In this article, we investigate the existence of infinitely many periodic solutions for some nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained using critical point theory.

2000 Mathematics Subject Classification: 34C37; 58E05; 70H05.

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

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 ∈ L¹(0, T; ℝ⁺), such that

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 [3-12] 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

with the norm

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

Define

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.,

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

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

(ii) If p₁(t) and p₂(t) ∈ C([0, T], ℝ⁺) and p₁(t) ≤ p₂(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].

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 ∈ L¹([0, T], ℝ^N) and v ∈ L¹([0, T], ℝ^N), if

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

Definition 2.4 [17]. Define

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

Definition 2.5 [13]. Define

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

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

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

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

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

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

Lemma 2.13 [19]. Let X be a reflexive infinite Banach space, ϕ ∈ C¹(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 R₂(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 ∈ C¹(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 k₀ > 0 so as to for each k ≥ k₀, 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

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

uniformly for a.e. t ∈ [0, T];

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

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 r₁ > p⁺ and M > 0 such that

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

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

for all v ∈ W T 1 , p ( t ) .

It follows from (3.1) that

where Ω₁:= {t ∈ [0, T]; |u_n(t)| ≤ r}, Ω₂:= [0, T] \ Ω₁ and C₀ is a positive constant.

However, from (3.2), we have

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

and ||u||_∞ ≤ C₁||u|| by Lemma 2.8, where C₁ is a positive constant.

Therefore, we have

i.e.,

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

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

For any u ∈ Y_k, let

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 C₂ such that

In view of (B3), there exist two positive constants M₁ and C₃ such that

for a.e. t ∈ [0, T] and |x| ≥ C₃.

It follows (3.7) and (3.8) that

where Ω₃:= {t ∈ [0, T]; |u(t)| ≥ C₃}, Ω₄:= [0, T] \ Ω₃ and C₄ is a positive constant.

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

For any u ∈ Z_k, let

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

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

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 M₂ and C₁₀ such that

uniformly for a.e. t ∈ [0, T] and |x| ≥ C₅.

When ||u|| ≥ 1, we conclude

where Ω₅:= {t ∈ [0, T]; |u(t)| ≥ C₅}, Ω₆:= [0, T] \ Ω₅ and C₆ is a positive constant.

Choosing r_k = 1/β_k, it is obvious that

then

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

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

Hence, we have

since c_k → ∞, we conclude

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

for all k ∈ ℕ and some positive constant C₇.

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

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

where C₁ is the same as in (3.3), and

for a.e. t ∈ [0, T] and |x| ≤ δ.

Let σ:= δ/C₁ and ||u|| = σ, since σ < 1, we have

by Lemmas 2.4 and 2.8.

Combining (3.15) and (3.16), we have

so we can choose σ small enough, such that

and this completes the proof of Proposition 3.1.

Proposition 3.2. For any finite dimensional subspace W of W T 1 , p ( t ) , there is r₂ = r₂(W) > 0 such that ϕ(u) ≤ 0 for u ∈ W \ B r 2 ( 0 ) , where B r 2 ( 0 ) is an open ball in W of radius r₂ centered at 0.

Proof. The proof of Proposition 3.2 is the same as the proof of the condition (A2) in the proof of Theorem 3.1.

We have proved the functional ϕ satisfies all the conditions of Lemma 2.13, ϕ has an unbounded sequence of critical values c_k = ϕ(u_k) by Lemma 2.13. Arguing as in the proof of Theorem 3.1, 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 → ∞. The proof of Theorem 3.2 is complete.

Proof of Theorem 3.3. First, we show that ϕ satisfies the (PS) c * for every c ∈ ℝ. Suppose 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 } is a bounded sequence, otherwise, { u n j } would be unbounded sequence, passing to a subsequence, still denoted by { u n j } such that u n j ≥ 1 and u n j → ∞ . Note that

However, from (3.17), we have

thus ||u_n|| is a bounded sequence in W T 1 , p ( t ) . Going, if necessary, to a subsequence, we can assume that u n j ⇀ u in W T 1 , p ( t ) . As X = ⋃ n j Y n j ¯ , we can choose v n j ∈ Y n j such that v n j → u . Hence

In view of (3.4) and (3.5), we can also conclude u n j → u , furthermore, we have φ ′ ( u n j ) → φ ′ ( u ) .

Let us prove ϕ'(u) = 0 below. Taking arbitrarily ω_k ∈ Y_k, notice when n_j ≤ k we have

Going to limit in the right side of above equation reaches

so ϕ'(u) = 0, this shows that ϕ satisfies the (PS) c * for every c ∈ ℝ.

For any finite dimensional subspace W ⊂ W T 1 , p ( t ) , there exists ε₁ > 0 such that

Otherwise, for any positive integer n, there exists u_n ∈ W \ {0} such that

Set v n ( t ) : = u n ( t ) u n ∈ W \ { 0 } , then ||v_n|| = 1 for all n ∈ ℕ and

Since dimW < ∞, it follows from the compactness of the unit sphere of W that there exists a subsequence, denoted also by {v_n}, such that {v_n} converges to some v₀ in W. It is obvious that ||v₀|| = 1.

By the equivalence of the norms on the finite dimensional space W, we have v_n → v₀ in L p - ( 0 , T ; ℝ N ) , i.e.,

By (3.20) and Hölder inequality, we have

Thus, there exist ξ₁, ξ₂ > 0 such that

In fact, if not, we have

for all positive integer n.

It implies that

as n → ∞, where C₆ is the same in (3.3). Hence v₀ = 0 which contradicts that ||v₀|| = 1. Therefore, (3.22) holds. Now let

and Ω n c = [ 0 , T ] \ Ω n = { t ∈ [ 0 , T ] : a ( t ) | v n ( t ) | γ ≥ 1 n } .

By (3.19) and (3.22), we have

for all positive integer n. Let n be large enough such that

then we have

for all large n, which is a contradiction to (3.21). Therefore, (3.18) holds.

For any u ∈ Z_k, let

then we conclude γ_k → 0 as k → ∞ as in the proof of Theorem 3.1.

Let ρ k : = ( 2 c p + γ k γ ) 1 p + - γ , where c : = ( ∫ 0 T a ( t ) p - p - - γ d t ) p - - γ p - , it is obvious that ρ_k → 0, as k → ∞. In view of (3.23), We conclude

so the condition (A7) in Lemma 2.15 is satisfied.

Furthermore, by (3.23), for any u ∈ Z_k with ||u|| ≤ ρ_k, we have

Therefore,

so we have

for ρ_k, γ_k → 0, as k → ∞.

For any u ∈ Y_k \ {0} with ||u|| ≤ 1,

where ε₁ is given in (3.18), and

Choosing 0 < r k < min { ρ k , ( p - ε 2 2 ) 1 p - - γ } , we conclude

i.e., the condition (A6) in Lemma 2.15 is satisfied. The proof of Theorem 3.3 is complete.