In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems x¨(t) + Bx˙(t) − A(t)x(t) + Hx(t, x(t)) = 0, where A(t) ∈ (R, RN) is a symmetric matrix for all t ∈ R, B = [bij] is an antisymmetric N × N constant matrix, and H(t, x) ∈ C 1 (R × Bδ , R) is only locally defined near the origin in x for some δ > 0. With the nonlinearity H(t, x) being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark’s theorem

Jiang Shan

Liu Guanggang

Xu Huijuan

English

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2022, No. 67, 1–8; https://doi.org/10.14232/ejqtde.2022.1.67 www.math.u-szeged.hu/ejqtde/Infinitely many homoclinic solutionsfor a class of damped vibration problemsHuijuan Xu, Shan Jiang and Guanggang LiuBSchool of Mathematical Sciences, Liaocheng University, Liaocheng 252000, P. R. ChinaReceived 18 May 2022, appeared 30 December 2022Communicated by Gabriele BonannoAbstract. In this paper, we consider the multiplicity of homoclinic solutions for thefollowing damped vibration problemsẍ(t) + Bẋ(t)− A(t)x(t) + Hx(t, x(t)) = 0,where A(t) ∈ (R, RN) is a symmetric matrix for all t ∈ R, B = [bij] is an antisymmetricN × N constant matrix, and H(t, x) ∈ C1(R × Bδ, R) is only locally defined near theorigin in x for some δ > 0. With the nonlinearity H(t, x) being partially sub-quadratic atzero, we obtain infinitely many homoclinic solutions near the origin by using a Clark’stheorem.Keywords: homoclinic solutions, Clark’s theorem, critical points, Palais–Smale condi-tion.2020 Mathematics Subject Classification: 34C25, 35A15, 35B38.1 IntroductionThe homoclinic orbit is an important kind of trajectory in dynamical systems recognized byPoincaré at the end of the 19th century. Their presence often means the occurrence of chaosor the bifurcation behavior of periodic orbits, see [4, 7, 10, 12, 14] and references therein. Inrecent decades, the existence and multiplicity of homoclinic orbits has been studied in depthvia variational methods. In this paper, we consider the existence of infinitely many homoclinicsolutions for the following damped vibration problemsẍ(t) + Bẋ(t)− A(t)x(t) + Hx(t, x(t)) = 0, (1.1)where x(t) ∈ C2(R, RN), A(t) = [aij(t)] is a symmetric and positive N × N matrix-valuedfunction with aij ∈ L∞(R, R)(∀i, j = 1, 2, . . . , N), B = [bij] is an antisymmetric N × N constantmatrix, H(t, x) ∈ C1(R × Bδ, R) with Bδ = {x ∈ RN | |x| ≤ δ} for some δ > 0, Hx(t, x) denoteits derivative with respect to the x variable.BCorresponding author. Email: lgg112@163.com2 H. J. Xu, S. Jiang and G. G. Liu,When B = 0, the system (1.1) is the classical second-order Hamiltonian systems which hasbeen extensively studied in the past, see [1, 5, 6, 8, 11, 13, 15, 16] and references therein. WhenB ̸= 0, many authors have studied the existence and multiplicity of homoclinic solutions for(1.1) under various growth conditions, see [2,3,17–19] and references therein. In [17], Wu andZhang obtained the existence and multiplicity of homoclinic solutions by using a symmetricmountain pass theorem and a generalized mountain pass theorem under the local (AR) su-perquadratic growth condition. In [2], by using a variant fountain theorem, Chen obtainedinfinitely many nontrivial homoclinic orbits for non-periodic damped vibration systems whenH(t, x) satisfies the subquadratic condition at infinity. In [19], Zhang and