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Existence of nonconstant periodic solutions for \(p(t)\)-Laplacian Hamiltonian system

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Abstract

The purpose of this paper is to consider the existence of periodic solutions for the \(p(t)\)-Laplacian Hamiltonian system. Some results are obtained by using the least action principle and the minimax methods.

Introduction

In this paper, we consider the following problem:

$$\begin{aligned} \textstyle\begin{cases} \frac{d}{dt}( \vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t))-\nabla F(t,u(t))=0, \quad t\in [0,T], \\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(T>0\), \(u\in R^{N}\). \(F(t,u)\) and \(p(t)\) satisfy the following conditions:

(\(F_{0}\)):

\(F:[0,T]\times R^{N}\rightarrow R\) is measurable and T-periodic in t for each \(u\in R^{N}\) and continuously differentiable in u for a.e. \(t\in [0,T]\), and there exist \(a\in C(R^{+}, R^{+})\) and \(b\in L^{1}([0,T], R^{+})\) such that

$$ \bigl\vert F(t,u) \bigr\vert \leq a\bigl( \vert u \vert \bigr)b(t), \qquad \bigl\vert \nabla F(t,u) \bigr\vert \leq a\bigl( \vert u \vert \bigr)b(t) $$

for all \(u\in R^{N}\) and a.e. \(t\in [0,T]\).

(P):

\(p(t)\in C([0,T],R^{+})\), \(p(t)=p(t+T)\) and

$$ 1< p^{-}:=\min p(t)\leq p^{+}:=\max p(t)< +\infty . $$

The nonlinear problems involving the \(p(t)\)-Laplace type operator are extremely attractive because they can be used to model dynamical phenomena which arise from the study of elastic mechanics. The detailed application backgrounds of the \(p(x)\)-Laplace type operators can be found in [1, 2] and the references therein. The \(p(t)\)-Laplacian system possesses more complicated nonlinearity than that of the p-Laplacian, for example, it is not homogeneous, this causes many troubles, and some classical theories and methods, such as the theory of Sobolev spaces, are not applicable.

In recent years, many researchers studied the periodic solutions of the system (1.1). Some existence results are obtained by using the least action principle and minimax methods in critical point theory. Many solvability conditions were given, such as the coercivity condition, the periodicity condition, the convexity condition, the boundedness condition, the subadditive condition and the sublinear condition. We refer to [3,4,5,6,7,8,9,10,11,12,13].

When \(F(t,x)=G(x)+H(t,x)\), the \(p(t)\)-Laplacian system has also been studied by many authors. Especially, in [13], the authors supposed that \(H(t,x)\) is \(p^{-}\)-sublinear, that is, there exist \(f,g\in L^{1}([0,T],R^{+})\) and \(\alpha \in [0,p^{-})\) such that

$$ \bigl\vert \nabla H(t,x) \bigr\vert \leq f(t) \vert x \vert ^{\alpha }+g(t) $$

for all \(x\in R^{N}\) and a.e. \(t\in [0,T]\), and there exist \(0\leq r<1/(p ^{+}T^{p^{-}})\) and \(1\leq \beta \leq p^{-}\) such that

$$ \bigl\langle \nabla G(x)-\nabla G(y),x-y\bigr\rangle \geq -r \vert x-y \vert ^{\beta } $$

for all \(x,y\in R^{N}\).

Moreover, they consider system (1.1) with \(F(t,x)\) which is the sum of a subconvex function and another function under suitable conditions, by the least action principle and the saddle point theorem, they obtain some existence results.

Motivated by the results mentioned above, we aim in this paper to study the system (1.1) with a potential \(F(t,x)\) which is also the sum of \(F_{1}(t,x)\) and \(F(x)\), where the conditions on \(F_{1}(t,x)\) and \(F(x)\) are more general and simple. By the least action principle, we obtained three existence results.

This paper is organized as follows. In Sect. 2, we give some necessary preliminary knowledge on variable exponent Sobolev spaces. In Sect. 3, we present our main results and completed the proof. In Sect. 4, we give some examples to illustrate our results.

Preliminary

For the convenience of readers, we first state some properties of the variable exponent Lebesgue–Sobolev spaces \(L^{p(t)}\) and \(W^{1,p(t)} _{T}\) (for details, see [14,15,16,17,18]). In the following, we use \(\vert \cdot \vert \) to denote the Euclidean norm in \(R^{N}\).

