Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory

Abstract

By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations 𝑥̇(𝑡)=−𝑓(𝑥(𝑡−𝜎)), where 𝜎∈(0,+∞) and 𝑓∈𝐶(ℝ^𝑛,ℝ^𝑛). It is shown that all the periodic solutions derived here are brought about by the time delay.

1. Introduction

This paper is concerned with the existence of periodic solutions to the system of delay differential equations 𝑥̇(𝑡)=−𝑓(𝑥(𝑡−𝜎)),(1) where 𝜎∈(0,+∞) and 𝑓∈𝐶(ℝ^𝑛,ℝ^𝑛).

Delay differential equations have widely been applied to describe the dynamics phenomena in both natural and manmade processes such as chemistry, physics, engineering, and economics. The existence of the periodic solutions for delay differential equations has been extensively investigated by using various methods, including fixed point theorems [1–5], Hopf bifurcation theorems [6–8], variational methods [9–14], the methods of differential inequalities [15–21], and other effective approaches (e.g., see [22–24]). In [25–31], the minimal periods of the periodic solutions to Lipschitzian differential equations are estimated through the Lipschitz constants (see Remark 4).

The use of variational methods in the study of 4𝜎-periodic solutions of system (1) having a variational structure was introduced in 2005 by Guo and Yu [9]. Assume that(F₁)𝑓 is odd in 𝑥; that is, 𝑓(−𝑥)=−𝑓(𝑥), for all 𝑥∈ℝ^𝑛;(F₂)there exists 𝐹∈𝐶¹(ℝ^𝑛,ℝ) such that 𝐹_𝑥(𝑥)=𝑓(𝑥), for all 𝑥∈ℝ^𝑛, where 𝐹_𝑥 denotes the gradient of 𝐹.

In [9], the authors obtained the multiplicity results for periodic solutions to (1) in the case that 𝑓 is asymptotically linear. Later, the existence of the periodic solution of (1) was investigated by using Morse theory and Galerkin methods [10]. For the other relative investigations, we refer the reader to [11–14].

Many practical problems, such as nonlinear population growth models and control systems working with potentially explosive chemical reactions, can be transformed into the form of (1). For example, by the change of variables 𝑦=𝑎tanh(𝑎𝑥), the following generalized food-limited population model 𝑦̇(𝑡)=−𝜃sign𝑦(𝑡−1)𝑦(𝑡−1)^𝛾𝑎²−𝑦²(𝑡)(2) is transformed equivalently into (1) with 𝑛=1, 𝑓(𝑥)=𝜃𝑎^𝛾sign(𝑥)|tanh(𝑎𝑥)|^𝛾, and 𝜎=1, where 𝜃 and 𝑎 are positive numbers. When 𝛾=1, 𝑓^′(0)=𝜃𝑎². It is known from [24] that, with the slope 𝑓^′(0) increasing and tending to infinity, the number of the periodic solutions of (2) increases and tends to infinity. Naturally, one would conjecture that when 0<𝛾<1, (2) possesses infinitely many periodic solutions, since in this case lim_𝑥→0𝑓^′(𝑥)=+∞.

Motivated by the above observation, in this paper, we study the existence of infinitely many periodic solutions to the system (1) under the assumptions (F₁), (F₂), and(F₃)there are 1<𝛼, 𝛽<2 and 𝑑₁,𝑟₀>0 such that(i)0<(𝑓(𝑥),𝑥)⩽𝛼𝐹(𝑥), for all 𝑥∈𝐵_𝑟0⧵{𝟎};(ii)𝑑₁|𝑓(𝑥)|^𝛽′⩽𝐹(𝑥), for all 𝑥∈𝐵_𝑟0, where 1/𝛽+1/𝛽^′=1, 𝐵_𝑟0={𝑥∈ℝ^𝑛∶|𝑥|⩽𝑟₀}.

Here and subsequently, (⋅,⋅), |⋅| denote the inner product and the standard norm in ℝ^𝑛, respectively, and the bold face 𝟎 represents the coordinate origin of 𝐑^𝑛. The main result of this paper is stated as follows.

Theorem 1. Assume that (𝐹₁)--(𝐹₃) hold. Then (1) possesses a sequence of nonconstant 4𝜎-periodic solutions {𝑥_𝑚} satisfying ‖𝑥_𝑚‖_∞→0 as 𝑚→∞.

