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# Infinitely many periodic solutions for some second-order differential systems with p(t)-Laplacian

Liang Zhang, Xian Hua Tang* and Jing Chen

### Author affiliations

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, P. R. China

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### Citation and License

Boundary Value Problems 2011, 2011:33  doi:10.1186/1687-2770-2011-33

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/33

 Received: 3 June 2011 Accepted: 14 October 2011 Published: 14 October 2011

© 2011 Zhang et al; licensee Springer.

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.

### Abstract

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.

##### Keywords:
p(t)-Laplacian; Periodic solutions; Critical point theory

### 1. Introduction

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

(1.1)

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

(A) p(0) = p(T) and , where q+ > 1 which satisfies 1/p- + 1/q+ = 1.

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

(A') F(t, x) is measurable in t for every x ∈ ℝN 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 ∈ ℝN and a.e. t ∈ [0, T].

The operator 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 , which is different from the usual space , 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 throughout this article, and we use 〈·, ·〉 and |·| to denote the usual inner product and norm in ℝN, respectively.

### 2. Preliminaries

In this section, we recall some known results in nonsmooth critical point theory, and the properties of space are listed for the convenience of readers.

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

with the norm

For , let u' denote the weak derivative of u, if and satisfies

Define

with the norm .

In this article, we will use the following equivalent norm on W1, p(t) ([0, T], ℝN), i.e.,

and some lemmas given in the following section have been proven under the norm of , 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 , which is the same with the usual norm in space Lp.

The space Lp(t) is a generalized Lebesgue space, and the space W1, 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]. Lp(t) and W1, p(t) are both Banach spaces with the norms defined above, when p- > 1, they are reflexive.

Lemma 2.2 [14]. (i) The space Lp(t) is a separable, uniform convex Banach space, its conjugate space is Lq(t), for any u Lp(t) and v Lq(t), we have

where .

(ii) If p1(t) and p2(t) ∈ C([0, T], ℝ+) and p1(t) ≤ p2(t) for any t ∈ [0, T], then , and the embedding is continuous.

Lemma 2.3 [14]. If we denote , ∀ u Lp(t), then

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

(ii) ;

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

(iv) For u ≠ 0, .

Similar to Lemma 2.3, we have

Lemma 2.4. If we denote , ∀ u W1,p(t), then

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

(ii) ;

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

(iv) For u ≠ 0, .

Defnition 2.2 [17].

with the norm .

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 by the following way.

Definition 2.3 [17]. Let u L1([0, T], ℝN) and v L1([0, T], ℝN), if

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

Definition 2.4 [17]. Define

with the norm .

Definition 2.5 [13]. Define

and to be the closure of in W1,p(t) ([0, T], ℝN).

Remark 2.2. From Definition 2.4, if , it is easy to conclude that .

Lemma 2.5 [13].

(i) is dense in ;

(ii) ;

(iii) If , then the derivative u' is also the T-weak derivative , i.e., .

Lemma 2.6 [17]. Assume that , then

(i) ,

(ii) u has its continuous representation, which is still denoted by , u(0) = u(T),

(iii) is the classical derivative of u, if .

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

Lemma 2.7 [13]. is a reflexive Banach space if p- > 1.

Obviously, there are continuous embeddings and . 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 :

(i) , 1 ≤ q ≤ ∞;

(ii) , where .

Lemma 2.10 [13]. If u, un Lp(t) (n = 1,2,...), then the following statements are equivalent to each other

(i) ;

(ii) ;

(iii) un u in measure in [0, T] and .

Lemma 2.11 [14]. The functional J defined by

is continuously differentiable on and J' is given by

(2.1)

and J' is a mapping of (S+), i.e., if un u weakly in and

then un has a convergent subsequence on .

Lemma 2.12 [18]. Since is a separable and reflexive Banach space, there exist and such that

and .

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

(2.2)

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 ,

(ii) for any finite-dimensional subspace W of X, there exists positive constants R2(W) such that ϕ(u) ≤ 0 for u W\Br(0), where Br(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 Xk, Yk, and Zk are defined by (2.2);

If for every k ∈ ℕ, there exists ρk > rk > 0 such that

(A2) , where ;

(A3) , as k → ∞, where ;

(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 ρk > rk > 0 such that

(A5) , as k → ∞;

(A6) ;

(A7) ;

(A8) ϕ satisfies the condition for every c ∈ [dk0, 0).

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

Remark 2.3. ϕ satisfies the condition means that if any sequence such that nj → ∞, and , then contains a subsequence converging to critical point of ϕ. It is obvious that if ϕ satisfies the condition, then ϕ satisfies the (PS)c condition.

### 3. Main results and proofs of the theorems

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 ℝN;

(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 ℝN.

Then system (1.1) has infinite solutions uk in for every positive integer k such that ||uk||→ +∞, 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) , and there exists r1 > p+ and M > 0 such that

Then system (1.1) has infinite solutions uk in for every positive integer k such that ||uk||→ +∞, 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 uk in 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 such that ϕ(un) is bounded and ϕ'(un) → 0 as n → ∞. First, we prove {un} is a bounded sequence, otherwise, {un} would be unbounded sequence, passing to a subsequence, still denoted by {un}, such that ||un|| ≥ 1 and ||un|| → ∞. Note that

(3.1)

for all .