Let \(p(t)\) satisfy the condition \((P)\) and define a generalized Lebesgue space

$$\begin{aligned} L^{p(t)}\bigl([0,T];R^{N}\bigr)=\biggl\{ u\in L^{1}\bigl([0,T];R^{N}\bigr): \int ^{T}_{0} \vert u \vert ^{p(t)}\,dt< \infty \biggr\} , \end{aligned}$$

with the norm

$$\begin{aligned} \vert u \vert _{L^{p(t)}}= \vert u \vert _{p(t)}=\inf \biggl\{ \lambda > 0: \int ^{T}_{0} \biggl\vert \frac{u}{ \lambda } \biggr\vert ^{p(t)}\,dt\leq 1\biggr\} . \end{aligned}$$

Define

$$\begin{aligned} C^{\infty }_{T}=C^{\infty }_{T} \bigl(R;R^{N}\bigr)=\bigl\{ u\in C^{\infty } \bigl(R;R^{N}\bigr)):\text{$u$ is $T$-periodic} \bigr\} . \end{aligned}$$

For \(u,v\in L^{1}([0,T];\mathbb{R}^{N})\), if

$$ \int _{0}^{T}\bigl(u(t),\varphi '(t)\bigr)\,dt=- \int _{0}^{T}\bigl(v(t),\varphi (t)\bigr)\,dt, \quad \forall \varphi \in C_{T}^{\infty }, $$

then \(v(t)\) is called a T-weak derivative of \(u(t)\) and is denoted by \(\dot{u}(t)\).

It has been proved that (see [19], page 6)

$$ \int _{0}^{T}\dot{u}(s)\,ds=0, $$

and there exists \(C\in \mathbb{R}^{N}\) such that

$$ u(t)= \int _{0}^{T}v(s)\,ds+C, \quad \text{a.e. } t\in [0,T], $$

and \(u(0)=u(T)=C\).

Define a generalized Sobolev space

$$\begin{aligned} W^{1,p(t)}_{T}\bigl([0,T];R^{N}\bigr)=\bigl\{ u \in L^{p(t)}\bigl([0,T];R^{N}\bigr):\dot{u} \in L^{p(t)}\bigl([0,T];R^{N}\bigr)\bigr\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert _{W^{1,p(t)}_{T}}= \Vert u \Vert = \vert u \vert _{p(t)}+ \vert \dot{u} \vert _{p(t)}. \end{aligned}$$

For \(u\in W^{1,p(t)}_{T}([0,T];R^{N})\), let

$$\begin{aligned} \bar{u}=\frac{1}{T} \int ^{T}_{0}u(s)\,ds, \qquad \tilde{u}(t)=u(t)-\bar{u}, \end{aligned}$$

and

$$ \widetilde{W}^{1,p(t)}_{T}\bigl([0,T];R^{N} \bigr)=\biggl\{ u\in W^{1,p(t)}_{T}\bigl([0,T];R ^{N}\bigr): \int ^{T}_{0}u(s)\,ds=0\biggr\} , $$

then

$$\begin{aligned} W^{1,p(t)}_{T}\bigl([0,T];R^{N}\bigr)= \widetilde{W}^{1,p(t)}_{T}\bigl([0,T];R^{N}\bigr) \oplus R^{N}. \end{aligned}$$

In the following we use \(L^{p(t)}\), \(W^{1,p(t)}_{T}\), \(\widetilde{W}^{1,p(t)} _{T}\) to denote the \(L^{p(t)}([0,T];R^{N})\), \(W^{1,p(t)}_{T}([0,T]; R ^{N})\), \(\widetilde{W}^{1,p(t)}_{T}([0,T];R^{N})\), respectively.

Lemma 2.1

([20])

There is a continuous embedding \(W^{1,p(t)}\hookrightarrow C([0,T],\mathbb{R}^{N})\), when \(p^{-}>1\), the embedding is compact. And for \(u\in \widetilde{W}^{1,p(t)}_{T}\), there is a constant C independent of u such that

$$ \Vert u \Vert _{\infty }\leq C \Vert u \Vert . $$

Lemma 2.2

([2, 20])

For every \(u\in \widetilde{W}^{1,p(t)}_{T}\), there exist constants \(c'_{1}\), \(c_{1}\) such that

$$\begin{aligned}& \Vert u \Vert _{\infty }\leq c_{1} \vert \dot{u} \vert _{p(t)}, \\& \Vert u \Vert _{\infty }\leq 2c'_{1} \biggl[\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{1}{p ^{-}}}+T^{\frac{1}{p^{-}}}\biggr], \end{aligned}$$

where \(c_{1},c'_{1}>0\).

Lemma 2.3

([20])

Let \(u=\bar{u}+\tilde{u}\in W ^{1,p(t)}_{T}\), then the norm \(\vert \widetilde{\dot{u}} \vert _{p(t)}\) is an equivalent norm on \(\widetilde{W}^{1,p(t)}_{T}\) and \(\vert \bar{u} \vert + \vert \dot{u} \vert _{p(t)}\) is an equivalent norm on \(W^{1,p(t)}_{T}\). Therefore, for \(u\in W^{1,p(t)}_{T}\),

$$ \Vert u \Vert \rightarrow \infty \quad \Rightarrow \quad \vert \overline{u} \vert + \int _{0}^{T} \bigl\vert u'(t) \bigr\vert ^{p(t)}\,dt \rightarrow \infty . $$

Lemma 2.4

([2])

  1. (i)