Example 2. When 0<𝛾<1, it is easy to check that 𝑓(𝑥)=𝜃𝑎sign(𝑥)|tanh(𝑎𝑥)|^𝛾 satisfies (F₁)--(F₃) with 𝛼=𝛽=1+𝛾; then (2) has a sequence of nonconstant 4-periodic solutions {𝑥_𝑚} satisfying ‖𝑥_𝑚‖_∞→0 as 𝑚→∞.

Remark 3. Let us compare the result here with that in the case of ordinary differential equations (ODE). Without the time delay, (1) reduces to the following system of ODE 𝑥^′(𝑡)=𝑓(𝑥(𝑡)).(3) Let 𝑥(𝑡)=𝑥(𝑡;𝑥₀) be the solution of (3) satisfying the initial condition 𝑥(0)=𝑥₀≠𝟎. Then the derivative of the Lyapunov function 𝑉(𝑥)=|𝑥|² along 𝑥(𝑡) reads 𝑑𝑉𝑑𝑡₍₃₎=−𝑓(𝑥(𝑡)),𝑥(𝑡).(4) From (F₃)-(i), we see that 𝑑𝑉/𝑑𝑡|₍₃₎<0 for 0<|𝑥|<𝑟₀, which implies that there is no any periodic orbit of (3) across 𝐵_𝑟0⧵{𝟎}; that is, the trivial solution is an isolated periodic solution. However, by the above theorem, with the time delay, the system (1) possesses infinitely many periodic solutions in any neighborhood of the origin.

Remark 4. Consider the following system of 𝑚th order functional differential equations: 𝑥^(𝑚)(𝑡)=𝑓(𝑥(𝜏(𝑡))),𝑡∈ℝ,(5) where 𝑓∶ℝ^𝑛↦ℝ^𝑛 satisfies the Lipschitz condition and 𝜏∶ℝ¹↦ℝ¹ is a measurable function. The lower bounds for the periods of the periodic solutions to (5) and their special forms are estimated in [25–31]. From this perspective, Theorem 1 complements the information in the case of non-Lipschitzian differential equations. For the unique solvability of the periodic problems on functional differential equations, we refer the reader to [1, 15–21].

The remainder of this paper is divided into two parts. In the next section, we state the preliminaries on the variational structure for (1). In the final section, the proof of Theorem 1 will be given via the ℤ₂-genus theory, together with an approximating argument.

2. Preliminaries

Let 𝐿²(𝑆¹,ℝ^𝑛) denote the set of 𝑛-tuples of 2𝜋-periodic functions which are square integrable. If 𝑥∈𝐿²(𝑆¹,ℝ^𝑛), it has a Fourier expansion 𝑥(𝑡)=𝑎₀+_𝑗∈ℕ𝑎_𝑗cos𝑗𝑡+𝑏_𝑗sin𝑗𝑡,(6) where 𝑎_𝑖,𝑏_𝑗∈ℝ^𝑛 and the series converges in the space 𝐿²(𝑆¹,ℝ^𝑛). For 𝑥∈𝐿²(𝑆¹,ℝ^𝑛) with its expansion (6), set 𝐻∶={𝑥∈𝐿²(𝑆¹,ℝ^𝑛)∣‖𝑥‖_𝐻<∞}, where ‖𝑥‖_𝐻∶=𝑎₀²+_𝑗∈ℕ1+𝑗𝑎_𝑗²+𝑏_𝑗².(7) Then 𝐻, equipped with the norm ‖⋅‖_𝐻, is a Sobolev space.

On the other hand, for 𝑥∈𝐻 with its expansion (6), set ‖𝑥‖∶=𝑎₀²+_𝑗∈ℕ𝑗𝑎_𝑗²+𝑏_𝑗².(8) Then 𝐻 possesses another norm ‖⋅‖ which is equivalent to ‖⋅‖_𝐻. In the following, we always employ ‖⋅‖ as the norm of 𝐻. The associated inner product with ‖⋅‖ is denoted by ⟨⋅,⋅⟩.

