - ;

(iii) ||u|| → 0 ⇔ I(u) → 0; ||u|| → ∞ ⇔ I(u) → ∞.

(iv) For u ≠ 0, u = λ ⇔ I ( u λ ) = 1 .

Defnition 2.2 17 .

C T ∞ = C T ∞ ( ℝ , ℝ N ) : = { u ∈ C ∞ ( ℝ , ℝ N ) : u  is T  - periodic }

with the norm u ∞ : = max t ∈ [ 0 , T ] | u ( t ) | .

For a constant p ∈ (1, ∞), using another conception of weak derivative which is called T-weak derivative, Mawhin and Willem gave the definition of the space W T 1 , p by the following way.

Definition 2.3 17 . Let u ∈ L ¹([0, T], ℝ ^N ) and v ∈ L ¹([0, T], ℝ ^N ), if

∫ 0 T v ϕ d t = - ∫ 0 T u ϕ ′ d t ∀ ϕ ∈ C T ∞ ,

then v is called a T-weak derivative of u and is denoted by u ˙ .

Definition 2.4 17 . Define

W T 1 , p ( [ 0 , T ] , ℝ N ) = { u ∈ L p ( [ 0 , T ] , ℝ N ) : u ˙ ∈ L p ( [ 0 , T ] , ℝ N ) }

with the norm u W T 1 , p = ( | u | p p + | u ˙ | p p ) 1 ∕ p .

Definition 2.5 13 . Define

W T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) = { u ∈ L p ( t ) ( [ 0 , T ] , ℝ N ) : u ˙ ∈ L p ( t ) ( [ 0 , T ] , ℝ N ) }

and H T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) to be the closure of C T ∞ in W ^1,p(t) ([0, T], ℝ ^N ).

Remark 2.2. From Definition 2.4, if u ∈ W T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) , it is easy to conclude that u ∈ W T 1 , p - ( [ 0 , T ] , ℝ N ) .

Lemma 2.5 13 .

(i) C T ∞ ( [ 0 , T ] , ℝ N ) is dense in W T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) ;

(ii) W T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) = H T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) : = { u ∈ W 1 , p ( t ) ( [ 0 , T ] , ℝ N ) : u ( 0 ) = u ( T ) } ;

(iii) If u ∈ H T 1 , 1 , then the derivative u' is also the T-weak derivative u ˙ , i.e., u ′ = u ˙ .

Lemma 2.6 17 . Assume that u ∈ W T 1 , 1 , then

(i) ∫ 0 T u ˙ d t = 0 ,

(ii) u has its continuous representation, which is still denoted by u ( t ) = ∫ 0 t u ˙ ( s ) d s + u ( 0 ) , u(0) = u(T),

(iii) u ˙ is the classical derivative of u, if u ˙ ∈ C ( [ 0 , T ] , ℝ N ) .

Since every closed linear subspace of a reflexive Banach space is also reflexive, we have

Lemma 2.7 13 . H T 1 , p ( t ) ( [ 0 , T ] , ℝ N ) is a reflexive Banach space if p ^- > 1.

Obviously, there are continuous embeddings L p ( t ) → L p - , W 1 , p ( t ) → W 1 , p - and H T 1 , p ( t ) → H T 1 , p - . By the classical Sobolev embedding theorem, we obtain

Lemma 2.8 13 . There is a continuous embedding

W 1 , p ( t ) ( or H T 1 , p ( t ) ) → C ( [ 0 , T ] , ℝ N ) ,

when p ^- > 1, the embedding is compact.

Lemma 2.9 13 . Each of the following two norms is equivalent to the norm in W T 1 , p ( t ) :

(i) | u ˙ | p ( t ) + | u | q , 1 ≤ q ≤ ∞;

(ii) | u ˙ | p ( t ) + | ū | , where ū = ( 1 ∕ T ) ∫ 0 T u ( t ) d t .

Lemma 2.10 13 . If u, u _n ∈ L ^p(t) (n = 1,2,...), then the following statements are equivalent to each other

(i) lim n → ∞ | u n - u | p ( t ) = 0 ;

(ii) lim n → ∞ ρ ( u n - u ) = 0 ;

(iii) u _n → u in measure in [0, T] and lim n → ∞ ρ ( u n ) = ρ ( u ) .

Lemma 2.11 14 . The functional J defined by

J ( u ) = ∫ 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t

is continuously differentiable on W T 1 , p ( t ) and J' is given by

⟨ J ′ ( u ) , v ⟩ = ∫ 0 T ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , v ˙ ( t ) ) d t ,

and J' is a mapping of (S ₊), i.e., if u_n ⇀ u weakly in W T 1 , p ( t ) and

limsup n → ∞ ( J ′ ( u n ) - J ′ ( u ) , u n - u ) ≤ 0 ,

then u _n has a convergent subsequence on W T 1 , p ( t ) .