    The space \((L^{p(t)}, \vert \cdot \vert )\) is a separable, reflexive, uniform convex Banach space, and its conjugate space is \(L^{q(t)}\), where \(\frac{1}{p(t)}+\frac{1}{q(t)}\). For any \(u\in L^{p(t)}\) and \(v\in L^{q(t)}\), we have

    $$ \biggl\vert \int _{0}^{T}u(t)v(t)\,dt \biggr\vert \leq 2 \vert u \vert _{p(t)} \vert v \vert _{q(t)}. $$
  2. (ii)

    If \(p_{1}(t), p_{2}(t)\in C([0,T],\mathbb{R}^{1})\) and \(1< p_{1}(t)\leq p_{2}(t)\) for any \(t\in [0,T]\), then \(L^{p_{2}(t)} \hookrightarrow L^{p_{1}(t)}\), and the embedding is continuous.

Main results of problem (1.1)

Definition 3.1

A function \(u(t)\in W_{T}^{1,p(t)}\) is called a weak solution of (1.1), if \(\vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t)\) has a weak derivative, still denoted by \(\frac{d}{dt}(\dot{u}(t)| ^{p(t)-2} \dot{u}(t))\), such that

$$\begin{aligned} \textstyle\begin{cases} \frac{d}{dt}( \vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t))-\nabla F(t,u(t))=0, \quad \text{a.e. } t\in [0,T], \\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0. \end{cases}\displaystyle \end{aligned}$$

Define a functional φ on \(W^{1,p(t)}_{T}\) by

$$\begin{aligned} \varphi (u)= \int ^{T}_{0}\frac{1}{p(t)}) \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T} _{0}F\bigl(t,u(t)\bigr)\,dt. \end{aligned}$$

Lemma 3.1

Suppose that assumptions \((F_{0})\) and \((P)\) hold, then the functional φ is continuously differentiable on \(W^{1,p(t)}_{T}\) and

$$\begin{aligned} \bigl\langle \varphi '(u),v\bigr\rangle = \int ^{T}_{0}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)-2} \dot{u}(t),\dot{v}(t)\bigr)\,dt+ \int ^{T}_{0}\bigl(\nabla F\bigl(t,u(t) \bigr),v(t)\bigr)\,dt \end{aligned}$$

for all \(u,v\in W^{1,p(t)}_{T}\). And if there exists \(u\in W^{1,p(t)} _{T}\) such that \(\langle \varphi '(u),v\rangle =0\) for all \(v\in W ^{1,p(t)}_{T}\), then u is a weak solution (1.1).

Proof

For convenience, define functionals J and H on \(W^{1,p(t)}_{T}\) by

$$ J(u)= \int ^{T}_{0}\frac{1}{p(t)}) \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt, \qquad H(u)= \int ^{T}_{0}F\bigl(t,u(t)\bigr)\,dt. $$

It is easy to see that ([18])

$$ \varphi (u)=J(u)+H(u), \qquad \bigl\langle \varphi '(u),v\bigr\rangle =\bigl\langle J'(u),v\bigr\rangle +\bigl\langle H'(u),v \bigr\rangle , $$

for all \(v\in W^{1,p(t)}_{T}\) and

$$ \bigl\langle J'(u),v\bigr\rangle = \int ^{T}_{0}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)-2}\dot{u}(t), \dot{v}(t)\bigr)\,dt. $$

It is easy to see that

$$ \bigl\langle H'(u),v\bigr\rangle = \int ^{T}_{0}\bigl(\nabla F\bigl(t,u(t) \bigr),v(t)\bigr)\,dt. $$

In fact

$$\begin{aligned} \frac{1}{s}\bigl[H(u+sv)-H(u)\bigr] =&\frac{1}{s} \int ^{T}_{0}\bigl[F\bigl(t,u(t)+sv(t)\bigr)-F \bigl(t,u(t)\bigr)\bigr]\,dt \\ =& \int ^{T}_{0}\bigl(\nabla F\bigl(t,u(t)+\theta sv(t)\bigr),v(t)\bigr)\,dt, \end{aligned}$$

where \(0<\theta <1\).

Since

$$ \bigl\vert \nabla F\bigl(t,u(t)+\theta sv(t)\bigr) \bigr\vert \leq a\bigl( \bigl\vert u(t)+\theta sv(t) \bigr\vert \bigr)b(t)\leq a _{0}b(t), $$

and \(\nabla F(t,u(t)+\theta sv(t))\rightarrow \nabla F(t,u(t))\), a.e. \(t\in [0,T]\), as \(s\rightarrow 0\).

Here \(a_{0}\) is some constant.

By the Lebegue dominated convergence theorem, we have

$$ \lim_{s\rightarrow 0}\frac{1}{s}\bigl[H(u+sv)-H(u)\bigr]= \int _{0}^{T}(\nabla F\bigl(t,u(t),v(t)\bigr) \,dt, $$

which implies that H is Gâteaux differentiable, and

$$ \bigl\langle H'(u),v\bigr\rangle = \int _{0}^{T}\bigl(\nabla F\bigl(t,u(t) \bigr),v(t)\bigr)\,dt, \forall v\in W_{T}^{1,p(t)}. $$

Next we will prove that the functional φ is continuous differentiable, it suffices to show that both of \(J'\) and \(H'\) are continuous on \(W^{1,p(t)}_{T}\).