Now set 𝐸∶={𝑥∈𝐻∣𝑥(⋅+𝜋)=−𝑥(⋅)}.(9) Then 𝐸 is a closed subspace of 𝐻 and the Fourier expansion of 𝑥∈𝐸 reduces to 𝑥(𝑡)=^∞_𝑗=1𝑎_2𝑗−1cos2𝑗−1𝑡+𝑏_2𝑗−1sin2𝑗−1𝑡.(10) Thus with 𝑥₁,𝑥₂∈𝐸 being expanded as 𝑥_𝑖(𝑡)=^∞_𝑗=1𝑎^(𝑖)_2𝑗−1cos2𝑗−1𝑡+𝑏^(𝑖)_2𝑗−1sin2𝑗−1𝑡,𝑖=1,2,(11) we have 𝑥₁,𝑥₂=^∞_𝑗=12𝑗−1𝑎⁽¹⁾_2𝑗−1,𝑎⁽²⁾_2𝑗−1+𝑏⁽¹⁾_2𝑗−1,𝑏⁽²⁾_2𝑗−1.(12)

For 𝑥,𝑦∈𝐸, we call 𝑦 a weak derivative of 𝑥 and denote it by 𝑥̇=𝑦 if ₀^2𝜋𝑥(𝑡),𝑧^′(𝑡)𝑑𝑡=−₀^2𝜋𝑦(𝑡),𝑧(𝑡)𝑑𝑡,∀𝑧∈𝐶^∞𝑆¹,ℝ^𝑛.(13) Further, for 𝑥∈𝐶^∞(𝑆¹,ℝ^𝑛)∩𝐸 with its expansion (10), define 𝐴(𝑥)1∶=2₀^2𝜋𝑥̇𝜋𝑡+2(𝑡)=1,𝑥𝑑𝑡2^∞_𝑗=1(−1)^𝑗2𝑗−1𝑎_2𝑗−1²+𝑏_2𝑗−1².(14) Then it is easy to check that |𝐴(𝑥)|⩽‖𝑥‖² for 𝑥∈𝐶^∞(𝑆¹,ℝ^𝑛)∩𝐸. Therefore 𝐴 extends to all of 𝐸 as a continuous quadratic form. This extension will still be denoted by 𝐴.

Let 𝐹∈𝐶¹(ℝ^𝑛,ℝ) and satisfy 𝐹(−𝑥)=𝐹(𝑥),𝐹(𝑥)⩽𝐶₁+𝐶₂|𝑥|^𝑠,𝑥∈ℝ^𝑛(15) for some 𝑠∈[1,∞). Define Φ(𝑥)∶=₀^2𝜋𝐹(𝑥(𝑡))𝑑𝑡,𝑥∈𝐸(16) and 𝐼(𝑥)=𝐴(𝑥)+Φ(𝑥), 𝑥∈𝐸. The following lemma is derived from [9, Lemma  2.2].

Lemma 5 (see [9]). Let 𝐹∈𝐶¹(ℝ^𝑛,ℝ) and satisfy (15). Then 𝐼∈𝐶¹(𝐸,ℝ) and 𝐼^′(𝑥)𝑦=𝐴^′(𝑥)𝑦+₀^2𝜋𝐹_𝑥(𝑥(𝑡)),𝑦(𝑡),𝑦∈𝐸,(17) where 𝐴^′(𝑥)𝑦=₀^2𝜋𝑥̇𝜋𝑡+2,𝑦(𝑡)𝑑𝑡,𝑦∈𝐸.(18) Moreover, the existence of 2𝜋-periodic solutions 𝑥(𝑡) for 𝑥^′(𝑡)=−𝐹_𝑥𝑥𝜋𝑡−2(19) satisfying 𝑥∈𝐸 is equivalent to the existence of critical points of functional 𝐼.

Let {𝑒₁,…,𝑒_𝑛} be the orthonormal basis of ℝ^𝑛. For 𝑘∈ℕ, set 𝐸₊(𝑘)∶=spancos[(4𝑘−1)𝑡]𝑒_𝑖,sin[(4𝑘−1)𝑡]𝑒_𝑖∶,𝐸𝑖=1,2,…,𝑛₋(𝑘)∶=spancos[(4𝑘−3)𝑡]𝑒_𝑖,sin[(4𝑘−3)𝑡]𝑒_𝑖∶𝑖=1,2,…,𝑛.(20) For 𝑙,𝑚∈ℕ∪{+∞}, define 𝑉^±_𝑙=⊕^𝑙_𝑘=1𝐸_±(𝑘),𝑉^𝑚_𝑙=𝑉⁻_𝑙⊕𝑉⁺_𝑚,(21) where the closure is of 𝐸 sense. Set 𝑉^±∶=𝑉^±_+∞; then 𝐸=𝑉⁺⊕𝑉⁻. In the rest of this paper, this decomposition will always be referred to when a point 𝑥∈𝐸 is written as 𝑥=𝑥⁺+𝑥⁻, where 𝑥^±∈𝑉^±.