Yuan studied theexistence of the homoclinic solutions via the genus properties in critical point theory whenH(t, x) is of subquadratic growth as |x| → +∞. In [3], Chen and Tang obtained infinitelymany homoclinic solutions for (1.1) by using a fountain theorem when H(t, x) satisfies a newsubquadratic condition. In [18], Zhu obtained the existence of nontrivial homoclinic solutionsusing the mountain pass theorem when H(t, x) satisfies asymptotically quadratic condition.In this paper, we study the existence of homoclinic solutions for (1.1) when the nonlinearityH(t, x) is only defined near the origin with respect to x and H(t, x) is partially subquadraticat zero. To the best of our knowledge, the existence of homoclinic solutions for dampedvibration systems in this case has not been considered before. Our work is motivated by [9],where the authors improved and extended Clark’s theorem and applied it to the problems onsolutions of elliptic equations and periodic solutions of Hamiltonian systems. Here by usingthe Clark’s theorem in [9], we prove that (1.1) has infinitely many homoclinic solutions nearthe origin. Furthermore, we make the following assumptions:(H1) H(t, x) ∈ C1(R × Bδ, R) is even in x, H(t, x) = H(t,−x) for all t ∈ R and x ∈ Bδ, andH(t, 0) = 0 for all t ∈ R;(H2) There exists constants α > 0, such that (A(t)x, x) ≥ α|x|2 and ∥B∥ < 2√α for all(t, x) ∈ (R, RN);(H3) There exist t0 ∈ R and r > 0 such that uniformly in t ∈ [t0 − r, t0 + r],lim|x|→0H(t, x)|x|2 = +∞;(H4) For all (t, x) ∈ R × Bδ,|Hx(t, x)| ≤ b(t),where b(t) : R → R is a function such that b ∈ Lξ(R) for some 1 ≤ ξ ≤ 2.Now, we state the main result as follows.Theorem 1.1. Assume that (H1)–(H4) hold, then (1.1) has infinitely many homoclinic solution xkwith ∥xk∥L∞ → 0 as k → ∞.Remark 1.2. Now we give some comparisons between our result and other results on the sys-tem (1.1). Firstly, in the previous works [2,3, 17–19], the authors needed to make assumptionsabout the behavior of the nonlinearity H(t, x) as |x| → +∞. They assumed that H(t, x) satis-fies the subquadratic condition, superquadratic condition or asymptotically quadratic condi-tion at infinity. Compared with these works, we do not need the behavior of the nonlinearityH(t, x) for |x| large. Secondly, our subquadratic conditions near zero are also weaker than therelated papers [2,3]. In [2,3], the authors assumed that H(t, x) satisfies lim|x|→0H(t,x)|x|2 = +∞ forInfinitely many homoclinic solutions for a class of damped vibration problems 3all t ∈ R. By contrast, we only assume that lim|x|→0H(t,x)|x|2 = +∞ in a interval t ∈ [t0 − r, t0 + r].Thirdly, in the literature [2,3,17–19], the authors did not give the information for the obtainedhomoclinic solutions. However, we can prove that the homoclinic solutions found here con-verge to the null solution in L∞ norm.Example 1.3. Let H(t, x) = η(t)|x|µ, where 1 < µ < 2, η(t) ∈ C∞(R, R) satisfies that η(t) = 1,∀|t| ≤ 1, and η(t) = 0, ∀|t| ≥ 2. It is not difficult to see that H(t, x) satisfies all conditionsof Theorem 1.1. It is worth noting that H(t, x) does not satisfies lim|x|→0H(t,x)|x|2 = +∞ for allt ∈ R.The remainder of this paper is organized as follows. In Section 2, we give the variationalframework for (1.1). In Section 3, we prove our main result in detail.2 PreliminariesIn this section, we establish