Periodic Solutions of a Class of Fourth-Order Superlinear Differential Equations

Abstract

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.

1. Introduction

The existence of periodic solutions of fourth-order differential equations has been studied by more and more researchers [1–6]. The application methods contain mainly Clark theorem [2–4], Cone theory [6], and so on.

For a single equation, Tersian and Chaparova [2] study the existence of infinitely many unbounded solutions, using symmetric mountain pass lemma: 𝑢^𝑖𝑣−𝑝𝑢^′′+𝑎(𝑥)𝑢−𝑏(𝑥)𝑢³𝑢(0)(𝐿)=0,𝑥∈𝐑,=𝑢=0,𝑢^′′(0)=𝑢^′′(𝐿)=0. (1.1)

It is a natural problem to wonder whether symmetric mountain pass lemma method may be applied not only to single equations but also to systems of differential equations. In this paper we study the existence of periodic solutions of the fourth-order equations, by making use of the classical variational techniques and symmetric mountain pass lemma 𝑢⁽⁴⁾−𝑐𝑢^′′(𝑥)()+𝑎𝑢−𝜕𝐹𝑥,𝑢,𝑣𝑣𝜕𝑢=0,0<𝑥<𝐿,₍₄₎−𝑑𝑣^′′(𝑥)()+𝑏𝑣−𝜕𝐹𝑥,𝑢,𝑣𝑢(0)𝜕𝑣=0,0<𝑥<𝐿,=𝑢^′′(0)=𝑢(𝐿)=𝑢^′′(𝐿)𝑣(0)=0,=𝑣^′′(0)=𝑣(𝐿)=𝑣^′′(𝐿)=0. (1.2)Through studying System (1.2), (1.1) of the corresponding conclusions are extended.

The paper is organized as follows. In Section 2, we consider the result of System (1.2) under certain conditions. In Section 3, we prove the main result of this paper and give an example.

2. Main Result

In this paper, we state our main result. First we give the following list of assumptions on the parameters in System (1.2):(𝐀)𝑎(𝑥)>0,𝑏(𝑥)>0,𝑐>−𝜋²/𝐿²,𝑑>−𝜋²/𝐿².(𝐅_𝟏)𝐹 is an even functional about (𝑢,𝑣). That is, 𝐹(𝑥,−𝑢,−𝑣)=𝐹(𝑥,𝑢,𝑣) for every (𝑢,𝑣)∈𝐑².(𝐅_𝟐)There exists 𝛽>2, as 𝑢²+𝑣²≠0, we have ()𝑢⋅𝜕𝐹𝑥,𝑢,𝑣()𝜕𝑢+𝑣⋅𝜕𝐹𝑥,𝑢,𝑣()𝜕𝑣≥𝛽𝐹𝑥,𝑢,𝑣>0forevery𝑥∈𝐑. (2.1)(𝐅_𝟑)𝐹(𝑥,𝑢,𝑣)=𝑜(𝑢²+𝑣²) with respect to 𝑥 consistently, as 𝑢²+𝑣²→0.

Denote 𝑎₁=min_{𝑥∈[0,𝐿]}𝑎(𝑥), 𝑎₂=max_{𝑥∈[0,𝐿]}𝑎(𝑥), 𝑏₁=min_{𝑥∈[0,𝐿]}𝑏(𝑥), 𝑏₂=max_{𝑥∈[0,𝐿]}𝑏(𝑥).

From condition (𝐀), we obtain 𝑎_𝑖>0,𝑏_𝑖>0, when 𝑖=1,2.

Remark 2.1. Let 𝑧=(𝑢,𝑣)∈𝐑², then condition (𝐅_𝟐) is transformed to (∇𝐹(𝑥,𝑧),𝑧)>𝛽𝐹(𝑥,𝑧)>0forevery𝑧≠0, (2.2) where (⋅,⋅) represents the usual inner product in 𝐑².

Remark 2.2. From (𝐅_𝟑), we obtain lim_|𝑧|→0𝐹(𝑥,𝑧)/|𝑧|²=0, where |⋅| represents normal norm in 𝐑². Besides, from the continuity of 𝐹, we obtain 𝐹(𝑥,0,0)=0.

Our main result is as follows.

Theorem 2.3. Suppose 𝑎(𝑥),𝑏(𝑥), and 𝐹 satisfy (𝐀), (𝐅_𝟏)−(𝐅_𝟑). Then System (1.2) has infinitely many distinct pairs of solutions 𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛), which are critical points of the functional 𝐼∶𝑋→𝐑, and 𝐼(𝑧_𝑛)→∞ as 𝑛→∞.