Remark 6. In view of (12), (14), and (18), we see that 𝐴(𝑥)=12𝑥⁺²−‖𝑥⁻‖²(22) and that 𝐴^′(𝑥)𝑦=𝑥⁺,𝑦⁺−𝑥⁻,𝑦⁻,𝑥,𝑦∈𝐸.(23)
The following lemma is derived from [32, Lemma  2.1].

Lemma 7 (see [32]). For each 𝑠∈[1,∞) there is 𝛾_𝑠>0 such that ‖𝑥‖_𝑠⩽𝛾_𝑠𝑚^−1/𝑠‖𝑥‖(24) for all 𝑥∈(𝑉^𝑚−1_𝑚−1)^⟂ with 𝑚⩾2, the orthogonal complement in 𝐸, where (and below) ‖⋅‖_𝑠 denotes the usual 𝐿^𝑠-norm.

3. Proof of Theorem 1

Without loss of generality we assume that 𝜎=𝜋/2 since, under the change of variables 𝑦(𝑡)=𝑥(2𝜎𝑡/𝜋), (1) can be transformed into the system 𝑦̇(𝑡)𝑓𝑦𝜋=−𝑡−2,1^′ where 𝑓(𝑦)=(2𝜎/𝜋)𝑓(𝑦) still satisfies (F₁--F₃) with 𝑓 being replaced by 𝑓.

Let 𝜒∈𝐶^∞(ℝ,[0,1]) be such that 𝜒(𝑠)=0 for 𝑠⩽𝑟₀/2, 𝜒(𝑠)=1 for 𝑠⩾𝑟₀, and 𝜒^′(𝑠)>0 for 𝑠∈(𝑟₀/2,𝑟₀). Define 𝐹∶ℝ^𝑛↦ℝ by 𝐹(𝑥)∶=1−𝜒(|𝑥|)𝐹(𝑥)+𝜒(|𝑥|)𝑀₀|𝑥|^𝛼,(25) where 𝑀₀=inf{𝐹(𝑥)/𝑟^𝛼₀∶𝑟₀/2⩽|𝑥|⩽𝑟₀}.

Let 𝛼^′>0 be such that 1/𝛼+1/𝛼^′=1. By (F₃) we get 0<𝐹_𝑥(𝑥),𝑥⩽𝛼𝐹(𝑥),∀𝑥∈ℝ^𝑛𝐹(𝑥)⩾𝐶,(26)₁𝐹_𝑥(𝑥)^𝛽′,|𝑥|𝐶⩽1,₁𝐹_𝑥(𝑥)^𝛼′,|𝑥|>1,(27) where (and below) 𝐶_𝑗's stand for positive constants.

Lemma 8. Let 𝐹∶ℝ^𝑛↦ℝ be defined by (25); then 1<𝛽<𝛼<2, 𝐹∈𝐶¹(ℝ^𝑛,ℝ), and 𝐶₂|𝑥|^𝛼⩽𝐹(𝑥)⩽𝐶₃|𝑥|^𝛽,|𝑥|𝐶⩽1,₃|𝑥|^𝛼,|𝑥|>1.(28)