Let \(u_{n},u\in W_{T}^{1,p(t)}\), such that \(\Vert u_{n}-u \Vert \rightarrow 0\) (\(n \rightarrow \infty \)). Obviously,

$$\begin{aligned} \Vert u_{n}-u \Vert \rightarrow 0 \quad \Leftrightarrow \quad \vert u_{n}-u \vert _{p(t)}\rightarrow 0 \quad \text{and} \quad \bigl\vert u'_{n}-u' \bigr\vert _{p(t)}\rightarrow 0. \end{aligned}$$
(3.1)

Note that the mapping defined by \(u\in L^{p(t)}\rightarrow \vert u(t) \vert ^{p(t)-2}u \in L^{q(t)}\), where \(\frac{1}{p(t)}+\frac{1}{q(t)}=1\), is a bounded and continuous mapping from \(L^{p(t)}\) into \(L^{q(t)}\) (see [15]). It follows from (3.1) that

$$ 2 \bigl\vert \bigl\vert u'_{n} \bigr\vert ^{p(t)-2}u'_{n}- \bigl\vert u' \bigr\vert ^{p(t)-2}u' \bigr\vert _{q(t)}\rightarrow 0, $$

which, combined with Lemma 2.4, shows

$$\begin{aligned} \bigl\Vert J'(u_{n})-J'(u) \bigr\Vert =&\sup_{ \Vert v \Vert \leq 1} \biggl\vert \int _{0}^{T}\bigl( \bigl\vert u'_{n} \bigr\vert ^{p(t)-2}u'_{n}- \bigl\vert u' \bigr\vert ^{p(t)-2}u',v'(t) \bigr) \biggr\vert \\ \leq &2\sup_{ \Vert v \Vert \leq 1} \bigl\vert \bigl\vert u'_{n} \bigr\vert ^{p(t)-2}u'_{n}- \bigl\vert u' \bigr\vert ^{p(t)-2}u' \bigr\vert _{q(t)} \bigl\vert v' \bigr\vert _{p(t)} \\ \leq &2 \bigl\vert \bigl\vert u'_{n} \bigr\vert ^{p(t)-2}u'_{n}- \bigl\vert u' \bigr\vert ^{p(t)-2}u' \bigr\vert _{q(t)}\rightarrow 0. \end{aligned}$$

Hence, \(J'\) is continuous.

Next, we prove the continuity of \(H'(u): W_{T}^{1,p(t)}\rightarrow (W _{T}^{1,p(t)})^{*}\).

Let \(u_{n},u\in W_{T}^{1,p(t)}\), such that \(\Vert u_{n}-u \Vert \rightarrow 0\) (\(n \rightarrow \infty \)).

From Lemma 2.1, it follows that

$$ u_{n},u\in C\bigl([0,T],\mathbb{R}^{N}\bigr)\quad \text{and} \quad \max_{0\leq t\leq T} \bigl\vert u_{n}(t)-u(t) \bigr\vert \rightarrow 0\quad (n\rightarrow \infty ). $$

From the condition \((F_{0})\), we have

$$\begin{aligned} \bigl\vert \nabla F\bigl(t,u_{n}(t)\bigr)-\nabla F\bigl(t,u(t)\bigr) \bigr\vert \rightarrow 0\quad (n\rightarrow \infty ), \text{a.e. } t\in [0,T], \end{aligned}$$
(3.2)

and \(a( \vert u_{n}(t) \vert )\rightarrow a( \vert u(t) \vert )\) in \(C([0,T];\mathbb{R}^{N})\).

Therefore, for n large enough,

$$ a\bigl( \bigl\vert u_{n}(t) \bigr\vert \bigr)\leq a\bigl( \bigl\vert u(t) \bigr\vert \bigr)+1. $$

Then we have

$$\begin{aligned} \bigl\vert \nabla F\bigl(t,u_{n}(t)\bigr)-\nabla F\bigl(t,u(t) \bigr) \bigr\vert \leq &\bigl(a\bigl( \bigl\vert u(t) \bigr\vert \bigr)+a \bigl( \bigl\vert u_{n}(t) \bigr\vert \bigr)\bigr)b(t) \\ \leq & \Bigl(2\max_{0\leq t\leq T}a\bigl( \bigl\vert u(t) \bigr\vert \bigr)+1\Bigr)b(t) \\ \leq & Cb(t). \end{aligned}$$
(3.3)

Here C is some constant.