In this paper, the existence of periodic solutions of a single equation in System (1.1) are extended to the case of equations, and also the cubic growth of nonlinear term is extended to a general form of superlinear growth.

3. Variational Structure and the Proof of Result

In this section, we prove the main result stated in Section 2.

3.1. Variational Structure

Denote 𝑋(𝐿)=𝐻²(0,𝐿)∩𝐻¹₀(0,𝐿)². (3.1) Then 𝑋(𝐿) is a Hilbert space. The norm is ‖𝑧‖²=‖𝑢‖²_𝑐+‖𝑣‖²_𝑑, (3.2) where ‖𝑢‖_𝑐=^𝐿₀𝑢^′′(𝑥)²+𝑐𝑢^′(𝑥)²+𝑎(𝑥)|𝑢(𝑥)|²𝑑𝑥^1/2,‖𝑣‖_𝑑=^𝐿₀𝑣^′′(𝑥)²+𝑑𝑣^′(𝑥)²+𝑏(𝑥)|𝑣(𝑥)|²𝑑𝑥^1/2, (3.3)𝑧=(𝑢,𝑣)∈𝑋(𝐿). The corresponding inner product are 𝑧₁,𝑧₂=^𝐿₀𝑧₁^′′,𝑧₂^′′+𝑐𝑢^′₁,𝑢^′₂+𝑑𝑣^′₁,𝑣^′₂+𝑎(𝑥)𝑢₁,𝑢₂+𝑏(𝑥)𝑣₁,𝑣₂𝑢𝑑𝑥,₁,𝑢_2𝑐=^𝐿₀𝑢₁^′′,𝑢₂^′′+𝑐𝑢^′₁,𝑢^′₂+𝑎(𝑥)𝑢₁,𝑢₂𝑣𝑑𝑥,₁,𝑣_2𝑑=^𝐿₀𝑣₁^′′,𝑣₂^′′+𝑑𝑣^′₁,𝑣^′₂+𝑏(𝑥)𝑣₁,𝑣₂𝑑𝑥. (3.4) For every 𝑧=(𝑢,𝑣)∈𝑋(𝐿), using Poincaré inequality [7], we obtain ^𝐿₀𝑢²𝐿𝑑𝑥≤²𝜋^2𝐿₀𝑢^′2𝑑𝑥,^𝐿₀𝑢^′2𝐿𝑑𝑥≤²𝜋^2𝐿₀𝑢^′′2𝑑𝑥. (3.5) Thus, we can define another norm ‖⋅‖₁ in 𝑋(𝐿). That is, for every 𝑧∈𝑋(𝐿), ‖𝑧‖₁=^𝐿₀𝑧^′′(𝑥)²𝑑𝑥^1/2. (3.6) The inner product in 𝑋(𝐿) as follows: 𝑧₁,𝑧₂₁=^𝐿₀𝑧₁^′′(𝑥),𝑧₂^′′(𝑥)𝑑𝑥,𝑧₁,𝑧₂∈𝑋(𝐿). (3.7) The two different norms (3.2) and (3.6) are equivalent in 𝑋(𝐿).

In this section we consider System (1.2). The Fréchet derivative of 𝐼 is given by the following: 𝐼()=1𝑢,𝑣2^𝐿₀𝑢^′′2+𝑐𝑢^′2+𝑎(𝑥)𝑢²+𝑣^′′2+𝑑𝑣^′2+𝑏(𝑥)𝑣²𝑑𝑥−^𝐿₀𝐹(𝑥,𝑢,𝑣)𝑑𝑥, (3.8) where 𝑧=(𝑢,𝑣)∈𝑋(𝐿).

Remark 3.1. In general, the growth of 𝐹 is limited by the differentiability of functional 𝐼, but we apply truncation techniques in [8]. First, introduce auxiliary functional and the auxiliary functional is Fréchet differentiable. Second, we use critical point theory to prove the existence of critical point of auxiliary functional, then prove the existence of the original equation. However, in order to avoid technical complexity, we assume directly functional 𝐼 is Fréchet differentiable.