Proof. From (25), it is easy to see that 𝐹∈𝐶¹(ℝ^𝑛,ℝ). Now we start to prove (28). Let 𝑀 be such a constant that |ln𝐹(𝑥)−𝛼ln|𝑥||⩽𝑀 for 𝑥∈𝑆₁≡𝜕𝐵₁. For 𝑥∈ℝ^𝑛, |𝑥|⩽1, set 𝑥₀=𝑥/|𝑥|; then 𝑥₀∈𝑆₁. Define 𝑔(𝑡)=ln𝐹(𝑡𝑥₀)−𝛼ln|𝑡𝑥₀|, 𝑡∈(0,1]. Then, by (26), 𝑔^′(𝑡)=𝐹_𝑥𝑡𝑥₀𝐹𝑡𝑥₀,𝑥₀−𝛼⩽0,(29) which implies that 𝑔(|𝑥|)⩾𝑔(1); that is, ln𝐹(𝑥)−𝛼ln|𝑥|⩾ln𝐹𝑥₀−𝛼ln𝑥₀⩾−𝑀.(30) It follows that 𝐹(𝑥)⩾𝑒^−𝑀|𝑥|^𝛼 for |𝑥|⩽1, which, combining with (25), leads to the inequality on the left hand of (28) with 𝐶₂ being chosen adequately.
Again, for 𝑥∈ℝ^𝑛, |𝑥|⩽1, set 𝑥₀=𝑥/|𝑥| and define ℎ(𝑡)=(𝐹(𝑡𝑥₀))^1/𝛽−𝑡|𝑥₀|/(𝛽𝐶^1/𝛽′₁), 𝑡∈[0,1]. Then by the first inequality in (27), ℎ^′(𝑡)=1𝛽𝐹_𝑥𝑡𝑥₀,𝑥₀𝐹𝑡𝑥₀^1/𝛽′−1𝐶^1/𝛽′₁𝑥₀⩽𝑥₀𝛽𝐹_𝑥𝑡𝑥₀𝐹𝑡𝑥₀^1/𝛽′−1𝐶^1/𝛽′₁<0.(31) Thus ℎ(|𝑥|)⩽ℎ(0)=0, which leads to 𝐹(𝑥)⩽𝐶^′₃|𝑥|^𝛽, where 𝐶^′₃=1/(𝛽𝐶^1/𝛽′₁)^𝛽. In the same way, from the second inequality in (27), we can arrive at 𝐹(𝑥)⩽𝐶₃^′′|𝑥|^𝛼 for |𝑥|>1, where the constant 𝐶₃^′′ only depends on 𝛼 and 𝐶₁. With 𝐶₃=max{𝐶^′₃,𝐶₃^′′}, the inequalities on the right hand of (28) hold. Thus we get (28), which implies that 𝐶₂|𝑥|^𝛼⩽𝐶₃|𝑥|^𝛽 for |𝑥|⩽1 and that 1<𝛽<𝛼<2. The proof is complete.

Now we consider the functional 𝐼(𝑥)=𝐴(𝑥)+₀^2𝜋𝐹(𝑥)𝑑𝑡,𝑥∈𝐸.(32)

Lemma 9. 𝐼 satisfies (𝑃𝑆) condition; that is, every sequence {𝑥_𝑘}⊂𝐸 such that {𝐼(𝑥_𝑘)} is bounded and 𝐼^′(𝑥_𝑘)→0 as 𝑘→∞ has a convergent subsequence.

Proof. By Lemma 5, for 𝑥∈𝐸, 𝐼^′(𝑥) is defined by 𝐼^′(𝑥)𝑦=𝐴^′(𝑥)𝑦+₀^2𝜋𝐹_𝑥(𝑥),𝑦𝑑𝑡,∀𝑦∈𝐸.(33) To verify that 𝐼 satisfies (PS) condition, we suppose |𝐼(𝑥_𝑘)|⩽𝐶₄ and 𝐼^′(𝑥_𝑘)→0 as 𝑘→∞. Note that, for large 𝑘, |𝐼^′(𝑥_𝑘)𝑥|⩽‖𝑥‖. Thus for large 𝑘 and 𝑥=𝑥_𝑘, from (32) and (33), 𝐶₄+‖𝑥‖(𝑥)−1⩾𝐼2𝐼^′(𝑥)𝑥=₀^2𝜋𝐹(𝑥)−12𝐹_𝑥(𝑥),𝑥𝑑𝑡.(34) Noticing that 1<𝛽<𝛼<2, we see from (25) that, for all 𝑥∈ℝ^𝑛, 𝐹_𝑥(𝑥),𝑥⩾𝐶₅max|𝑥|^𝛼,|𝑥|^𝛽−𝐶₆,(35) which, combining with (26) and (34), implies 𝐶₄+‖𝑥‖⩾𝛼⁻¹−2⁻¹₀^2𝜋𝐹_𝑥(𝑥),𝑥𝑑𝑡⩾𝐶₇max‖𝑥‖^𝛼_𝛼,‖𝑥‖^𝛽_𝛽−𝐶₈,𝑥‖(36)max‖^𝛼_𝛼,‖𝑥‖^𝛽_𝛽⩽𝐶₉(‖𝑥‖+1).(37)
Next for large 𝑘, taking 𝑥=𝑥_𝑘 and 𝜍=𝑥⁺_𝑘 in ₀^2𝜋𝐹_𝑥(𝑥),𝜍𝑑𝑡+𝐴^′(𝑥)𝜍=𝐼^′(𝑥)𝜍⩽‖𝜍‖(38) and using (23), (27), and (28) and the Hölder inequality (1/𝛼+1/𝛼^′=1, 1/𝛽+1/𝛽^′=1), we get 𝑥⁺²⩽₀^2𝜋𝐹_𝑥(𝑥),𝑥⁺𝑑𝑡+𝑥⁺⩽𝐶_{10|𝑥(𝑡)|>1}|𝑥|^𝛼/𝛼′𝑥⁺𝑑𝑡+_{|𝑥(𝑡)|⩽1}|𝑥|^𝛽/𝛽′𝑥⁺+𝑥𝑑𝑡₊⩽𝐶₁₀‖𝑥‖^𝛼/𝛼′_𝛼𝑥⁺_𝛼+‖𝑥‖^𝛽/𝛽′_𝛽𝑥⁺_𝛽+𝑥⁺⩽𝐶₁₁‖𝑥‖^𝛼/𝛼′_𝛼+‖𝑥‖^𝛽/𝛽′_𝛽+1𝑥⁺,(39) where the last inequality holds since 𝐸 is compactly embedded in 𝐿^𝑠(𝑆¹,ℝ^𝑛) for 𝑠⩾1. It follows from (36) that 𝑥⁺⩽𝐶₁₂‖𝑥‖^1/𝛼′+‖𝑥‖^1/𝛽′+1.(40) Similarly, (40) works with 𝑥⁺ being replaced by 𝑥⁻. Combining these inequalities shows ‖𝑥‖⩽𝐶₁₃‖𝑥‖^1/𝛼′+‖𝑥‖^1/𝛽′+1,(41) which implies that {𝑥_𝑘} is bounded in 𝐸.
Let Φ be defined by (16). By [33, Proposition B.37], {Φ^′(𝑥_𝑘)} is precompact in 𝐸. Moreover, from (23) and (33), 𝐼^′𝑥_𝑘=𝑥⁺_𝑘−𝑥⁻_𝑘+Φ^′𝑥_𝑘.(42) It follows that {𝑥_𝑘} has a convergent subsequence. The proof is complete.