the variational framework for (1.1) and give a preliminary result.Let E = H1(R, RN) be a Hilbert space where the function is from R to RN with the innerproduct⟨x, y⟩0 =∫R((x(t), y(t)) + (ẋ(t), ẏ(t)))dt, ∀x, y ∈ E0, (2.1)where (·, ·) means the standard inner product in RN . The corresponding norm is∥x∥0 =( ∫R(|x(t)|2 + |ẋ(t)|2)dt) 12, ∀x ∈ E0. (2.2)For simplicity, we define a new norm on E. Let∥x∥ =( ∫R[|ẋ|2 + (A(t)x(t), x(t))− (Bẋ(t), x(t))]dt) 12, ∀x ∈ E. (2.3)And the corresponding inner product is denoted by ⟨·, ·⟩. Now we show that the norms ∥ · ∥and ∥ · ∥0 are equivalent. Since ∥B∥ < 2√α from (H2), then∥B∥22α < 2. Hence we can choose aconstant ε0 such that∥B∥22α< ε0 < 2. (2.4)SetC0 = min{1 − ε02, α − ∥B∥22ε0}. (2.5)By (2.4), we see that C0 > 0. Then by (H2) and mean inequality, we have∥x∥2 =∫R[|ẋ|2 + (A(t)x(t), x(t))− (Bẋ(t), x(t))]dt≥∫R[(1 − ε02)|ẋ(t)|2 + (α − ∥B∥22ε0)|x|2]dt≥ C0∥x∥20.(2.6)4 H. J. Xu, S. Jiang and G. G. Liu,On the other hand,∥x∥2 =∫R[|ẋ|2 + (A(t)x(t), x(t))− (Bẋ(t), x(t))]dt≤∫R[|ẋ(t)|2 + ∥A(t)∥L∞(R)|x|2 + ∥B∥(|ẋ(t)|2 + |x|2)]dt≤ C1∥x∥20,(2.7)where C1 = (1 + ∥A(t)∥L∞(R) + ∥B∥) is a constant. Therefore, the norms ∥ · ∥ and ∥ · ∥0 areequivalent.To obtain the homoclinic solution of (1.1), we consider the following systemsẍ(t) + Bẋ(t)− A(t)x(t) + Ĥx(t, x(t)) = 0, (2.8)where Ĥ ∈ C1(R×RN , R) satisfies that Ĥ is even in u, Ĥ(t, x) = H(t, x) for t ∈ R and |x| < δ2 ,and Ĥ(t, x) = 0 for t ∈ R and |x| > δ.Define the functional Φ on E byΦ(x) =12∫R[|ẋ|2 + (A(t)x(t), x(t))− (Bẋ(t), x(t))]dt −∫RĤ(t, x(t))dt=12∥x∥2 −∫RĤ(t, x(t))dt.(2.9)By (H1), Φ ∈ C1(E, R) and the critical points of Φ correspond to the homoclinic solutions of(2.8) (see [17]). We can get that⟨Φ′(x), y⟩ =∫R[(ẋ(t), ẏ(t)) + (A(t)x(t), y(t))− (Bẋ(t), y(t))]dt−∫R(Ĥx(t, x(t)), y(t))dt.(2.10)Now we introduce a Clark’s theorem established by Liu and Wang [9]. Clark’s theoremis a classical theorem in the critical point theory and has a large number of applications indifferential equations. In [9], Liu and Wang improved and extended Clark’s theorem, andapplied it to elliptic equations and Hamiltonian systems.Let X be a Banach space, Φ ∈ C1(X, R). We say that Φ satisfies (PS) condition if anysequence {xj} such that Φ(xj) is bounded and Φ′(xj) → 0 as j → ∞ contains a convergentsubsequence.Theorem 2.1 ([9]). Assume Φ satisfies the (PS) condition, is even and bounded from below, andΦ(0) = 0. If for any k ∈ N, there exists a k-dimensional subspace Xk of X and ρk > 0 such thatsupXk ⋂ Sρk Φ < 0, where Sρ = {x ∈ X | ∥x∥ = ρ}, then at least one of the following conclusionsholds.(1) There exists a sequence of critical points {xk} satisfying Φ(xk) < 0 for all n and ∥xk∥ → 0 ask → ∞.(2) There exists r > 0 such that for any 0 < a < r there exists a critical point x such that ∥x∥ = aand Φ(x) = 0.Remark 2.2. Clearly, under the assumptions of Theorem 2.1 there exist infinitely many criticalpoints xk of Φ that satisfies Φ(xk) ≤ 0, Φ(xk) → 0 and ∥xk∥ → 0 as k → ∞.Infinitely many homoclinic solutions for a class of damped vibration problems 53 Proof of the main resultIn this section, we use Theorem 2.1 to prove the main result of this paper.Proof. Step 1. We prove that Φ is bounded from below. Let ∥ · ∥Lp(R) denote the norm ofLp(R, RN)(1 ≤ p ≤ ∞). By (H4), we have that|Ĥ(t, x)| ≤ b(t)|x|, ∀(t, x) ∈ R × RN , (3.1)where b ∈ Lξ(R) is