Then from (3.2) and (3.3), together with the Lebesgue dominated convergence theorem, we get

$$ \int _{0}^{T} \bigl\vert \nabla F \bigl(t,u_{n}(t)\bigr)-\nabla F\bigl(t,u(t)\bigr) \bigr\vert \,dt \rightarrow 0\quad (n \rightarrow \infty ). $$

Therefore,

$$\begin{aligned}& \bigl\Vert H'(u_{n})-H'(u) \bigr\Vert _{(W_{T}^{1,p(t)})^{*}} \\& \quad = \sup_{ \Vert v \Vert \leq 1,v\in W_{T}^{1,p(t)}} \biggl\vert \int _{0}^{T}\bigl(\nabla F\bigl(t,u _{n}(t)\bigr)-\nabla F\bigl(t,u(t)\bigr),v(t)\bigr)\,dt \biggr\vert \\& \quad \leq \sup_{ \Vert v \Vert \leq 1,v\in W_{T}^{1,p(t)}} \int _{0}^{T} \bigl\vert \nabla F\bigl(t,u _{n}(t)\bigr)-\nabla F\bigl(t,u(t)\bigr) \bigr\vert \cdot \bigl\vert v(t) \bigr\vert \,dt \\& \quad \leq C \int _{0}^{T} \bigl\vert \nabla F \bigl(t,u_{n}(t)\bigr)-\nabla F\bigl(t,u(t)\bigr) \bigr\vert \,dt \rightarrow 0\quad (n\rightarrow \infty ). \end{aligned}$$

So \(H'\) is continuous.

Suppose that \(u\in W_{T}^{1,p(t)}\) such that

$$ \bigl\langle \varphi '(u),v\bigr\rangle =0, \quad \forall v\in W_{T}^{1,p(t)}, $$

then

$$\begin{aligned} \int _{0}^{T}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)-2}\dot{u}(t),v(t)\bigr)\,dt=- \int _{0}^{T}\bigl( \nabla F\bigl(t,u(t) \bigr),v(t)\bigr)\,dt \end{aligned}$$
(3.4)

for all \(v\in W_{T}^{1,p(t)}\), obviously, also for all \(v\in C_{T} ^{\infty }\).

From \((F_{0})\) and Lemma 2.1, \(\nabla F(t,u(t))\in L^{1}\), therefore, (3.4) implies that \(\nabla F(t,u(t))\) is the weak derivative \(\frac{d}{dt}( \vert \dot{u}(t) \vert ^{p(t)-2}\dot{u}(t))\) of \(\vert \dot{u}(t) \vert ^{p(t)-2} \dot{u}(t)\), and

$$\begin{aligned}& \frac{d}{dt}\bigl( \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)-2}\dot{u}(t)\bigr)=\nabla F\bigl(t,u(t)\bigr),v(t)), \quad \text{a.e. }t \in [0,T], \\& \bigl\vert \dot{u}(0) \bigr\vert ^{p(0)-2}\dot{u}(0)= \bigl\vert \dot{u}(T) \bigr\vert ^{p(T)-2}\dot{u}(T), \end{aligned}$$

since \(p(0)=p(T)\), it follows that \(\dot{u}(0)=\dot{u}(T)\), so \(u(t)\) is the weak solution of (1.1).

The proof is completed. □

For the reader’s convenience, we give a definition.

Definition 3.2

([21])

A function \(F: R^{N} \rightarrow R\) is said to be \((\lambda ,\mu )\)-subconvex if

$$ F\bigl(\lambda (x+y)\bigr)\leq \mu \bigl(F(x)+F(y)\bigr) $$

for some \(\lambda ,\mu >0\) and all \(x,y\in R^{N}\).

Theorem 3.1

Let \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy \((F_{0})\) and the following conditions:

  1. (i)

    \(F_{1}(t,\cdot )\) is \((\lambda p^{-},\mu p^{-})\)-subconvex for a.e. \(t \in [0,T]\), where

    $$ \lambda p^{-}>1, \quad\quad 1< 2\mu p^{-}< \bigl(\lambda p^{-}\bigr)^{p^{-}}; $$
  2. (ii)

    there exists a constant \(0\leq r<\frac{p^{-}}{(4c_{1})^{p^{-}}Tp ^{+}}\) such that

    $$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\geq -r \vert x-y \vert ^{p^{-}} $$

    for all \(x,y\in R^{N}\) and a.e. \(t\in [0,T]\);

  3. (iii)
    $$ \frac{1}{\mu p^{-}} \int _{0}^{T} F_{1}\bigl(t,\lambda p^{-}x\bigr)\,dt+ \int _{0} ^{T} F_{2}(x)\,dt \rightarrow +\infty $$

    as \(\vert x \vert \rightarrow +\infty \).

Then problem (1.1) has at least one nontrivial solution which minimizes φ on \(W_{T}^{1,p(t)}\).