Lemma 10. For each 𝑙∈ℕ, there are 𝜌_𝑙>0,  𝑎_𝑙>0, and 0<𝑏_𝑙→0 such that (a)𝐼(𝑥)⩾0,   for all 𝑥∈𝐵_𝜌𝑙∩𝑉_𝑙^+∞ and inf𝐼(𝜕𝑆_𝜌𝑙∩𝑉_𝑙^+∞)⩾𝑎_𝑙, where 𝐵_𝜌𝑙={𝑥∈𝐸∶‖𝑥‖⩽𝜌_𝑙};(b)sup𝐼((𝑉^+∞_𝑙−1)^⟂)⩽𝑏_𝑙.

Proof. Noticing that 𝑉_𝑙^+∞=𝑉⁻_𝑙⊕𝑉⁺_+∞ and that dim(𝑉⁻_𝑙)<∞, we have, for 𝑥∈𝐵_𝜌𝑙∩𝑉_𝑙^+∞, ‖𝑥‖_𝛼=sup_𝑆1𝑥(𝑡),𝑦(𝑡)𝑑𝑡∣𝑦∈𝐿^𝛼′𝑆¹,ℝ^𝑛,𝑦_𝛼′=1⩾sup_𝑆1𝑥(𝑡),𝑦(𝑡)𝑑𝑡∣𝑦∈𝑉⁻_𝑙,𝑦_𝛼′=1=sup_𝑆1𝑥⁻(𝑡),𝑦(𝑡)𝑑𝑡∣𝑦∈𝑉⁻_𝑙,𝑦_𝛼′=‖𝑥=1₋‖_𝛼⩾𝜂_𝑙‖𝑥⁻‖,(43) where 𝜂_𝑙 is a positive constant depending on 𝑙. It follows by (28) that, for 𝑥∈𝐵_𝜌𝑙∩𝑉_𝑙^+∞, 𝐼(𝑥)⩾12𝑥⁺²+𝐶₂𝜂_𝑙‖𝑥⁻‖^𝛼−12‖𝑥⁻‖²=12𝑥⁺²+𝐶₂𝜂_𝑙−12‖𝑥⁻‖^2−𝛼‖𝑥⁻‖^𝛼,(44) which implies (a) by setting 𝜌_𝑙∶=min{1,(𝐶₂𝜂_𝑙)^1/(2−𝛼)} and 𝑎_𝑙=𝜌²_𝑙(1+𝐶₂𝜂_𝑙)/2.
Let 𝑥∈(𝑉^+∞_𝑙−1)^⟂. By (28) and Lemma 7, 𝐼(𝑥)⩽𝐶₃‖𝑥‖^𝛼_𝛼+‖𝑥‖^𝛽_𝛽−12‖𝑥‖²⩽𝐶₁₆𝑙⁻¹‖𝑥‖^𝛼+‖𝑥‖^𝛽−12‖𝑥‖²⩽𝑏_𝑙∶=sup_𝑠≥0𝑔(𝑠),(45) where 𝑔(𝑠)∶=𝐶₁₆𝑙⁻¹(𝑠^𝛼+𝑠^𝛽)−𝑠²/2. Noticing 1<𝛽⩽𝛼<2, one can see that 𝑏_𝑙→0 as 𝑙→∞ and (b) follows. The proof is complete.

