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Multiple periodic solutions for damped vibration systems with superquadratic terms at infinity

Abstract

By using variational methods, we obtain infinitely many nontrivial periodic solutions for a class of damped vibration systems with superquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature. Besides, some examples are given to illustrate our results.

Introduction and the main result

In this paper, we study the existence of infinitely many nontrivial periodic solutions for the following damped vibration system:

$$ \textstyle\begin{cases} \ddot{u}+(q(t)I_{N\times N}+B)\dot{u}+(\frac{1}{2}Bq(t)-A(t))u+ F_{u}(t,u)=0,\quad t\in\mathbb{R},\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \quad T>0, \end{cases} $$
(1.1)

where \(u=u(t)\in C^{2}(\mathbb{R},\mathbb{R}^{N})\), \(I_{N\times N}\) is the \(N\times N\) identity matrix, \(q(t)\in L^{1}(\mathbb {R};\mathbb{R})\) is T-periodic and satisfies \(\int_{0}^{T} q(t)\,dt=0\), \(A(t)=[a_{ij}(t)]\) is a T-periodic symmetric \(N\times N\) matrix-valued function with \(a_{ij}\in L^{\infty}(\mathbb{R};\mathbb{R})\) (\(\forall i,j=1, 2,\ldots, N\)), \(B=[b_{ij}]\) is an antisymmetric \(N\times N\) constant matrix.

When \(B=0\) (zero matrix), the authors [6] studied the special case of (1.1) and obtained the existence and multiplicity of periodic solutions. When \(B\neq0\), the author [2] obtained infinitely many periodic solutions of (1.1) with \(F(t,u)\) satisfying the asymptotically quadratic condition:

$$ \lim_{ \vert u \vert \rightarrow\infty}\frac{F(t,u)}{ \vert u \vert ^{2}}=V(t) \quad\mbox{uniformly for } t \in[0,T], \mbox{ where } \inf_{t\in[0,T]}V(t)>0, $$
(1.2)

the authors [4] obtained one existence result and two multiplicity results with \(F(t,u)\) satisfying the superquadratic condition:

$$ 0< \mu F(t,u)\leq\bigl(F_{u}(t,u),u\bigr),\quad\forall t\in[0,T], \forall \vert u \vert \geq r, $$
(1.3)

where \(\mu> 2\) and \(r\geq0\) are some constants, \((\cdot,\cdot)\) and \(|\cdot|\) denote the inner product and the associated norm in \(\mathbb {R^{N}}\); the author [3] used a more general superquadratic condition (\(\lim_{|u|\rightarrow\infty}\frac{F(t,u)}{|u|^{2}}=+\infty\) uniformly for \(t\in[0,T]\)) and obtained infinitely many periodic solutions for (1.1).

By the more general superquadratic condition used in [3] and some weaker conditions, we also obtain infinitely many periodic solutions for (1.1). Our main result reads as follows.

Theorem 1.1

System (1.1) has infinitely many nontrivialT-periodic solutions if\(F(t,u)\)isT-periodic in t and even inu, and the following conditions hold:

\({(F_{1})}\):

\(F(t,u)\)is measurable intfor every\(u\in\mathbb {R^{N}}\)and continuously differentiable inufor a.e. \(t\in[0,T]\), and there exist\(a\in C(\mathbb{R^{+}},\mathbb{R^{+}})\), \(b\in L^{1}([0,T];\mathbb{R^{+}})\)such that

$$\bigl\vert F(t,u) \bigr\vert \leq a\bigl( \vert u \vert \bigr)b(t), \qquad \bigl\vert F_{u}(t,u) \bigr\vert \leq a\bigl( \vert u \vert \bigr)b(t), \quad \forall(t,u)\in[0,T]\times\mathbb{R}^{N}. $$
\({(F_{2})}\):

\(F(t,u)\geq0\), \(\forall(t,u)\in[0,T]\times\mathbb{R}^{N}\), and

$$\lim_{ \vert u \vert \rightarrow\infty}\frac{F(t,u)}{ \vert u \vert ^{2}}=+\infty\quad\textit {uniformly for } t\in[0,T]. $$
\({(F_{3})}\):

There exists\(\alpha>2\)such that

$$\lim_{ \vert u \vert \rightarrow\infty}\frac{F_{u}(t,u)}{ \vert u \vert ^{\alpha-1}}< +\infty \quad\textit{uniformly for } t\in[0,T]. $$
\({(F_{4})}\):

There are constants\(b>0\)and\(1\leq\beta\in(\alpha -2,+\infty)\)such that

$$\lim_{ \vert u \vert \rightarrow\infty}\inf\frac{(F_{u}(t,u),u)-2F(t,u)}{ \vert u \vert ^{\beta}}>b \quad\textit{uniformly for } t\in[0,T]. $$

To compare our result with the most related result [3], we firstly describe the result in [3].