Proof

Let \(\alpha =\log _{\lambda p^{-}}(2\mu p^{-})\), then \(0<\alpha <p^{-}\). For \(\vert x \vert >1\), there exists a positive integer n such that

$$ n-1< \log _{\lambda p^{-}} \vert x \vert \leq n. $$

Furthermore, we have \(\vert x \vert ^{\alpha }\geq (\lambda p^{-})^{(n-1)\alpha }=(2\mu p^{-})^{n-1}\) and \(\vert x \vert \leq (\lambda p^{-})^{n}\). Then from (i) and condition \((F_{0})\), it follows that

$$\begin{aligned} F_{1}(t,x) =&F_{1}\biggl(t,\lambda p^{-} \biggl(\frac{x}{2\lambda p^{-}}+\frac{x}{2 \lambda p^{-}}\biggr)\biggr) \leq 2\mu p^{-}F_{1}\biggl(t,\frac{x}{2\lambda p^{-}}\biggr) \leq \cdot \cdot \cdot \\ \leq &\bigl(2\mu p^{-}\bigr)^{n}F_{1} \biggl(t,\frac{x}{(2\lambda p^{-})^{n}}\biggr) \leq 2 \mu p^{-} \vert x \vert ^{\alpha }a_{0}b(t), \end{aligned}$$

for a.e. \(t\in [0,T]\) and all \(\vert x \vert >1\), where \(a_{0}=\max_{0\leq s \leq 1}a(s)\).

Moreover, we have

$$ F_{1}(t,x)\leq \bigl(2\mu p^{-} \vert x \vert ^{\alpha }+1\bigr)a_{0}b(t) $$

for a.e. \(t\in [0,T]\) and all \(x\in R^{N}\), \(0<\alpha <p^{-}\).

By (i) and Lemma 2.2, it is obvious that

$$\begin{aligned} \int _{0}^{T}F_{1}\bigl(t,u(t) \bigr)\,dt \geq &\frac{1}{\mu p^{-}} \int _{0}^{T}F _{1}\bigl(t,\lambda p^{-}\overline{u}\bigr)\,dt- \int _{0}^{T}F_{1}(t,- \widetilde{u})\,dt \\ \geq &\frac{1}{\mu p^{-}} \int _{0}^{T}F_{1}\bigl(t,\lambda p^{-} \overline{u}\bigr)\,dt-\bigl(2\mu p^{-} \Vert \widetilde{u} \Vert ^{\alpha }_{\infty }+1\bigr)a _{0} \int _{0}^{T}b(t)\,dt \\ \geq &\frac{1}{\mu p^{-}} \int _{0}^{T}F_{1}\bigl(t,\lambda p^{-} \overline{u}\bigr)\,dt-c_{2}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{ \alpha }{p^{-}}}-c_{3} \end{aligned}$$
(3.5)

for all \(u\in W_{T}^{1,p(t)}\) and some constants \(c_{2}\), \(c_{3}\).

Using (ii) and Lemma 2.2, we have

$$\begin{aligned} \int _{0}^{T}\bigl(F_{2}(u)-F_{2}( \overline{u})\bigr)\,dt =& \int _{0}^{T} \int _{0} ^{1}\bigl(\nabla F_{2}( \overline{u}+s\widetilde{u}),\widetilde{u}\bigr)\,ds\,dt \\ =& \int _{0}^{T} \int _{0}^{1}\frac{1}{s}\bigl(\nabla F_{2}(\overline{u}+s \widetilde{u})-\nabla F_{2}( \overline{u}),s\widetilde{u}\bigr)\,ds\,dt \\ \geq &- \int _{0}^{T} \int _{0}^{1}\frac{r}{s} \vert s \widetilde{u} \vert ^{p^{-}}\,ds\,dt \\ =&-r \int _{0}^{T}\frac{1}{p^{-}} \vert \widetilde{u} \vert ^{p^{-}}\,dt \\ \geq &-\frac{rT}{p^{-}} \Vert \widetilde{u} \Vert _{\infty }^{p^{-}} \\ \geq &-\frac{2^{2p^{-}}rTc_{1}^{p^{-}}}{p^{-}} \int _{0}^{T} \vert \dot{u} \vert ^{p(t)}\,dt +c_{4} \end{aligned}$$
(3.6)

for all \(u\in W_{T}^{1,p(t)}\) and a constant \(c_{4}\).

Combining (iii), (3.5) and (3.6), we get

$$\begin{aligned} \varphi (u) =& \int ^{T}_{0}\frac{1}{p(t)}) \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}F\bigl(t,u(t)\bigr)\,dt \\ \geq &\frac{1}{p^{+}} \int ^{T}_{0} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}F _{1}\bigl(t,u(t) \bigr)\,dt + \int ^{T}_{0}\bigl(F_{2}(u)-F_{2}( \overline{u})\bigr)\,dt+ \int ^{T} _{0}F_{2}(\overline{u}) \,dt \\ \geq &\biggl(\frac{1}{p^{+}}-\frac{(4c_{1})^{p^{-}}rT}{p^{-}}\biggr) \int ^{T}_{0} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+\biggl(\frac{1}{\mu p^{-}} \int _{0}^{T}F_{1}\bigl(t,\lambda p^{-}\overline{u}\bigr)\,dt+ \int _{0}^{T}F_{2}(\overline{u}) \,dt\biggr) \\ & {} -c_{2}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{\alpha }{p^{-}}}+c _{5} \end{aligned}$$

for all \(u\in W_{T}^{1,p(t)}\) and some constants \(c_{2}\), \(c_{5}\), it follows from (iii), \(0<\alpha <p^{-}\), \(0\leq r<\frac{p^{-}}{(4c_{1})^{p ^{-}}Tp^{+}}\) and Lemma 2.3, \(\varphi (u)\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow +\infty \). Applying Theorem 1.1 and Corollary 1.1 in Mawhin–Willem [19], we complete the proof. □