In fact, for every 𝑧=(𝑢,𝑣)∈𝑋(𝐿),𝑧=(𝑢,𝑣)∈𝑋(𝐿), we obtain 𝐼^′(𝑧),𝑧=𝐼^′_𝑢(𝑢,𝑣),𝑢+𝐼^′_𝑣(𝑢,𝑣),𝑣, (3.9) where 𝐼^′_𝑢(𝑢,𝑣),𝑢=^𝐿₀𝑢^′′𝑢^′′+𝑐𝑢^′𝑢^′+𝑎(𝑥)𝑢()𝑢−𝜕𝐹𝑥,𝑢,𝑣𝜕𝑢𝑢𝐼𝑑𝑥,_′𝑣(𝑢,𝑣),𝑣=^𝐿₀𝑣^′′𝑣^′′+𝑐𝑣^′𝑣^′+𝑎(𝑥)𝑣()𝑣−𝜕𝐹𝑥,𝑢,𝑣𝜕𝑣𝑣𝑑𝑥. (3.10) and 𝐼^′_𝑢(𝑢,𝑣),𝐼^′_𝑣(𝑢,𝑣)∈[𝐻²(0,𝐿)∩𝐻¹₀(0,𝐿)]^∗, 𝐼^′(𝑧)∈𝑋(𝐿)^∗.

It is similar to the discussion of [8], the solutions of System (1.2) corresponds to the critical point of the functional 𝐼, so we need to discuss the critical point of functional 𝐼. In order to prove Theorem 2.3, we introduce below definition and lemma.

Definition 3.2 (see [9]). Let X be a real Banach space, 𝐼∈𝐶¹(𝑋,𝑅), 𝐼 is a Fréchet continuously differentiable functional in 𝑋(𝐿). 𝐼 is said to be satisfying Palais-Smale (PS) condition if any sequence {𝑢_𝑛}⊂𝑋 for which {𝐼(𝑢_𝑛)} is bounded and {𝐼^′(𝑢_𝑛)}→0 as 𝑗→∞, possesses a convergent subsequence.

Lemma 3.3 (see [8]). Let 𝑋 be an infinite dimensional Banach space and (𝑋_𝑛)_𝑛 be a sequence of finite dimensional subspaces of 𝑋 such that dim𝑋_𝑛=𝑛, 𝑋₁⊂𝑋₂⊂⋯⊂𝑋_𝑛⊂𝑋,^∞_𝑛=1𝑋_𝑛=𝑋. (3.11) Let 𝐼∈𝐶¹(𝑋,𝑅) be an even functional, 𝐼(0)=0, and 𝐼 satisfy (𝑃𝑆) condition. Suppose that(𝐀_𝟏) there are constants 𝜌,𝛼>0 such that 𝐼|_𝜕𝐵𝜌≥𝛼, and(𝐀_𝟐) for every 𝑛 there is an 𝑅_𝑛>0 such that 𝐼≤0 on 𝑋_𝑛⧵𝐵_𝑅𝑛.
Then 𝐼 possesses infinitely many pairs of critical points with unbounded sequence of critical values.

3.2. The Proof of Result

Step 1 (Functional 𝐼 satisfies (PS) condition). Let {𝑧_𝑛}={(𝑢_𝑛,𝑣_𝑛)} be a (PS) sequence in 𝑋, that is, {𝐼(𝑧_𝑛)} is bounded and 𝐼^′(𝑧_𝑛)→0, as 𝑛→∞. Suppose that {𝑧_𝑛} is unbounded in 𝑋, that is, ‖𝑧_𝑛‖→∞ as 𝑛→∞. Since 𝐼𝑧_𝑛+1𝛾𝐼^′𝑧_𝑛𝑧_𝑛≥𝐼𝑧_𝑛−1𝛾𝐼^′𝑧_𝑛,𝑧_𝑛=1𝛾𝑧_𝑛², (3.12) it follows that 𝐼𝑧_𝑛𝑧_𝑛²+𝐼^′𝑧_𝑛𝛾𝑧_𝑛≥1𝛾, (3.13) where 𝛾≥4. Letting 𝑛→∞ in (3.13), we have a contradiction with ‖𝑧_𝑛‖→∞ as 𝑛→∞.
Therefore {𝑧_𝑛} is a bounded sequence in 𝑋(𝐿). Passing if necessary to a subsequence we may assume that {𝑧_𝑛} is weakly convergent to a function 𝑧∈𝑋(𝐿), 𝑧_𝑛⇀𝑧 in 𝑋(𝐿), and 𝑧_𝑛→𝑧 in 𝐶[(0,𝐿)].
From the Lebesgue theorem, 𝑧∈𝑋(𝐿), 𝑧_𝑛⇀𝑧 in 𝑋(𝐿), and 𝑧_𝑛→𝑧 in 𝐶[(0,𝐿)], letting 𝑛→∞ in (3.9) 𝐼^′𝑧_𝑛,𝑧_𝑛=𝑧_𝑛²−^𝐿₀𝜕𝐹𝑥,𝑢_𝑛,𝑣_𝑛𝑢𝜕𝑢^𝑛𝑑𝑥−𝜕𝐹𝑥,𝑢_𝑛,𝑣_𝑛𝑣𝜕𝑣^𝑛𝐼𝑑𝑥,_′𝑧_𝑛,𝑧=𝑧_𝑛,𝑧−^𝐿₀𝜕𝐹𝑥,𝑢_𝑛,𝑣_𝑛𝜕𝑢𝑢𝑑𝑥−𝜕𝐹𝑥,𝑢_𝑛,𝑣_𝑛𝜕𝑣𝑣𝑑𝑥, (3.14) we obtain lim_𝑛→∞𝑧_𝑛²=^𝐿₀𝜕𝐹(𝑥,𝑢,𝑣)()𝜕𝑢𝑢𝑑𝑥+𝜕𝐹𝑥,𝑢,𝑣‖𝑧‖𝜕𝑣𝑣𝑑𝑥=². (3.15) From (3.15) and 𝑧∈𝑋(𝐿), 𝑧_𝑛⇀𝑧 in 𝑋(𝐿), we have ‖𝑧_𝑛−𝑧‖→0 as 𝑛→∞.