It follows from (3.1) that

(3.2)

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

However, from (3.2), we have

Thus ||un|| is a bounded sequence in .

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

(3.3)

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

Therefore, we have

(3.4)

i.e.,

(3.5)

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

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

For any u Yk, let

(3.6)

and it is easy to verify that ||·||* defined by (3.6) is a norm of Yk. Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant C2 such that

(3.7)

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

(3.8)

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 dk such that

(3.9)

For any u Zk, let

(3.10)

then we conclude βk → 0 as k → ∞.

In fact, it is obvious that βk βk + 1 > 0, so βk β ≥ 0 as k → ∞. For every k ∈ ℕ, there exists uk Zk such that

(3.11)

As is reflexive, {uk} has a weakly convergent subsequence, still denoted by {uk}, such that uk u. We claim u = 0.

In fact, for any fm ∈ {fn: n = 1, 2...,}, we have fm(uk) = 0, when k > m, so

for any fm ∈ {fn: n = 1, 2 ...,}, therefore u = 0.

By Lemma 2.8, when uk ⇀ 0 in , then uk → 0 strongly in C([0, T]; ℝN). So, we conclude β = 0 by (3.11).

In view of (B2), there exist two positive constants M2 and C10 such that

(3.12)

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 rk = 1/βk, it is obvious that

then

(3.13)

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

In view of (3.9), let ρk:= max{dk, rk + 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 ck = ϕ(uk) by Lemma 2.14, we only need to show ||uk||→ ∞ as k → ∞.

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

Hence, we have

(3.14)

since ck → ∞, 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 ck → ∞. This completes the proof of Theorem 3.1.

Proof of Theorem 3.2. To prove {un} has a convergent subsequence in space 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 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

(3.15)

for a.e. t ∈ [0, T] and |x| ≤ δ.

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

(3.16)

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 , there is r2 = r2(W) > 0 such that ϕ(u) ≤ 0 for , where 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 ck = ϕ(uk) by Lemma 2.13. Arguing as in the proof of Theorem 3.1, system (1.1) has infinite solutions {uk} in for every positive integer k such that ||uk||→ +∞, as k → ∞. The proof of Theorem 3.2 is complete.

Proof of Theorem 3.3. First, we show that ϕ satisfies the for every c ∈ ℝ. Suppose nj → ∞, and , then is a bounded sequence, otherwise, would be unbounded sequence, passing to a subsequence, still denoted by such that and . Note that

(3.17)

However, from (3.17), we have

thus ||un|| is a bounded sequence in . Going, if necessary, to a subsequence, we can assume that in . As , we can choose such that . Hence

In view of (3.4) and (3.5), we can also conclude , furthermore, we have .

Let us prove ϕ'(u) = 0 below. Taking arbitrarily ωk Yk, notice when nj k we have

Going to limit in the right side of above equation reaches

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

For any finite dimensional subspace , there exists ε1 > 0 such that

(3.18)

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

Set , then ||vn|| = 1 for all n ∈ ℕ and

(3.19)

Since dimW < ∞, it follows from the compactness of the unit sphere of W that there exists a subsequence, denoted also by {vn}, such that {vn} 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 vn v0 in , i.e.,

(3.20)

By (3.20) and Hölder inequality, we have

(3.21)

Thus, there exist ξ1, ξ2 > 0 such that

(3.22)

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 .

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 Zk, let

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

(3.23)

Let , where , it is obvious that ρk → 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 Zk with ||u|| ≤ ρk, we have

Therefore,

so we have

for ρk, γk → 0, as k → ∞.

For any u Yk \ {0} with ||u|| ≤ 1,

where ε1 is given in (3.18), and

Choosing , we conclude

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

### 4. Example

In this section, we give three examples to illustrate our results.

Example 4.1. In system (1.1), let and

Choose

so it is easy to verify that all the conditions (B1)-(B4) are satisfied. Then by Theorem 3.1, system (1.1) has infinite solutions {uk} in for every positive integer k such that ||uk||→ +∞, as k → ∞.

Example 4.2. In system (1.1), let F(t, x) = |x|8 and

We choose , r = 2, μ' = 8, r1 = 7, Q' = 1 and M = 1, so it is easy to verify that all the conditions of Theorem 3.2 are satisfied. Then by Theorem 3.2, so system (1.1) has infinite solutions {uk} in for every positive integer k such that ||uk||→ +∞, as k → ∞.

Example 4.3. In system (1.1), let F(t, x) = a(t)|x|3 where

and

It is easy to verify that all the conditions of Theorem 3.3 are satisfied. Then by Theorem 3.3, so system (1.1) has infinite solutions {uk} in for every positive integer k.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All the authors typed, read, and approved the final manuscript.

### Acknowledgements

The authors thank the anonymous referees for valuable suggestions and comments which led to improve this article. This Project is supported by the National Natural Science Foundation of China (Grant No. 11171351).

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