Theorem 1.2

([3])

System (1.1) has infinitely many nontrivialT-periodic solutions if\(F\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})\)isT-periodic in t and even inu, and it satisfies the following conditions:

\({(\mathit{SF}_{1})}\):

\(F(t,0)=0\), \(\forall t\in[0,T]\), and there are two constants\(d_{1}>0\)and\(\alpha_{1}>2\)such that

$$\bigl\vert F_{u}(t,u) \bigr\vert \leq d_{1}\bigl(1+ \vert u \vert ^{\alpha_{1}-1}\bigr), \quad\forall(t,u)\in [0,T]\times \mathbb{R}^{N}. $$
\({(\mathit{SF}_{2})}\):

\(\frac{1}{2}(F_{u}(t,u),u)\geq F(t,u)\geq0\)for all\((t,u)\in [0,T]\times\mathbb{R}^{N}\)and

$$\lim_{ \vert u \vert \rightarrow\infty}\frac{F(t,u)}{ \vert u \vert ^{2}}=+\infty\quad\textit {uniformly for } t\in[0,T]. $$
\({(\mathit{SF}_{3})}\):

There is a constant\(b>0\)such that

$$\lim_{ \vert u \vert \rightarrow\infty}\inf\frac{(F_{u}(t,u),u)-2F(t,u)}{ \vert u \vert ^{\alpha _{1}}}>b \quad\textit{uniformly for } t\in[0,T]. $$

Remark 1.1

The method of Theorem 1.1 is based on the fountain theorem of Bartsch [1], which is essentially different from the variant fountain theorem developed by Zou [7] used in [3]. Our result extends and improves the result in [3]. The reasons are as follows:

  1. (1)

    We only need F to satisfy \((F_{1})\) rather than \(F\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})\).

  2. (2)

    We remove the condition \(\frac{1}{2}(F_{u}(t,u),u)\geq F(t,u)\) for all \((t,u)\in[0,T]\times\mathbb{R}^{N}\) in \((\mathit{SF}_{2})\).

  3. (3)

    Condition \((F_{3})\) is weaker than \((\mathit{SF}_{1})\). Indeed, condition \((\mathit{SF}_{1})\) implies

    $$\bigl\vert F(t,u)-F(t,0) \bigr\vert = \biggl\vert \int_{0}^{1}\bigl(F_{u}(t,su),u\bigr)\,ds \biggr\vert \leq d_{1}\bigl( \vert u \vert + \vert u \vert ^{\alpha_{1}}\bigr),\quad\forall(t,u)\in[0,T]\times\mathbb{R}^{N}, $$

    it follows from the continuity of \(F(\cdot,0)\) that (for all \(\alpha \ge\alpha_{1}\))

    $$\begin{gathered} \lim_{ \vert u \vert \rightarrow\infty}\sup\frac{ \vert F(t,u) \vert }{ \vert u \vert ^{\alpha}} \\ \quad\leq\lim_{ \vert u \vert \rightarrow\infty}\sup\frac{d_{1}( \vert u \vert + \vert u \vert ^{\alpha_{1}})+\max_{t\in[0,T]} \vert F(t,0) \vert }{ \vert u \vert ^{\alpha}} < +\infty\quad \mbox{uniformly for } t\in[0,T]. \end{gathered} $$
  4. (4)

    The constant β in our condition \((F_{4})\) is more general than \(\alpha_{1}\) in \((\mathit{SF}_{3})\).

Example 1.1

The following example is given to illustrate our result. Let

$$F(t,u):=h(t) \vert u \vert ^{2}\ln\bigl(1+ \vert u \vert ^{2}\bigr),\quad u\in\mathbb{R}^{N},t\in[0,T], $$

where \(h\in L^{\infty}(0,T;\mathbb{R}^{+})\) with \(\inf_{t\in [0,T]}h(t)>0\). Then

$$F_{u}(t,u)=2h(t)\ln\bigl(1+ \vert u \vert ^{2}\bigr)u+ \frac{2h(t) \vert u \vert ^{2}u}{1+ \vert u \vert ^{2}}. $$

It is not hard to check that the function satisfies our conditions \((F_{1})\)\((F_{4})\).

The rest of our paper is organized as follows. In Sect. 2, we establish variational framework associated with (1.1) and give some preliminary lemmas, which are useful in the proof of our result, and then we give the detailed proof of our main result.

Variational frameworks and the proof of Theorem 1.1

Let \(\|\cdot\|_{p}\) denote the norm of \(L^{p}([0,T];\mathbb{R}^{N})\) for any \(p\in[1,\infty]\). Let

$$\begin{aligned} W:= &{}\bigl\{ u=u(t): [0,T]\rightarrow\mathbb{R}^{N}| u\mbox{ is absolutely continuous}, u(0)=u(T), \mbox{ and}\\ &\dot{u}\in L^{2} \bigl([0,T];\mathbb{R}^{N}\bigr) \bigr\} \end{aligned} $$

with the inner product

$$(u,v)_{W}:= \int_{0}^{T} \bigl[(u,v) +(\dot{u},\dot{v}) \bigr] \,dt, \quad \forall u,v\in W. $$

The corresponding norm is defined by \(\|u\|_{W}=(u,u)_{W}^{1/2}\). Obviously, W is a Hilbert space.