Remark 3.4. 𝛾 is the largest sum of the order of 𝑢 and 𝑣.

Step 2 (Geometric conditions). Let 𝑒₁=(1,0),𝑒₂=(0,1), then {𝑒₁,𝑒₂} constitutes a pair of standard orthogonal base in 𝐑^𝟐. Let us define 𝑋_2𝑚 to be the subspace of 𝑋(𝐿)𝑋_2𝑚=spansin𝑘𝜋𝑥𝐿𝑒_𝑖,𝑖=1,2,𝑘=1,2,…,𝑚, (3.16) for every 𝑚∈𝐍. We have dim𝑋_2𝑚=2𝑚, 𝑋₁⊂𝑋₂⋯⊂𝑋_2𝑚⊂𝑋,⋃^∞_𝑛=1𝑋_𝑛=𝑋.
For a given constant 𝜌>0, define a bounded closed set 𝐾⊂𝑋_2𝑚𝐾=𝑧=(𝑢,𝑣)∈𝑋_2𝑚∣𝑧=^𝑚_𝑘=1𝛼_𝑘sin𝑘𝜋𝑥𝐿𝑒₁+𝛽_𝑘sin𝑘𝜋𝑥𝐿𝑒₂,^𝑚_𝑘=1𝛼²_𝑘+𝛽²_𝑘=𝜌². (3.17)

Define mapping 𝐻∶𝑋_2𝑚→𝐑^2𝑚. For any 𝑧∈𝑋_2𝑚, we obtain 𝐻(𝑧)=𝛼₁,𝛽₁,𝛼₂,𝛽₂,…,𝛼_𝑚,𝛽_𝑚𝜌. (3.18) It is clear that 𝐻 is a linear odd mapping. For every 𝑧∈𝑋_2𝑚, we have ‖𝑧‖²₁=^𝐿₀𝑢^′′(𝑥)²+𝑣(𝑥)^′′2=𝜋𝑑𝑥₄2𝐿^3𝑚_𝑘=1𝑘⁴𝛼²_𝑘+𝛽²_𝑘. (3.19)

So 𝜌²𝜋⁴2𝐿³|𝐻(𝑧)|²≤|𝑧|²₁≤𝜌²(𝑚𝜋)⁴2𝐿³|𝐻(𝑧)|². (3.20) From (3.20), we obtain 𝐻 is an odd homeomorphism from 𝑋_2𝑚 to 𝐑^2𝑚. Then 𝐻 is an odd homeomorphism from 𝐾 to 𝑆^2𝑚−1, since 𝐻(𝐾)=𝑆^2𝑚−1.

On one hand, from functional (3.8) and using Sobolev’s embedding theorem, we obtain 𝐼(𝑧)≥12‖𝑧‖²−𝜀|𝑧|²𝐿≥12‖𝑧‖²𝜋−𝜀²𝐿‖𝑧‖². (3.21) Thus condition (𝐀_𝟏) is fulfilled if 𝜀=𝐿/4𝜋², 𝜌=‖𝑧‖/2.