Lemma 2.12 18 . Since W T 1 , p ( t ) is a separable and reflexive Banach space, there exist { e n } n = 1 ∞ ⊂ W T 1 , p ( t ) and { f n } n = 1 ∞ ⊂ ( W T 1 , p ( t ) ) * such that

f n ( e m ) = δ n , m = { 1 , n = m , 0 , n ≠ m ,

W T 1 , p ( t ) = span { e n : n = 1 , 2 , … } ¯ and ( W T 1 , p ( t ) ) * = span { f n : n = 1 , 2 , … } ¯ W * .

For k = 1, 2,..., denote

X k = span { e k } , Y k = ⊕ j = 1 k X j , Z k = ⊕ j = k ∞ X j ¯ .

Lemma 2.13 19 . Let X be a reflexive infinite Banach space, ϕ ∈ C ¹(X, ℝ) is an even functional with the (C) condition and ϕ(0) = 0. If X = Y ⊕ V with dimY < ∞, and ϕ satisfies

(i) there are constants σ, α > 0 such that φ | ∂ B σ ∩ V ≥ α ,

(ii) for any finite-dimensional subspace W of X, there exists positive constants R ₂(W) such that ϕ(u) ≤ 0 for u ∈ W\B _r (0), where B _r (0) is an open ball in W of radius r centered at 0. Then ϕ possesses an unbounded sequence of critical values.

Lemma 2.14 15 . Suppose

(A1) ϕ ∈ C ¹(X, ℝ) is an even functional, then the subspace X _k , Y _k , and Z _k are defined by (2.2);

If for every k ∈ ℕ, there exists ρ _k > r _k > 0 such that

(A2) a k : = max u ∈ Y k , u = ρ k φ ( u ) ≤ 0 , where Y k : = ⊕ j = 0 k X j ;

(A3) b k : = inf u ∈ Z k , u = r k φ ( u ) → ∞ , as k → ∞, where Z k : = ⊕ j = k ∞ X j ¯ ;

(A4) ϕ satisfies the (PS) _c condition for every c > 0.

Then ϕ has an unbounded sequence of critical values.

Lemma 2.15 16 . Assume (A1) is satisfied, and there is a k ₀ > 0 so as to for each k ≥ k ₀, there exist ρ _k > r _k > 0 such that

(A5) d k : = inf u ∈ Z k , u ≤ ρ k φ ( u ) → 0 , as k → ∞;

(A6) i k : = max u ∈ Y k , u = r k φ ( u ) < 0 ;

(A7) inf u ∈ Z k , u = ρ k φ ( u ) ≥ 0 ;

(A8) ϕ satisfies the (PS) c * condition for every c ∈ [d _k0, 0).

Then ϕ has a sequence of negative critical values converging to 0.

Remark 2.3. ϕ satisfies the (PS) c * condition means that if any sequence { u n j } ⊂ X such that n _j → ∞, u n j ∈ Y n j , φ ( u n j ) → c and ( φ | Y n j ) ′ ( u n j ) → 0 , then { u n j } contains a subsequence converging to critical point of ϕ. It is obvious that if ϕ satisfies the (PS) c * condition, then ϕ satisfies the (PS) _c condition.

Theorem 3.1. Let F(t, x) satisfies the condition (A'), and suppose the following conditions hold:

(B1) there exist β > p ⁺ and r > 0 such that

β F ( t , x ) ≤ ( ∇ F ( t , x ) , x )

for a.e. t ∈ [0, T] and all |x| ≥ r in ℝ ^N ;

(B2) there exist positive constants μ > p ⁺ and Q > 0 such that

limsup | x | → + ∞ F ( t , x ) | x | μ ≤ Q

uniformly for a.e. t ∈ [0, T];

(B3) there exists μ' > p ⁺ and Q' > 0 such that

liminf | x | → + ∞ F ( t , x ) | x | μ ′ ≥ Q ′

uniformly for a.e. t ∈ [0, T];

(B4) F(t, x) = F(t, -x) for t ∈ [0, T] and all x in ℝ ^N .

Then system (1.1) has infinite solutions u _k in W T 1 , p ( t ) for every positive integer k such that ||u _k ||_∞ → +∞, as k → ∞.