Let

$$Q(t):= \int_{0}^{t}q(s)\,ds $$

and

$$\Vert u \Vert _{0}:= \biggl( \int_{0}^{T}e^{Q(t)} \bigl( \vert u \vert ^{2} + \vert \dot{u} \vert ^{2} \bigr)\, dt \biggr)^{1/2}, \quad u\in W. $$

Obviously, the norm \(\|\cdot\|_{0}\) is equivalent to the usual one \(\| \cdot\|_{W}\) on W. We denote by \(\langle\cdot,\cdot\rangle_{0}\) the inner product corresponding to \(\|\cdot\|_{0}\) on W.

The corresponding functional for problem (1.1) is defined by

$$ \varPhi(u):=\frac{1}{2} \int_{0}^{T}e^{Q(t)} \bigl[ \vert \dot{u} \vert ^{2}+(Bu,\dot {u})+\bigl(A(t)u,u\bigr) \bigr]\,dt - \int_{0}^{T}e^{Q(t)}F(t,u)\,dt, \quad u\in W. $$
(2.1)

Let \(L: W\to W^{\ast}\) (\(W^{*}\) is the dual space of W) be an operator defined by

$$Lu(v):= \int_{0}^{T}e^{Q(t)} \biggl[(B\dot{u},v)+ \frac {1}{2}q(t) (Bu,v) \biggr]\,dt, \quad\forall u,v\in W. $$

We can identify \(W^{\ast}\) with W by Riesz representation theorem, so Lu can be viewed as a function belonging to W such that

$$Lu(v)=\langle Lu,v\rangle_{0}, \quad\forall u,v\in W. $$

It is not hard to check that L is a bounded linear operator on W. From the discussion in [4], we get that L is self-adjoint and compact on W. Since B is an antisymmetric \(N\times N\) constant matrix, it follows from integration by parts that

$$\langle Lu,u\rangle_{0}= \int_{0}^{T}e^{Q(t)} \biggl[(B\dot{u},u)+ \frac {1}{2}q(t) (Bu,u) \biggr]\,dt = \int_{0}^{T}e^{Q(t)} \bigl[(B\dot{u},u) \bigr]\,dt. $$

We define an operator \(K: W\to W^{\ast}\) by

$$\langle Ku,v\rangle_{0}=\langle Lu,v\rangle_{0}+ \int_{0}^{T}e^{Q(t)} \bigl( \bigl(I_{N\times N}-A(t)\bigr)u,v \bigr)\,dt, \quad\forall u,v\in W. $$

Then it is easy to check that K is a bounded self-adjoint linear operator. Therefore, the definitions of \(\langle\cdot,\cdot\rangle_{0}\) and K imply that Φ defined in (2.1) can be rewritten as

$$\varPhi(u)=\frac{1}{2}\bigl\langle (I-K)u,u\bigr\rangle _{0}- \int _{0}^{T}e^{Q(t)}F(t,u)\,dt, \quad u\in W, $$

where I denotes the identity operator.

By the classical spectral theory, we have the decomposition

$$W=W^{0} \oplus W^{-}\oplus W^{+}, $$

where \(W^{0}:=\ker(I-K)\), \(W^{+}\) and \(W^{-}\) are the positive and negative spectral subspaces of \(I-K\) in W, respectively. Besides, \(W^{-}\) and \(W^{0}\) are finite dimensional (see [4]). Obviously, we can define a new equivalent inner product \(\langle\cdot ,\cdot\rangle\) on W with corresponding norm \(\|\cdot\|\) such that

$$\bigl\langle (I-K)u,u\bigr\rangle _{0}=\pm \Vert u \Vert ^{2}, \quad\forall u\in W^{\pm}. $$

Then we have

$$\begin{aligned} \varPhi(u)&=\frac{1}{2}\bigl\langle (I-K)u,u \bigr\rangle _{0}-I(u) \\ &=\frac{1}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2} \bigr)-I(u), \quad u\in W, \end{aligned} $$

where \(I(u):=\int_{0}^{T}e^{Q(t)}F(t,u)\,dt\). Then, by the assumptions of F, we know that I and Φ are continuously differentiable and

$$\varPhi'(u)v=\bigl\langle u^{+},v^{+}\bigr\rangle -\bigl\langle u^{-},v^{-}\bigr\rangle -I'(u)v, \quad I'(u)v= \int_{0}^{T}e^{Q(t)} \bigl(F_{u}(t,u),v \bigr)\,dt, $$

where \(u,v\in W=W^{-} \oplus W^{0}\oplus W^{+}\) with \(u=u^{-}+u^{0}+u^{+}\) and \(v=v^{-}+v^{0}+v^{+}\); besides, the T-periodic solutions of (1.1) are the critical points of the \(C^{1}\) functional \(\varPhi: W\rightarrow\mathbb{R}\) (see [4]).

