Abstract
In this article, we investigate the existence of infinitely many periodic solutions for some nonautonomous secondorder 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 theory1. Introduction
Consider the secondorder 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 , 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
for all x ∈ ℝ^{N }and a.e. t ∈ [0, T].
The operator is said to be p(t)Laplacian, and becomes pLaplacian when p(t) ≡ p (a constant). The p(t)Laplacian possesses more complicated nonlinearity than pLaplacian; 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 electrorheological 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 [312] for an overview on this subject.
In 2003, Fan and Fan [13] studied the ordinary p(t)Laplacian system and introduced a generalized OrliczSobolev 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 AmbrosettiRabinowitz 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 ℝ^{N}, 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
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 , 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 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
(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 , and the embedding is continuous.
Lemma 2.3 [14]. If we denote , ∀ u ∈ L^{p(t)}, then
(i) u_{p(t) }< 1 (= 1; > 1) ⇔ ρ(u) < 1 (= 1; > 1);
(iii) u_{p(t) }→ 0 ⇔ ρ(u) → 0; u_{p(t) }→ ∞ ⇔ ρ(u) → ∞.
Similar to Lemma 2.3, we have
Lemma 2.4. If we denote , ∀ u ∈ W^{1,p(t)}, then
(i) u < 1 (= 1; > 1) ⇔ I(u) < 1 (= 1; > 1);
(iii) u → 0 ⇔ I(u) → 0; u → ∞ ⇔ I(u) → ∞.
Defnition 2.2 [17].
For a constant p ∈ (1, ∞), using another conception of weak derivative which is called Tweak derivative, Mawhin and Willem gave the definition of the space 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
then v is called a Tweak derivative of u and is denoted by .
Definition 2.4 [17]. Define
Definition 2.5 [13]. Define
and to be the closure of in W^{1,p(t) }([0, T], ℝ^{N}).
Remark 2.2. From Definition 2.4, if , it is easy to conclude that .
Lemma 2.5 [13].
(iii) If , then the derivative u' is also the Tweak derivative , i.e., .
Lemma 2.6 [17]. Assume that , then
(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 :
Lemma 2.10 [13]. If u, u_{n }∈ L^{p(t) }(n = 1,2,...), then the following statements are equivalent to each other
(iii) u_{n }→ 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
and J' is a mapping of (S_{+}), i.e., if u_{n }⇀ u weakly in and
then u_{n }has a convergent subsequence on .
Lemma 2.12 [18]. Since is a separable and reflexive Banach space, there exist and such that
For k = 1, 2,..., denote
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 ,
(ii) for any finitedimensional 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
(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
(A8) ϕ satisfies the 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 condition means that if any sequence such that n_{j }→ ∞, 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 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 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 wellknown AmbrosettiRabinowitz condition (see [19]), which was introduced in the context of semilinear 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 superlinear 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 r_{1 }> p^{+ }and M > 0 such that
Then system (1.1) has infinite solutions u_{k }in 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 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 itembyitem, then ϕ has an unbounded sequence of critical values.
Proof of Theorem 3.1. Let 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
It follows from (3.1) that
where Ω_{1}:= {t ∈ [0, T]; u_{n}(t) ≤ r}, Ω_{2}:= [0, T] \ Ω_{1 }and C_{0 }is a positive constant.
However, from (3.2), we have
Thus u_{n} is a bounded sequence in .
By Lemma 2.8, the sequence {u_{n}} has a subsequence, also denoted by {u_{n}}, such that
and u_{∞ }≤ C_{1}u by Lemma 2.8, where C_{1 }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 C_{2 }such that
In view of (B3), there exist two positive constants M_{1 }and C_{3 }such that
for a.e. t ∈ [0, T] and x ≥ C_{3}.
It follows (3.7) and (3.8) that
where Ω_{3}:= {t ∈ [0, T]; u(t) ≥ C_{3}}, Ω_{4}:= [0, T] \ Ω_{3 }and C_{4 }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 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 , 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
uniformly for a.e. t ∈ [0, T] and x ≥ C_{5}.
When u ≥ 1, we conclude
where Ω_{5}:= {t ∈ [0, T]; u(t) ≥ C_{5}}, Ω_{6}:= [0, T] \ Ω_{5 }and C_{6 }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 C_{7}.
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 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 C_{1 }is the same as in (3.3), and
for a.e. t ∈ [0, T] and x ≤ δ.
Let σ:= δ/C_{1 }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 , there is r_{2 }= r_{2}(W) > 0 such that ϕ(u) ≤ 0 for , where is an open ball in W of radius r_{2 }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 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 for every c ∈ ℝ. Suppose n_{j }→ ∞, and , then is a bounded sequence, otherwise, would be unbounded sequence, passing to a subsequence, still denoted by such that and . Note that
However, from (3.17), we have
thus u_{n} 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 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 for every c ∈ ℝ.
For any finite dimensional subspace , there exists ε_{1 }> 0 such that
Otherwise, for any positive integer n, there exists u_{n }∈ W \ {0} such that
Set , 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 v_{0 }in W. It is obvious that v_{0} = 1.
By the equivalence of the norms on the finite dimensional space W, we have v_{n }→ v_{0 }in , i.e.,
By (3.20) and Hölder inequality, we have
Thus, there exist ξ_{1}, ξ_{2 }> 0 such that
In fact, if not, we have
for all positive integer n.
It implies that
as n → ∞, where C_{6 }is the same in (3.3). Hence v_{0 }= 0 which contradicts that v_{0} = 1. Therefore, (3.22) holds. Now let
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 , where , 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 ε_{1 }is given in (3.18), and
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 {u_{k}} in for every positive integer k such that u_{k}_{∞ }→ +∞, as k → ∞.
Example 4.2. In system (1.1), let F(t, x) = x^{8 }and
We choose , r = 2, μ' = 8, r_{1 }= 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 {u_{k}} in for every positive integer k such that u_{k}_{∞ }→ +∞, 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 {u_{k}} 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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