On the other hand, as −𝐹(𝑥,𝑢,𝑣)<0, then there exists 𝜎, such that −𝐹(𝑥,𝑢,𝑣)<−(1/4)𝜎‖𝑧‖⁴₁.

Denote 𝐴(𝑛)=(𝑛𝜋/𝐿)⁴+𝑝(𝑛𝜋/𝐿)²+𝑎. From functional (3.8), we obtain 𝐼(𝑧)≤12^𝐿₀𝑧^′′2+𝑝𝑧^′2+𝑎𝑧²𝑑𝑥−^𝐿₀𝐹()≤𝐿𝑥,𝑧𝑑𝑥4𝐴(𝑛)‖𝑧‖²₁−^𝐿₀𝐹()≤𝐿𝑥,𝑧𝑑𝑥4𝐴(𝑛)‖𝑧‖²₁−𝐿4𝜎‖𝑧‖⁴₁, (3.22) where 𝑝=max{𝑐,𝑑},𝑎=max{𝑎₂,𝑏₂}. Here choosing 𝑅_𝑛=‖𝑧‖₁≥𝐴(𝑛)/𝜎, we obtain 𝐼(𝑧)≤0. (3.23) So (𝐀_𝟐) holds. The proof of Theorem 2.3 is completed.

Example 3.5. In System (1.2), consider the problem: 𝐹(𝑥,𝑢,𝑣)=𝑝₀(𝑥)𝑢^𝑛+𝑝₁(𝑥)𝑢^𝑛−1𝑣+⋯+𝑝_𝑖(𝑥)𝑢^𝑛−𝑖𝑣^𝑖+⋯+𝑝_𝑛−1(𝑥)𝑢𝑣^𝑛−1+𝑝_𝑛(𝑥)𝑣^𝑛, (3.24) where 𝑝_𝑖(𝑥)≥0, but there exists at least one 𝑝_𝑖(𝑥)≠0, 𝑛 is an even and 𝑛≥4, 𝑖=0,1,2,…,𝑛.

It is obvious that 𝐹(𝑥,−𝑢,−𝑣)=𝐹(𝑥,𝑢,𝑣) and 𝐹(𝑥,𝑢,𝑣)=𝑜(𝑢²+𝑣²) as 𝑢²+𝑣²→0.

For the superlinear property, we calculate that ()𝑢⋅𝜕𝐹𝑥,𝑢,𝑣()𝜕𝑢+𝑣⋅𝜕𝐹𝑥,𝑢,𝑣𝜕𝑣=𝑛𝑝₀(𝑥)𝑢^𝑛+(𝑛−1)𝑝₁(𝑥)𝑢^𝑛−1𝑣+⋯+(𝑛−𝑖)𝑝_𝑖(𝑥)𝑢^𝑛−𝑖𝑣^𝑖+⋯+𝑝_𝑛−1(𝑥)𝑢𝑣^𝑛−1+𝑝₁(𝑥)𝑢^𝑛−1𝑣+⋯+𝑖𝑝_𝑖(𝑥)𝑢^𝑛−𝑖𝑣^𝑖+⋯+(𝑛−1)𝑝_𝑛−1(𝑥)𝑢𝑣^𝑛−1+𝑛𝑝_𝑛(𝑥)𝑣^𝑛()().=𝑛𝐹𝑥,𝑢,𝑣≥4𝐹𝑥,𝑢,𝑣 (3.25) Therefore, there exists 𝛽=4>2, as 𝑢²+𝑣²≠0, we have ()𝑢⋅𝜕𝐹𝑥,𝑢,𝑣()𝜕𝑢+𝑣⋅𝜕𝐹𝑥,𝑢,𝑣()𝜕𝑣≥4𝐹𝑥,𝑢,𝑣>0forevery𝑥∈𝐑. (3.26)

So 𝐹 satisfies the conditions (𝐅_𝟏)−(𝐅_𝟑). We only choose 𝑎(𝑥)>0,𝑏(𝑥)>0, 𝑐>−𝜋²/𝐿²,𝑑>−𝜋²/𝐿², then the condition (𝐀) is satisfied. Therefore, System (1.2) has infinitely many distinct pairs of solutions by using Theorem 2.3.

Acknowledgment

This work is supported by the National Natural Science Foundation of China no. 11126063 and no. 111010981.