Since the embedding of W into \(C(0,T;\mathbb{R}^{N})\) is compact, there exists a constant \(C>0\) such that

$$ \Vert u \Vert _{\infty}\leq C \Vert u \Vert ,\quad\forall u\in W, $$
(2.2)

where \(\|u\|_{\infty}=\max_{t\in[0,T]}|u(t)|\). Besides, by the Sobolev embedding theorem, we get directly the following lemma.

Lemma 2.1

W is compactly embedded in\(L^{p}([0,T];\mathbb{R}^{N})\), \(\forall p\in [1,+\infty]\).

In order to prove Theorem 1.1, we state the fountain theorem of Bartsch (see [1, 5]). Let W be a Banach space with the norm \(\|\cdot\|\) and \(W:=\overline {\bigoplus_{m\in\mathbb{N}}X_{m}}\) with \(\dim X_{m}<\infty\) for any \(m\in \mathbb{N}\). Set

$$ Y_{k}:=\bigoplus_{m=1}^{k}X_{m}, \qquad Z_{k}:=\overline{\bigoplus_{m=k}^{\infty}X_{m}}. $$
(2.3)

Lemma 2.2

(Fountain theorem)

We assume that\(\varPhi\in C^{1}(X,\mathbb{R})\)satisfies the Cerami condition, \(\varPhi(-u)=\varPhi(u)\). For almost every\(k\in N\), there exist\(\rho_{k}>r_{k}>0\)such that

  1. (i)

    \(a_{k}:=\inf_{u\in Z_{k},\|u\|=r_{k}}\varPhi(u)\rightarrow +\infty\)as\(k\rightarrow\infty\).

  2. (ii)

    \(b_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}\varPhi(u)\leq0\).

ThenΦhas a sequence of critical values tending to +∞.

Remark 2.1

Under the Palais–Smale (PS) condition, the fountain theorem is established in [1, 5]. Because the deformation theorem holds true under the Cerami condition, we know the fountain theorem is still valid under the Cerami condition. Here, if any sequence \(u_{n}\subset X\) such that \(\{\varPhi(u_{n})\}\) is bounded and \(\|\varPhi{'}(u_{n})\|(1+\|u_{n}\|)\rightarrow0\) as \(n\rightarrow \infty\) has a convergent sequence, we say that \(\varPhi\in C^{1}(X, {R})\) satisfies the Cerami condition, such a subsequence is then called a Cerami sequence.

Lemma 2.3

If assumptions\((F_{1})\), \((F_{3})\), and\((F_{4})\)hold, thenΦsatisfies the Cerami condition \((C)\).

Proof

We assume that, for any sequence \(\{u_{n}\}\subset W\), \(\{\varPhi (u_{n})\}\) is bounded and \(\|\varPhi{'}(u_{n})\| {(1+\|u_{n}\| )}\rightarrow0\).

Part 1. We firstly prove the boundedness of \(\{u_{n}\}\). There is a constant \(M>0\) such that

$$ \bigl\vert \varPhi(u_{n}) \bigr\vert \leq M\quad\mbox{and}\quad \bigl\Vert \varPhi{'}(u_{n}) \bigr\Vert \bigl(1+ \Vert u_{n} \Vert \bigr)\leq M,\quad\forall n\in\mathbb{N}^{+}. $$
(2.4)

It follows from \((F_{4})\) that there exist \(c_{1}>0\) and \(M_{1}>1\) such that

$$ \bigl(F_{u}(t,u),u\bigr)-2F(t,u)\geq c_{1} \vert u \vert ^{\beta},\quad\forall \vert u \vert \geq M_{1}, \forall t\in[0,T]. $$
(2.5)

By \((F_{1})\), one has that

$$ \bigl\vert \bigl(F_{u}(t,u),u\bigr)-2F(t,u) \bigr\vert \leq c_{2}b(t),\quad\forall \vert u \vert \leq M_{1},\forall t \in[0,T], $$
(2.6)

where \(c_{2}=(M_{1}+2)\max_{s\in[0,M_{1}]}a(s)\). Combining (2.5) and (2.6), we get that

$$ \bigl(F_{u}(t,u),u\bigr)-2F(t,u)\geq c_{1} \vert u \vert ^{\beta}-c_{1}M^{\beta }_{1}-c_{2}b(t), \quad\forall \vert u \vert \in\mathbb{R}^{N},\forall t\in[0,T]. $$
(2.7)

It follows from \(e^{Q(t)}\geq d_{1}\) for some constant \(d_{1}>0\) (\(\forall t\in[0,T]\)), (2.4), (2.7), the definitions of \(\varPhi (u)\) and \(\varPhi{'}(u)\) that

$$ \begin{aligned}[b] 3M&\geq2\varPhi(u_{n})-\bigl\langle \varPhi{'}(u_{n}),u_{n}\bigr\rangle \\ &= \int_{0}^{T}e^{Q(t)} \bigl[ \bigl(F_{u}(t,u_{n}),u_{n} \bigr)-2F(t,u_{n}) \bigr]\,dt \\ &\geq d_{1}c_{1} \int_{0}^{T} \vert u_{n} \vert ^{\beta}\,dt-d_{1}c_{1}M_{1}^{\beta }T-d_{1}c_{2} \int_{0}^{T}b(t)\,dt, \quad\forall n\in \mathbb{N}^{+}. \end{aligned} $$
(2.8)