In the following, let Σ denote the family of closed (in 𝐸) subsets of 𝐸⧵{𝟎} symmetric with respect to the origin, and 𝛾∶Σ↦ℕ∪{0,∞},(46) the ℤ₂-genus map (see [33]). For 𝑙,𝑚∈ℕ, set Σ^𝑚_𝑙=𝐴∈Σ∶𝐴⊂𝑉^𝑚_+∞,𝛾(𝐴)⩾𝑛(𝑙+𝑚),(47) and define 𝑐_𝑙,𝑚=sup_{𝐴∈Σ𝑚𝑙}inf_𝑥∈𝐴𝐼(𝑢).(48)

Lemma 11. For all 𝑙,𝑚∈ℕ, 𝑐_𝑙,𝑚 is a critical value of 𝐼 and 𝑎_𝑙⩽𝑐_𝑙,𝑚⩽𝑏_𝑙.(49)

Proof. We first prove that (49) holds. For each 𝑚∈ℕ, let 𝜌_𝑙 be chosen as that in Lemma 10; then it follows by Lemma 10(a) that inf𝐼(𝜕𝑆_𝜌𝑙∩𝑉_𝑙^+∞)⩾𝑎_𝑙. Denote 𝐴=𝜕𝑆_𝜌𝑙∩𝑉^𝑚_𝑙; then 𝛾(𝐴)=𝑛(𝑚+𝑙) and 𝐴∈Σ^𝑚_𝑙. Since 𝐴⊂𝜕𝑆_𝜌𝑙∩𝑉_𝑙^+∞, we have 𝑐_𝑙,𝑚⩾inf_𝐴𝑥∈𝐼(𝑥)⩾inf𝐼𝜕𝑆_𝜌𝑙∩𝑉_𝑙^+∞⩾𝑎_𝑙.(50)
On the other hand, for every 𝐴∈Σ^𝑚_𝑙, by the property of genus, 𝐴∩(𝑉^+∞_𝑙−1)^⟂≠Ø, which, from Lemma 10(b), leads to inf_𝑥∈𝐴𝐼(𝑥)⩽𝑏_𝑙 for every 𝐴∈Σ^𝑚_𝑙. Thus 𝑐_𝑙,𝑚⩽𝑏_𝑙 and (49) holds.
By (F₁) and (25), 𝐹(𝑥) is even with respect to 𝑥, which implies that 𝐼 is even. We claim that 𝑐=𝑐_𝑙,𝑚 is a critical point of 𝐼. Otherwise, there exists 𝜖>0, such that there is no any critical point in the interval (𝑐−𝜖,𝑐+𝜖). By the definition of 𝑐_𝑙,𝑚, there exists 𝐴∈Σ^𝑚_𝑙, such that inf_𝑥∈𝐴𝐼(𝑥)>𝑐−𝜖.(51) For 𝑎∈ℝ, denote 𝐼^𝑎={𝑥∈𝐸∶𝐼(𝑥)⩾𝑎}. Use a positive rather than a negative gradient flow [33, Remark  A.17], we get 𝜂∈𝐶([0,1]×𝐸,𝐸) such that 𝜂(1,⋅) is odd and 𝜂1,𝐼^𝑐−𝜖⊂𝐼^𝑐+𝜖.(52) Since 𝐴⊂𝐼^𝑐−𝜖, we have 𝜂(1,𝐴)⊂𝐼^𝑐+𝜖; that is, inf_{()𝑥∈𝜂1,𝐴}𝐼(𝑥)⩾𝑐+𝜖.(53)
On the other hand, by the property of genus, we know that 𝛾(𝜂(1,𝐴))∈Σ^𝑚_𝑙, which, by the definition of 𝑐, leads to 𝑐⩾inf_{()𝑥∈𝜂1,𝐴}𝐼(𝑥)⩾𝑐+𝜖.(54) This contradiction implies that 𝑐_𝑙,𝑚 is a critical value of 𝐼. The proof is complete.