Remark 3.1. Suppose that F(t, ·) is continuously differentiable in x and p(t) ≡ p, then condition (B1) reduces to the well-known Ambrosetti-Rabinowitz condition (see 19 ), which was introduced in the context of semi-linear elliptic problems. This condition implies that F(t, x) grows at a superquadratic rate as |x| → ∞. This kind of technical condition often appears as necessary to use variational methods when we solve super-linear differential equations such as elliptic problems, Dirac equations, Hamiltonian systems, wave equations, and Schrödinger equations.

Theorem 3.2. Assume that F(t, x) satisfies (A'), (B1), (B3), and (B4) and the following assumption:

(B5) ∫ 0 T F ( t , 0 ) d t = 0 , and there exists r ₁ > p ⁺ and M > 0 such that

limsup | x | → 0 | F ( t , x ) | | x | r 1 ≤ M .

Then system (1.1) has infinite solutions u _k in W T 1 , p ( t ) for every positive integer k such that ||u _k ||_∞ → +∞, as k → ∞.

Theorem 3.3. Assume that F(t, x) satisfies the following assumption:

(B6) F(t, x):= a(t)|x| ^γ , where a(t) ∈ L ^∞ (0, T; ℝ⁺) and 1 < γ < p ^- is a constant. Then system (1.1) has infinite solutions u _k in W T 1 , p ( t ) for every positive integer k.

The proof of Theorem 3.1 is organized as follows: first, we show the functional ϕ defined by

φ ( u ) = ∫ 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + ∫ 0 T 1 p ( t ) | u ( t ) | p ( t ) d t - ∫ 0 T F ( t , u ( t ) ) d t

satisfies the (PS) condition, then we verify for ϕ the conditions in Lemma 2.14 item-by-item, then ϕ has an unbounded sequence of critical values.

Proof of Theorem 3.1. Let { u n } ⊂ W T 1 , p ( t ) such that ϕ(u _n ) is bounded and ϕ'(u _n ) → 0 as n → ∞. First, we prove {u _n } is a bounded sequence, otherwise, {u _n } would be unbounded sequence, passing to a subsequence, still denoted by {u _n }, such that ||u _n || ≥ 1 and ||u _n || → ∞. Note that

⟨ φ ′ ( u ) , v ⟩ = ∫ 0 T ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , v ˙ ( t ) ) d t + ∫ 0 T ( | u ( t ) | p ( t ) - 2 u ( t ) - ∇ F ( t , u ( t ) ) , v ( t ) ) d t

for all v ∈ W T 1 , p ( t ) .

It follows from (3.1) that

  ∫ 0 T ( β p ( t ) − 1 ) ( | u ˙ n ( t ) | p ( t ) + | u n ( t ) | p ( t ) ) d t = β φ ( u n ) − 〈 φ ′ ( u n ) , u n 〉 + ∫ 0 T [ β F ( t , u n ( t ) )   − ( ∇ F ( t , u n ( t ) ) , u n ( t ) ) ] d t = β φ ( u n ) − 〈 φ ′ ( u n ) , u n 〉 + ∫ Ω 1 [ β F ( t , u n ( t ) ) − ( ∇ F ( t , u n ( t ) ) , u n ( t ) ) ] d t + ∫ Ω 2 [ β F ( t , u n ( t ) ) − ( ∇ F ( t , u n ( t ) ) , u n ( t ) ) ] d t ≤ β φ ( u n ) − 〈 φ ′ ( u n ) , u n 〉 + ∫ Ω 1 [ β F ( t , u n ( t ) ) − ( ∇ F ( t , u n ( t ) ) , u n ( t ) ) ] d t ≤ β φ ( u n ) − 〈 φ ′ ( u n ) , u n 〉 + C 0 ,

where Ω₁:= {t ∈ [0, T]; |u _n (t)| ≤ r}, Ω₂:= [0, T] \ Ω₁ and C ₀ is a positive constant.

However, from (3.2), we have

β φ ( u n ) + C 0 ≥ β p + - 1 u n p - - φ ′ ( u n ) u n ,

Thus ||u _n || is a bounded sequence in W T 1 , p ( t ) .

By Lemma 2.8, the sequence {u _n } has a subsequence, also denoted by {u _n }, such that

u n ⇀ u weakly in W T 1 , p ( t ) and u n → u strongly in C ( [ 0 , T ] ; ℝ N )

and ||u||_∞ ≤ C ₁||u|| by Lemma 2.8, where C ₁ is a positive constant.