Theorem 3.2

Let \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy \((F_{0})\) and the following conditions:

  1. (i)

    there exist \(f(t),m(t)\in L^{1}([0,T];R^{+})\) and \(\gamma \in [0,p ^{-}-1)\) such that

    $$ \bigl\vert \nabla F_{1}(t,x) \bigr\vert \leq f(t) \vert x \vert ^{\gamma }+m(t) $$

    for all \(x\in R^{N}\) and a.e. \(t\in [0,T]\);

  2. (ii)

    there exist constants \(0\leq r<\frac{p^{-}}{p^{+}T(4c_{1})^{p ^{-}}}\), such that

    $$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\geq -r \vert x-y \vert ^{p^{-}} $$

    for all \(x,y\in R^{N}\) and a.e. \(t\in [0,T]\);

  3. (iii)
    $$ \liminf_{ \vert x \vert \rightarrow \infty }\frac{1}{ \vert x \vert ^{\gamma q}} \int _{0}^{T}F(t,x)\,dt \geq A_{1}c(\varepsilon ) $$

    as \(\vert x \vert \rightarrow +\infty \), where \(\frac{1}{p^{-}}+\frac{1}{q}=1\), \(A_{1}\), \(c(\varepsilon )\) are defined in the proof.

Then problem (1.1) has at least one nontrivial solution which minimizes φ on \(W_{T}^{1,p(t)}\).

Proof

For convenience, we denote

$$ A_{1}=2^{\gamma } \int _{0}^{T}f(t)\,dt, \qquad A_{2}= \int _{0}^{T}m(t)\,dt. $$

By condition (i), the ε-Young inequality, Sobolev’s inequality and Lemma 2.2, for any \(u\in W_{T}^{1,p(t)}\) and some constants \(c_{6}\), \(c_{7}\), \(c_{8}\), we have

$$\begin{aligned}& \biggl\vert \int _{0}^{T}(F_{1}\bigl(t,u(t) \bigr)-F_{1}(t,\overline{u})\,dt \biggr\vert \\& \quad = \biggl\vert \int _{0}^{T} \int _{0}^{1}\bigl(\nabla F_{1} \bigl(t,\overline{u}+s \widetilde{u}(t)\bigr),\widetilde{u}(t)\bigr)\,ds\,dt \biggr\vert \\& \quad \leq \int _{0}^{T} \int _{0}^{1}f(t) \bigl\vert \overline{u}+s \widetilde{u}(t) \bigr\vert ^{ \gamma } \bigl\vert \widetilde{u}(t) \bigr\vert \,ds\,dt + \int _{0}^{T} \int _{0}^{1}m(t) \bigl\vert \widetilde{u}(t) \bigr\vert \,ds\,dt \\& \quad \leq 2^{\gamma }\bigl( \vert \overline{u} \vert ^{\gamma }+ \Vert \widetilde{u} \Vert _{\infty }^{\gamma } \bigr) \Vert \widetilde{u} \Vert _{\infty } \int _{0}^{T}f(t)\,dt+ \Vert \widetilde{u} \Vert _{\infty } \int _{0}^{T}m(t)\,dt \\& \quad = A_{1} \vert \overline{u} \vert ^{\gamma }\cdot \Vert \widetilde{u} \Vert _{\infty }+A _{1} \Vert \widetilde{u} \Vert _{\infty }^{\gamma +1}+A_{2} \Vert \widetilde{u} \Vert _{\infty } \\& \quad \leq A_{1}\bigl(\varepsilon \Vert \widetilde{u} \Vert _{\infty }^{p^{-}}+c(\varepsilon ) \vert \overline{u} \vert ^{\gamma q}\bigr)+c_{6}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{ \gamma +1}{p^{-}}}+c_{7}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{1}{p ^{-}}}+c_{8} \\& \quad \leq A_{1}\varepsilon (4c_{1})^{p^{-}} \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+A _{1}c(\varepsilon ) \vert \overline{u} \vert ^{\gamma q} \\& \quad \quad{} +c_{6}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{\gamma +1}{p^{-}}}+c _{7}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{1}{p^{-}}}+c_{8}, \end{aligned}$$
(3.7)

where ε satisfies \(0<\varepsilon <\frac{1}{A_{1}}(\frac{1}{(4c _{1})^{p^{-}}p^{+}}-\frac{rT}{p^{-}})\), \(c(\varepsilon )=(\varepsilon p^{-})^{-\frac{q}{p^{-}}}\cdot \frac{1}{q}\).