By (2.8) and \(b\in L^{1}([0,T];\mathbb{R^{+}})\), we have

$$ \int_{0}^{T} \vert u_{n} \vert ^{\beta}\,dt\leq D,\quad\forall n\in\mathbb{N}^{+} $$
(2.9)

for some \(D>0\). Let \(\varPi_{n}=\{t\in[0,T]:|u_{n}|\geq M_{1}\}\), then we have

$$\int_{\varPi_{n}} \vert u_{n} \vert ^{\beta}\,dt \leq D_{1},\quad\forall n\in \mathbb{N}^{+} $$

for some \(D_{1}>0\). Since \(\beta\geq1\), we also have

$$ \int_{\varPi_{n}} \vert u_{n} \vert \,dt\leq D_{1},\quad\forall n\in\mathbb{N}^{+}. $$
(2.10)

For any \(n\in N\), let \(\chi_{n}:\mathbb{R}\rightarrow\mathbb{R}\) be the indicator of \(\varPi_{n}\), that is,

$$\chi_{n}(t):= \textstyle\begin{cases} 1, & t\in\varPi_{n},\\ 0,& t\notin\varPi_{n}, \end{cases}\displaystyle \quad\forall n\in \mathbb{N}^{+}. $$

Then, by the definition of \(\varPi_{n}\) and (2.10), we have

$$\bigl\Vert (1-\chi_{n})u_{n} \bigr\Vert _{\infty}\leq M_{1}\quad\mbox{and}\quad \Vert \chi _{n}u_{n} \Vert _{1}\leq D_{1},\quad \forall n\in\mathbb{N}^{+}. $$

It follows from the equivalence of any two norms on a finite-dimensional space \(W^{0}\oplus W^{-}\) that

$$\begin{aligned} \bigl\Vert u_{n}^{-} \bigr\Vert _{2}^{2}&=\bigl(u_{n}^{-},u_{n} \bigr)_{2} \\ &=\bigl(u_{n}^{-},(1-\chi_{n})u_{n} \bigr)_{2}+\bigl(u_{n}^{-},\chi_{n}u_{n} \bigr)_{2} \\ &\leq \bigl\Vert (1-\chi_{n})u_{n} \bigr\Vert _{\infty}\cdot \bigl\Vert u_{n}^{-} \bigr\Vert _{1}+ \bigl\Vert u_{n}^{-} \bigr\Vert _{\infty}\cdot \Vert \chi_{n}u_{n} \Vert _{1} \\ &\leq\bigl(h_{1} \bigl\Vert (1-\chi_{n})u_{n} \bigr\Vert _{\infty}+h_{2} \Vert \chi_{n}u_{n} \Vert _{1}\bigr) \bigl\Vert u_{n}^{-} \bigr\Vert _{2} \\ &\leq(h_{1} M_{1}+h_{2}D_{1}) \bigl\Vert u_{n}^{-} \bigr\Vert _{2}, \quad\forall n\in \mathbb{N}^{+} \end{aligned} $$

for some \(h_{1}, h_{2}>0\). Therefore,

$$\bigl\Vert u_{n}^{-} \bigr\Vert _{2}\leq h_{1} M_{1}+h_{2}D_{1}, \quad\forall n\in \mathbb{N}^{+}, $$

which together with the equivalence of any two norms on a finite-dimensional space \(W^{0}\oplus W^{-}\) implies that

$$ \bigl\Vert u_{n}^{-} \bigr\Vert \leq D_{2}, \quad\forall n\in\mathbb{N}^{+} $$
(2.11)

for some \(D_{2}>0\).

It follows from \((F_{3})\) that there exist \(c_{3}>0\) and \(M_{2}>0\) such that

$$F(t,u)\leq c_{3} \vert u \vert ^{\alpha},\quad\forall \vert u \vert \geq M_{2},\forall t\in[0,T]. $$

By \((F_{1})\), one has that

$$\bigl\vert F(t,u) \bigr\vert \leq c_{4}b(t),\quad\forall \vert u \vert \leq M_{2},\forall t\in[0,T], $$

where \(c_{4}=\max_{s\in[0,M_{2}]}a(s)\). Hence, we obtain

$$ F(t,u)\leq c_{3} \vert u \vert ^{\alpha}+c_{4}b(t), \quad\forall \vert u \vert \in\mathbb {R}^{N},\forall t\in[0,T]. $$
(2.12)