Therefore, we have

⟨ φ ′ ( u n ) - φ ′ ( u ) , u n - u ⟩ → 0 as n → ∞ ,

i.e.,

〈 φ ′ ( u n ) − φ ′ ( u ) , u n − u 〉 = ∫ 0 T ( ∇ F ( t , u n ( t ) ) − ∇ F ( t , u ( t ) ) , u n ( t ) − u ( t ) ) d t + ∫ 0 T ( | u n ( t ) | p ( t ) − 2 u n ( t ) − | u ( t ) | p ( t ) − 2 u ( t ) , u n ( t ) − u ( t ) ) d t + ∫ 0 T ( | u ˙ n ( t ) | p ( t ) − 2 u ˙ n ( t ) − | u ˙ ( t ) | p ( t ) − 2 u ˙ ( t ) , u ˙ n ( t ) − u ˙ ( t ) ) d t .

By (3.4) and (3.5), we get 〈J'(u) - J'(u _n ), u - u _n 〉 → 0, i.e.,

∫ 0 T ( | u ˙ n ( t ) | p ( t ) - 2 u ˙ n ( t ) - | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , u ˙ n ( t ) - u ˙ ( t ) ) d t → 0 ,

so it follows Lemma 2.11 that {u _n } admits a convergent subsequence.

For any u ∈ Y _k , let

u * : = ( ∫ 0 T | u ( t ) | μ ′ d t ) 1 ∕ μ ′ ,

and it is easy to verify that ||·||_* defined by (3.6) is a norm of Y _k . Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant C ₂ such that

C 2 u ≤ u * for u ∈ Y k .

In view of (B3), there exist two positive constants M ₁ and C ₃ such that

F ( t , x ) ≥ M 1 | x | μ ′

for a.e. t ∈ [0, T] and |x| ≥ C ₃.

It follows (3.7) and (3.8) that

φ ( u ) = ∫ 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + ∫ 0 T 1 p ( t ) | u ( t ) | p ( t ) d t − ∫ 0 T F ( t , u ( t ) ) d t ≤ 1 p − ( ‖ u ‖ p + + 1 ) − ∫ Ω 3 F ( t , u ( t ) ) d t − ∫ Ω 4 F ( t , u ( t ) ) d t ≤ 1 p − ( ‖ u ‖ p + + 1 ) − M 1 ∫ Ω 3 | u ( t ) | μ ′ d t − ∫ Ω 4 F ( t , u ( t ) ) d t = 1 p − ( ‖ u ‖ p + + 1 ) − M 1 ∫ 0 T | u ( t ) | μ ′ d t + M 1 ∫ Ω 4 | u ( t ) | μ ′ d t − ∫ Ω 4 F ( t , u ( t ) ) d t ≤ 1 p − ( ‖ u ‖ p + + 1 ) − C 2 μ ′ M 1 ‖ u ‖ μ ′ + C 4 ,

where Ω₃:= {t ∈ [0, T]; |u(t)| ≥ C ₃}, Ω₄:= [0, T] \ Ω₃ and C ₄ is a positive constant.

Since μ' > p ⁺, there exist positive constants d _k such that

φ ( u ) ≤ 0 for all u ∈ Y k and u ≥ d k .

For any u ∈ Z _k , let

u μ : = ( ∫ 0 T | u ( t ) | μ d t ) 1 ∕ μ and β k : = sup u ∈ Z k , u = 1 u μ ,

then we conclude β _k → 0 as k → ∞.

In fact, it is obvious that β _k ≥ β _{k + 1} > 0, so β _k → β ≥ 0 as k → ∞. For every k ∈ ℕ, there exists u _k ∈ Z _k such that

u k = 1 and u k μ > β k ∕ 2 .

As W T 1 , p ( t ) is reflexive, {u _k } has a weakly convergent subsequence, still denoted by {u _k }, such that u_k ⇀ u. We claim u = 0.

In fact, for any f _m ∈ {f _n : n = 1, 2...,}, we have f _m (u _k ) = 0, when k > m, so

f m ( u k ) → 0 , as k → ∞

for any f _m ∈ {f _n : n = 1, 2 ...,}, therefore u = 0.

By Lemma 2.8, when u_k ⇀ 0 in W T 1 , p ( t ) , then u _k → 0 strongly in C([0, T]; ℝ ^N ). So, we conclude β = 0 by (3.11).

In view of (B2), there exist two positive constants M ₂ and C ₁₀ such that

F ( t , x ) ≤ M 2 | x | μ

uniformly for a.e. t ∈ [0, T] and |x| ≥ C ₅.