Now we are in a position to give the following proof.

Proof of Theorem 1. In view of Lemma 11, let 𝑥_𝑙,𝑚∈𝑉^𝑚_+∞ be such that 𝐼𝑥_𝑙,𝑚=𝑐_𝑙,𝑚,𝐼^′𝑥_𝑙,𝑚=0.(55) Then by (PS) condition, along a subsequence as 𝑚→∞, 𝑥_𝑙,𝑚→𝑥_𝑙∈𝐸 such that 𝑎_𝑙⩽𝐼𝑥_𝑙⩽𝑏_𝑙,𝐼^′𝑥_𝑙=0,(56) which implies that 𝑥_𝑙 is nonzero. Moreover, by Lemma 5, 𝑥^′_𝑙(𝑡)=−𝐹_𝑥𝑥_𝑙𝜋𝑡−2.(57)
We claim that, for sufficiently large 𝑙, 𝑥_𝑙 solves (1). In fact, from (26) and (56) 𝑏_𝑙⩾𝐼𝑥_𝑙=𝐼𝑥_𝑙−12𝐼^′𝑥_𝑙𝑥_𝑙⩾𝛼1−2₀^2𝜋𝐹𝑥_𝑙𝑑𝑡.(58) By (27), (58), and Hölder inequality 𝑥⁺_𝑙²=₀^2𝜋𝐹_𝑥𝑥_𝑙,𝑥⁺_𝑙⩽𝑥𝑑𝑡_{+𝑙𝛼02𝜋}𝐹_𝑥𝑥_𝑙^𝛼′𝑑𝑡^1/𝛼′+𝑥⁺_𝑙𝛽0^2𝜋𝐹_𝑥𝑥_𝑙^𝛽′𝑑𝑡^1/𝛽′⩽𝐶₁₇𝑥⁺_𝑙0^2𝜋𝐹𝑥_𝑙𝑑𝑡^1/𝛼′+𝐶₁₈𝑥⁺_𝑙0^2𝜋𝐹𝑥_𝑙𝑑𝑡^1/𝛽′⩽𝐶₁₇𝑏^1/𝛼′_𝑙+𝐶₁₈𝑏^1/𝛽′_𝑙𝑥⁺.(59) Similarly, the above inequality works with 𝑥⁺_𝑙 replaced by 𝑥⁻_𝑙. These inequalities yield 𝑥_𝑙≤𝐶₁₉𝑏^1/𝛼′_𝑙+𝐶₂₀𝑏^1/𝛽′_𝑙.(60) Since 𝑏_𝑙→0 as 𝑙→∞, it follows that 𝑥_𝑙⟶0as𝑙⟶∞.(61)
Furthermore, from (27) and (57), we have 𝑥̇₀^2𝜋_𝑙(𝑡)²𝑑𝑡=₀^2𝜋𝐹_𝑥𝑥_𝑙²𝑑𝑡⩽𝐶₂₁₀^2𝜋𝐹𝑥_𝑙^2/𝛽′𝑑𝑡+₀^2𝜋𝐹𝑥_𝑙^2/𝛼′𝑑𝑡.(62) It follows from (58) that ‖𝑥̇_𝑙‖₂→0 as 𝑙→∞. Recalling (61), we get 𝑥_𝑙𝑊1,2⟶0as𝑙⟶∞,(63) which implies that ‖𝑥_𝑙‖_∞→0 as 𝑙→0. Thus for 𝑚 sufficiently large, ‖𝑥_𝑙‖_∞<𝑟₀/2 and therefor 𝐹_𝑥(𝑥_𝑙)=𝐹_𝑥(𝑥_𝑙). It follow from (57) that, for 𝑙 sufficiently large, 𝑥_𝑙 solves (1). In addition, by (1) and (F₃)(i), the only constant solution of (1) is the trivial solution. Then (56) yields that 𝑥_𝑙 is nonconstant and the proof of Theorem 1 is complete.

Acknowledgments

The authors are grateful to the referees’ careful reviewing and helpful comments. The work was supported by BJJW (KZ201310028031 and KM2014).