By (2.4), (2.11), (2.12), and \(e^{Q(t)}\leq d_{2}\) for some constant \(d_{2}>0\) (\(\forall t\in[0,T]\)), we have

$$ \begin{aligned}[b] \frac{1}{2} \Vert u_{n} \Vert ^{2}&=\varPhi(u_{n})+ \bigl\Vert u_{n}^{-} \bigr\Vert + \int _{0}^{T}e^{Q(t)}F(t,u_{n}) \,dt \\ &\leq M+D_{2}+d_{2}c_{3} \int_{0}^{T} \vert u_{n} \vert ^{\alpha}\,dt+d_{2}c_{4} \int _{0}^{T}b(t)\,dt,\quad\forall n\in \mathbb{N}^{+}. \end{aligned} $$
(2.13)

If \(\beta>\alpha\), Hölder’s inequality and (2.9) imply that

$$ \int_{0}^{T} \vert u_{n} \vert ^{\alpha}\,dt\leq T^{(\beta-\alpha)/\beta}\biggl( \int _{0}^{T} \vert u_{n} \vert ^{\beta}\,dt\biggr)^{\alpha/\beta}< +\infty,\quad\forall n\in \mathbb{N}^{+}. $$
(2.14)

It follows from (2.13), (2.14), and \(b\in L^{1}([0,T];\mathbb{R^{+}})\) that \((u_{n})\) is bounded.

If \(\beta\leq\alpha\), using (2.2), we have that

$$\begin{aligned} \int_{0}^{T} \vert u_{n} \vert ^{\alpha}\,dt&= \int_{0}^{T} \vert u_{n} \vert ^{\beta } \vert u_{n} \vert ^{\alpha-\beta}\,dt \\ &\leq \Vert u_{n} \Vert _{\infty}^{\alpha-\beta} \int_{0}^{T} \vert u_{n} \vert ^{\beta}\,dt \\ &\leq C^{\alpha-\beta} \Vert u_{n} \Vert ^{\alpha-\beta} \int_{0}^{T} \vert u_{n} \vert ^{\beta }\,dt,\quad\forall n\in\mathbb{N}^{+}. \end{aligned} $$

Noting the fact that \(\alpha-\beta<2\), it follows from (2.9), (2.13), and \(b\in L^{1}([0,T];\mathbb{R^{+}})\) that \((u_{n})\) is bounded.

Part 2. Then we prove that the sequence \(\{u_{n}\}\) has a convergent sequence. The boundedness of \(\{u_{n}\}\) implies that \(u_{n}\rightharpoonup u\) in W. First, we prove

$$ \int_{0}^{T}e^{Q(t)} \bigl(F_{u}(t,u_{n}),u_{n}-u \bigr)\,dt\rightarrow 0,\quad n\rightarrow\infty. $$
(2.15)

Note that Lemma 2.1 implies that \(u_{n}\rightarrow u\) in \(L^{p}\) and there is a constant \(c_{5}>0\), we have

$$ \Vert u_{n}-u \Vert _{p}\rightarrow0,\qquad \Vert u_{n} \Vert _{p}\leq c_{5} \Vert u \Vert , \quad \forall1\leq p< \infty. $$
(2.16)

If \(|u|\geq M_{3}\), \(M_{3}>0\), it follows from \((F_{3})\) that there exists \(c_{6}>0\) such that

$$ F_{u}(t,u)\leq c_{6} \vert u \vert ^{\alpha-1}, \quad\forall \vert u \vert \geq M_{3},\forall t\in[0,T]. $$
(2.17)

According to the boundedness of \((u_{n})\) and Lemma 2.1, we have \(\|u_{n}\|<\infty\). It follows from \(e^{Q(t)}\leq d_{2}\), (2.16), (2.17), and Hölder’s inequality that

$$ \begin{aligned}[b] &\int_{0}^{T}e^{Q(t)} \bigl(F_{u}(t,u_{n}),u_{n}-u \bigr)\,dt \\ &\quad\leq d_{2} \int_{0}^{T}\bigl(c_{6} \vert u \vert ^{\alpha-1}\bigr) (u_{n}-u)\,dt \\ &\quad\leq c_{6}d_{2} \Vert u_{n}-u \Vert _{\alpha}\cdot \Vert u_{n} \Vert _{\alpha}^{\alpha -1} \\ &\quad\leq c_{5}^{\alpha-1}c_{6}d_{2} \Vert u_{n}-u \Vert _{\alpha}\cdot \Vert u_{n} \Vert ^{\alpha-1}\rightarrow0. \end{aligned} $$
(2.18)

If \(|u|\leq M_{3}\), by \((F_{1})\), one has that

$$ \bigl\vert F_{u}(t,u) \bigr\vert \leq c_{7}b(t),\quad \forall \vert u \vert \leq M_{3},\forall t\in[0,T], $$
(2.19)

where \(c_{7}=\max_{s\in[0,M_{3}]}a(s)\). By Lemma 2.1, (2.19), \(e^{Q(t)}\leq d_{2}\), and \(b\in L^{1}([0,T];\mathbb {R^{+}})\), we obtain

$$ \begin{aligned}[b] &\int_{0}^{T}e^{Q(t)} \bigl(F_{u}(t,u_{n}),u_{n}-u \bigr)\,dt \\ &\quad\leq d_{2} \int_{0}^{T}c_{7}b(t) (u_{n}-u)\,dt \\ &\quad\leq c_{7}d_{2} \Vert u_{n}-u \Vert _{\infty} \int_{0}^{T}b(t)\,dt\rightarrow0. \end{aligned} $$
(2.20)