When ||u|| ≥ 1, we conclude

φ ( u ) = ∫ 0 T 1 p ( t ) | u ( t ) | p ( t ) d t + ∫ 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t - ∫ 0 T F ( t , u ( t ) ) d t (1) ≥ 1 p + ∫ 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t - ∫ Ω 5 F ( t , u ( t ) ) d t - ∫ Ω 6 F ( t , u ( t ) ) d t (2) ≥ 1 p + u p - - M 2 ∫ 0 T | u ( t ) | μ d t + M 2 ∫ Ω 6 | u ( t ) | μ d t - ∫ Ω 6 F ( t , u ( t ) ) d t (3) ≥ 1 p + u p - - M 2 β k μ u μ - C 6 , (4) (5)

where Ω₅:= {t ∈ [0, T]; |u(t)| ≥ C ₅}, Ω₆:= [0, T] \ Ω₅ and C ₆ is a positive constant.

Choosing r _k = 1/β _k , it is obvious that

r k → ∞ as k → ∞ ,

then

b k : = inf u ∈ Z k , u = r k φ ( u ) → ∞ as k → ∞ ,

i.e., the condition (A3) in Lemma 2.14 is satisfied.

In view of (3.9), let ρ _k := max{d _k , r _k + 1}, then

a k : = max u ∈ Y k , u = ρ k φ ( u ) ≤ 0 ,

and this shows the condition of (A2) in Lemma 2.14 is satisfied.

We have proved the functional ϕ satisfies all the conditions of Lemma 2.14, then ϕ has an unbounded sequence of critical values c _k = ϕ(u _k ) by Lemma 2.14, we only need to show ||u _k ||_∞ → ∞ as k → ∞.

In fact, since u _k is a critical point of the functional ϕ, we have

∫ 0 T | u ˙ k ( t ) | p ( t ) d t + ∫ 0 T | u k ( t ) | p ( t ) d t - ∫ 0 T ( ∇ F ( t , u k ( t ) ) , u k ( t ) ) d t = 0 .

Hence, we have

c k = φ ( u k ) = ∫ 0 T 1 p ( t ) | u ˙ k ( t ) | p ( t ) d t + ∫ 0 T 1 p ( t ) | u k ( t ) | p ( t ) d t - ∫ 0 T F ( t , u k ( t ) ) d t , (1)  ≤ 1 p - ∫ 0 T | u ˙ k ( t ) | p ( t ) d t + 1 p - ∫ 0 T | u k ( t ) | p ( t ) d t - ∫ 0 T F ( t , u k ( t ) ) d t , (2)  = ∫ 0 T ( ∇ F ( t , u k ( t ) ) , u k ( t ) ) d t - ∫ 0 T F ( t , u k ( t ) ) d t , (3)  (4) 

since c _k → ∞, we conclude

u k ∞ → ∞ as k → ∞

by (3.14). In fact, if not, going to a subsequence if necessary, we may assume that

u k ∞ ≤ C 7

for all k ∈ ℕ and some positive constant C ₇.

Combining (A') and (3.14), we have

c k ≤ ∫ 0 T ( ∇ F ( t , u k ( t ) ) , u k ( t ) ) d t - ∫ 0 T F ( t , u k ( t ) ) d t , ≤ ( C 7 + 1 ) max 0 ≤ s ≤ C 7 a ( s ) ∫ 0 T b ( t ) d t ,

which contradicts c _k → ∞. This completes the proof of Theorem 3.1.

Proof of Theorem 3.2. To prove {u _n } has a convergent subsequence in space W T 1 , p ( t ) is the same as that in the proof of Theorem 3.1, thus we omit it. It is obvious that ϕ is even and ϕ(0) = 0 under condition (B5), and so we only need to verify other conditions in Lemma 2.13.

Proposition 3.1. Under the condition (B5), there exist two positive constants σ and α such that ϕ(u) ≥ α for all u ∈ W ̃ T 1 , p ( t ) and ||u|| = σ.

Proof. In view of condition (B5), there exist two positive constants ε and δ such that

0 < ε < C 1 and 0 < δ < ε ,

where C ₁ is the same as in (3.3), and

| F ( t , x ) | ≤ ( M + ε ) | x | r 1

for a.e. t ∈ [0, T] and |x| ≤ δ.

Let σ:= δ/C ₁ and ||u|| = σ, since σ < 1, we have

u p + ≤ I ( u ) and u ∞ ≤ C 1 u .

by Lemmas 2.4 and 2.8.

Combining (3.15) and (3.16), we have

φ ( u ) = ∫ 0 T 1 p ( t ) | u ( t ) | p ( t ) d t + ∫ 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t - ∫ 0 T F ( t , u ( t ) ) d t (1) ≥ 1 p + ∫ 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t - ( M + ε ) ∫ 0 T | u ( t ) | r 1 d t (2) ≥ 1 p + u p + - ( M + ε ) T C 1 r 1 u r 1 (3) = 1 p + - ( M + ε ) T C 1 r 1 σ r 1 - p + σ p + , (4) (5)