from (H4). If ξ = 1, we have∫RĤ(t, x(t))dt ≤∫Rb(t)|x|dt ≤ ∥x∥L∞(R)∫Rb(t)dt ≤ C′1∥x∥∥b(t)∥L1(R), (3.2)where the Sobolev inequality ∥x∥L∞(R) ≤ C′1∥x∥ has been used. If 1 < ξ ≤ 2, by the Hölderinequality and the Sobolev inequality, we have∫RĤ(t, x(t))dt ≤( ∫R(b(t))ξ) 1ξ( ∫R|x|ξξ−1 dt) ξ−1ξ≤ C′ξ∥x∥∥b(t)∥Lξ (R). (3.3)Then, by (3.2), (3.3) we can see that∫RĤ(t, x(t))dt ≤ C′ξ∥x∥∥b(t)∥Lξ (R). (3.4)Therefore by (2.3) and (3.4), we haveΦ(x) =12∫R[|ẋ|2 + (A(t)x(t), x(t)) + (Bx(t), ẋ(t))]dt−∫RĤ(t, x(t))dt≥ 12∥x∥2 − C′ξ∥x∥∥b(t)∥Lξ (R).(3.5)Consequently, Φ is bounded from below.Step 2. We prove that Φ(x) satisfies the (PS) condition. Let {xn} be a (PS) sequence, thatis Φ(xn) is bounded and Φ′(xn) → 0 as n → ∞. By (3.5), we see that {xn} is bounded inE. Hence, there exists a subsequence of {xn} (for simplicity still denoted by {xn}) and somex0 ∈ E such that xn ⇀ x0 in E, and xn → x0 strongly in Cloc(R1) as n → ∞. Then Φ′(x0) = 0.Notice that∥xn − x0∥2 = ⟨(Φ′(xn)− Φ′(x0)), (xn − x0)⟩+∫R((Ĥx(t, xn)− Ĥx(t, x0)), (xn − x0))dt (3.6)Since xn ⇀ x0 in E and Φ′(xn) → 0 as n → ∞, we have⟨(Φ′(xn)− Φ′(x0)), (xn − x0)⟩ → 0 as n → ∞. (3.7)6 H. J. Xu, S. Jiang and G. G. Liu,By (3.1), the Hölder inequality and the Sobolev inequality, for every R > 0 we have∣∣∣∣∫R((Ĥx(t, xn)− Ĥx(t, x0)), (xn − x0))dt∣∣∣∣≤∫R|Ĥx(t, xn)− Ĥx(t, x0)||xn − x0|dt≤∫R\[−R,R]|Ĥx(t, xn)− Ĥx(t, x0)|(|xn|+ |x0|)dt+∫ R−R|Ĥx(t, xn)− Ĥx(t, x0)||xn − x0|dt≤ 2∫R\[−R,R]b(t)(|xn|+ |x0|)dt +∫ R−R|Ĥx(t, xn)− Ĥx(t, x0)||xn − x0|dt≤ 2∥b(t)∥Lξ (R\[−R,R])(∥xn∥+ ∥x0∥) +∫ R−R|Ĥx(t, xn)− Ĥx(t, x0)||xn − x0|dt.(3.8)For any ε > 0, since b(t) ∈ Lξ(R) and {xn} is bounded in E, there exists R0 > 0 large enoughsuch that(∥xn∥+ ∥x0∥)∥b(t)∥Lξ (R\[−R0,R0]) <ε4, ∀n ∈ Z+. (3.9)On the other hand, since xn → x0 strongly in C([−R0, R0]), there must exist n0 ∈ Z+ suchthat for n ≥ n0 ∫ R0−R0|Ĥx(t, xn)− Ĥx(t, x0)||xn − x0|dt <ε2. (3.10)Then by (3.8),(3.9) and (3.10), for n ≥ n0 we have∣∣∣∣∫R(Ĥx(t, xn)− Ĥx(t, x0), xn − x0)dt∣∣∣∣ < ε,which implies that ∣∣∣∣∫R(Ĥx(t, xn)− Ĥx(t, x0), xn − x0)dt∣∣∣∣ → 0 as n → ∞. (3.11)Hence, by (3.6), (3.7) and (3.11), we have xn → x0 in E as n → ∞. Therefore, Φ(x) satisfies the(PS) condition.Step 3. We show that for every k ∈ N, there exists a k-dimensional subspace Xk of X and ρk >0 such that supXk ⋂ Sρk Φ < 0. Let Xk be a k-dimensional subspace of C∞0 ([t0 − r, t0 + r]). SinceXk is a finite dimensional space and the norms in finite dimensional space are all equivalent,there exists a positive constant Ck > 0 such that∥x∥2 ≤ Ck∥x∥2L2 , ∀x ∈ Xk. (3.12)By (H3) and the definition of Ĥ(t, x), there exists a constant 0 < δk < δ2 such that for t ∈[t0 − r, t0 + r] and x ∈ Bδk , we haveĤ(t, x) ≥ Ck|x|2. (3.13)Infinitely many homoclinic solutions for a class of damped vibration problems 7Recall the Sobolev inequality ∥x∥L∞(R) ≤ C′1∥x∥, we take ρk =δkC′1. Then for any x ∈ Sρk , wehave ∥x∥L∞ < δk. Thus by (2.9), (3.12) and (3.13), for any x ∈ Xk ∩ Sρk we haveΦ(x) ≤ 12∥x∥2 − Ck∥x∥2L2(R)< −12∥x∥2= −12ρ2k < 0,which implies that supXk ⋂ Sρk Φ < 0. 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Infinitely many homoclinic solutions for a class of damped vibration problems

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Infinitely many homoclinic solutions for a class of damped vibration problems

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