Combining (2.18) and (2.20), we can see that (2.15) holds. Therefore, by (2.15), \(\varPhi {'}(u_{n})\rightarrow0\), \(u_{n}\rightharpoonup u\) in W, and the definition of \(\varPhi{'}\), we have

$$ \begin{aligned}[b] 0&=\lim_{n\rightarrow\infty}\bigl\langle \varPhi{'}(u_{n}),u_{n}-u\bigr\rangle \\ &=\lim_{n\rightarrow\infty}(u_{n},u_{n}-u) -\lim _{n\rightarrow\infty} \int_{0}^{T}e^{Q(t)} \bigl(F_{u}(t,u_{n}),u_{n}-u \bigr)\,dt \\ &=\lim_{n\rightarrow\infty} \Vert u_{n} \Vert ^{2}- \Vert u \Vert ^{2}-0. \end{aligned} $$
(2.21)

That is,

$$ \lim_{n\rightarrow\infty} \Vert u_{n} \Vert = \Vert u \Vert . $$
(2.22)

It follows from \(u_{n}\rightharpoonup u\) in W that

$$\Vert u_{n}-u \Vert ^{2}=(u_{n}-u,u_{n}-u) \rightarrow0, $$

so \(\{u_{n}\}\) has a convergent subsequence in W. Thus Φ satisfies the Cerami condition. □

Since \(\dim W^{0}\) and \(\dim W^{-}\) are finite, we can choose an orthonormal basis \(\{e_{m}\}_{m=1}^{k_{1}}\) of \(W^{0}\), an orthonormal basis \(\{e_{m}\}_{m=k_{1}+1}^{k_{2}}\) of \(W^{-}\), and an orthonormal basis \(\{e_{m}\}_{m=k_{2}+1}^{\infty}\) of \(W^{+}\), where \(1\leq k_{1}< k_{2}\) and \(k_{1}+1\leq k_{2}<\infty\). Then \(\{e_{m}\}_{m=1}^{\infty}\) is an orthonormal basis of W. Let \(X_{m}:=\mathbb{R}e_{m}\), then \(Y_{k}\) and \(Z_{k}\) can be defined as (2.3).

Lemma 2.4

If\(Z_{k}=\overline{\bigoplus_{m\geq k}X_{m}}\), then

$$\beta_{k}:=\sup_{u\in Z_{k}, \Vert u \Vert =1} \Vert u \Vert _{\infty}\rightarrow0\quad \textit{as }k\rightarrow\infty. $$

Proof

It is clear that \(0<\beta_{k+1}\leq\beta_{k}\), so \(\beta_{k}\rightarrow \beta\geq0\), \(k\rightarrow\infty\). For every \(k\in N\), there exists \(u_{k}\in Z_{k}\) such that \(\|u_{k}\|=1\) and \(\|u_{k}\|_{\infty}>\frac{1}{2}\beta_{k}\). By the definition of \(Z_{k}\), \(u_{k}\rightharpoonup0\) in W, then by Lemma 2.1 in [3] and Rellich’s embedding theorem (see [5]), \(u_{k}\rightarrow0\) in \(Z_{k}\). Therefore, we have \(\beta =0\), that is, \(\beta_{k}\rightarrow0\). □

Proof of Theorem 1.1

For the Hilbert space W, define \(Y_{k}\) and \(Z_{k}\) as in (2.3). According to Lemma 2.3 and the evenness of \(F(t,\cdot )\), we know that Φ satisfies the Cerami condition \((C)\) and \(\varPhi(-u)=\varPhi(u)\). It remains to verify conditions (i) and (ii) of Lemma 2.2.

Verification of (i). Taking \(r_{k}=\beta_{k}^{-1}\), using Lemma 2.4, one has that

$$r_{k}\rightarrow+\infty\quad\mbox{as }k\rightarrow\infty. $$

Choose k large enough such that \(Z_{k}\subset W^{+}\) and

$$r_{k}\geq\biggl(4\max_{s\in[0,1]}a(s) \int_{0}^{T}e^{Q(t)}b(t)\,dt \biggr)^{1/2}. $$

Now, for \(u\in Z_{k}\) with \(\|u\|=r_{k}\) and (\(F_{1}\)), we have that

$$\begin{aligned} \varPhi(u)&=\frac{1}{2} \Vert u \Vert ^{2}- \int_{0}^{T}e^{Q(t)}F(t,u)\,dt \\ &\geq\frac{1}{2} \Vert u \Vert ^{2}-\max _{s\in[0, \Vert u \Vert _{\infty}]}a(s) \int _{0}^{T}e^{Q(t)}b(t)\,dt \\ &\geq\frac{1}{2} \Vert u \Vert ^{2}-\max _{s\in[0,\beta_{k} \Vert u \Vert ]}a(s) \int _{0}^{T}e^{Q(t)}b(t)\,dt \\ &\geq\frac{1}{2} \Vert u \Vert ^{2}-\max _{s\in[0,1]}a(s) \int_{0}^{T}e^{Q(t)}b(t)\, dt \\ &\geq\frac{ r_{k}^{2}}{4}, \end{aligned} $$

which implies that

$$a_{k}=\inf_{u\in Z_{k},\|u\|=r_{k}}\varPhi(u)\geq\frac {r_{k}^{2}}{4} \rightarrow+\infty \quad\mbox{as }k\rightarrow\infty. $$

Verification of (ii). Similar to Lemma 2.3 in [2], we get that, for a finite-dimensional subspace \(Y_{k}\subset W\), for any \(k\in\mathbb{N}\), there exists a constant \(\epsilon_{k}>0\) such that

$$ m\bigl(\varLambda_{u}^{k}\bigr)\geq\epsilon_{k}, \quad\forall u\in Y_{k}\setminus\{ 0\}, $$
(2.23)

where \(m(\cdot)\) denotes the Lebesgue measure in \(\mathbb{R}\), \(\varLambda _{u}^{k}:=\{t\in[0,T]:|u|\geq\epsilon_{k}\|u\|\}\). Note that \(e^{Q(t)}\geq d_{1}\) for some constant \(d_{1}>0\) (\(\forall t\in[0,T]\)). By \((F_{2})\), for any \(k\in\mathbb{N}\), there exists a constant \(S_{k}>0\) such that

$$ F(t,u)\geq\frac{ \vert u \vert ^{2}}{d_{1}\epsilon_{k}^{3}}, \quad\forall t\in [0,T],\forall \vert u \vert \geq S_{k}. $$
(2.24)

Hence, using (2.23), (2.24), \((F_{2})\), and \(e^{Q(t)}\geq d_{1}\), we have that, for any \(k\in\mathbb{N}\) and \(u\in Y_{k}\) with \(\|u\|\geq S_{k}/\epsilon_{k}\),

$$\begin{aligned} \varPhi(u)&\leq\frac{1}{2} \bigl\Vert u^{+} \bigr\Vert ^{2}- \int_{0}^{T}e^{Q(t)}F(t,u)\,dt \\ &\leq\frac{1}{2} \Vert u \Vert ^{2}-\biggl( \int_{[0,T]\setminus\varLambda _{u}^{k}}e^{Q(t)}F(t,u)\,dt + \int_{\varLambda_{u}^{k}}e^{Q(t)}F(t,u)\,dt\biggr) \\ &\leq\frac{1}{2} \Vert u \Vert ^{2}- \int_{\varLambda_{u}^{k}}e^{Q(t)}F(t,u)\,dt \\ &\leq\frac{1}{2} \Vert u \Vert ^{2}-d_{1} \int_{\varLambda_{u}^{k}}\frac { \vert u \vert ^{2}}{d_{1}\epsilon_{k}^{3}}\,dt \\ &\leq\frac{1}{2} \Vert u \Vert ^{2}-\frac{d_{1}\epsilon_{k}^{2} \Vert u \Vert ^{2}}{d_{1}\epsilon_{k}^{3}}m \bigl(\varLambda_{u}^{k}\bigr) \\ &\leq\frac{1}{2} \Vert u \Vert ^{2}- \Vert u \Vert ^{2} \\ &=-\frac{1}{2} \Vert u \Vert ^{2}. \end{aligned}$$
(2.25)

Now, for any \(k\in\mathbb{N}\), if we choose

$$\Vert u \Vert =\rho_{k}>\max\{\gamma_{k},S_{k}/ \epsilon_{k}\}, $$

then (2.25) implies that

$$b_{k=}\max_{u\in Y_{k},\|u\|=\rho_{k}}\varPhi(u)\leq-\frac{1}{2}\rho _{k}^{2}\leq0,\quad\forall k\in\mathbb{N}. $$

Consequently, by Lemma 2.2, system (1.1) has infinitely many nontrivial T-periodic solutions. □

Conclusion

We obtain infinitely many nontrivial periodic solutions for a class of damped vibration systems with superquadratic terms at infinity. By using some weaker conditions, our results extend and improve some existing results in the literature.

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Research supported by the National Natural Science Foundation of China (No. 11771182) and the Natural Science Foundation of Shandong Province (No. ZR2017JL005).

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Li, P., Chen, G. Multiple periodic solutions for damped vibration systems with superquadratic terms at infinity. Bound Value Probl 2019, 191 (2019). https://doi.org/10.1186/s13661-019-01306-2

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MSC

  • 34C25
  • 70H05

Keywords

  • Damped vibration systems
  • Periodic solutions
  • Superquadratic