Combining (3.6), (3.7) and (iii), we obtain

$$\begin{aligned} \varphi (u) =& \int ^{T}_{0}\frac{1}{p(t)}) \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}F\bigl(t,u(t)\bigr)\,dt \\ \geq &\frac{1}{p^{+}} \int ^{T}_{0} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}\bigl(F _{1} \bigl(t,u(t)\bigr)-F_{1}(t,\overline{u})\bigr)\,dt \\ & {} + \int ^{T}_{0}\bigl(F_{2}(u)-F_{2}( \overline{u})\bigr)\,dt+ \int ^{T}_{0}F(t, \overline{u})\,dt \\ \geq &\biggl(\frac{1}{p^{+}}-\frac{(4c_{1})^{p^{-}}rT}{p^{-}}-A_{1} \varepsilon (4c_{1})^{p^{-}}\biggr) \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt -c_{6}\biggl( \int _{0} ^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{\gamma +1}{p^{-}}} \\ & {} -c_{7}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt\biggr)^{\frac{1}{p^{-}}}+ \vert \overline{u} \vert ^{\gamma q}\biggl(\frac{1}{ \vert \overline{u} \vert ^{\gamma q}} \int _{0} ^{T}F(t,\overline{u}) \,dt-A_{1}c(\varepsilon )\biggr)+c_{8} \end{aligned}$$

for all \(u\in W_{T}^{1,p(t)}\) and a constant \(c_{8}\), it follows from (iii) and Lemma 2.3, \(\varphi (u)\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow +\infty \). Applying Theorem 1.1 and Corollary 1.1 in Mawhin–Willem [19], we complete the proof. □

Theorem 3.3

Let \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) satisfy \((F_{0})\) and the following conditions:

  1. (i)

    there exist \(k(t),h(t)\in L^{1}([0,T];R^{+})\) and \(\beta \in [0,p ^{-})\) such that

    $$ F_{1}(t,x)\geq - k(t) \vert x \vert ^{\beta }+h(t) $$

    for all \(x\in R^{N}\) and a.e. \(t\in [0,T]\);

  2. (ii)

    there exist constants \(0\leq r<\frac{p^{-}}{(4c_{1})^{p^{-}}Tp ^{+}}\) such that

    $$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\geq -r \vert x-y \vert ^{p^{-}} $$

    for all \(x,y\in R^{N}\) and a.e. \(t\in [0,T]\);

  3. (iii)
    $$ \liminf_{ \vert x \vert \rightarrow \infty }\frac{1}{ \vert x \vert ^{\beta }} \int _{0}^{T}F _{2}(x)\,dt\geq B $$

    as \(\vert x \vert \rightarrow +\infty \), where \(B=2^{\beta }\int _{0}^{T}k(t)\,dt\).

Then problem (1.1) has at least one nontrivial solution which minimizes φ on \(W_{T}^{1,p(t)}\).

Proof

By condition (i) and Sobolev’s inequality and Lemma 2.2, we get

$$\begin{aligned} \int _{0}^{T}F_{1}\bigl(t,u(t) \bigr)\,dt \geq & \int _{0}^{T}\bigl[-k(t) \bigl\vert u(t) \bigr\vert ^{\beta }+h(t)\bigr]\,dt \\ \geq &-2^{\beta }\bigl( \vert \overline{u} \vert ^{\beta }+ \bigl\Vert \widetilde{u}(t) \bigr\Vert ^{ \beta }_{\infty } \bigr) \int _{0}^{T}k(t)\,dt+ \int _{0}^{T}h(t)\,dt \\ =&-B \vert \overline{u} \vert ^{\beta }-B \bigl\Vert \widetilde{u}(t) \bigr\Vert ^{\beta }_{\infty }+ \int _{0}^{T}h(t)\,dt \\ \geq &-B(4c_{1})^{p_{-}}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\biggr)^{\frac{ \beta }{p^{-}}}-B \vert \overline{u} \vert ^{\beta }+c_{9} \end{aligned}$$
(3.8)

for all \(u\in W_{T}^{1,p(t)}\) and a constant \(c_{9}\), where \(B=2^{\beta }\int _{0}^{T}k(t)\,dt\).

It follows from (3.6) and (3.8) that

$$\begin{aligned} \varphi (u) =& \int ^{T}_{0}\frac{1}{p(t)} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}F\bigl(t,u(t)\bigr)\,dt \\ \geq &\frac{1}{p^{+}} \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt+ \int ^{T}_{0}F _{1}\bigl(t,u(t) \bigr)\,dt + \int ^{T}_{0}\bigl(F_{2}(u)-F_{2}( \overline{u})\bigr)\,dt+ \int ^{T} _{0}F_{2}(\overline{u}) \,dt \\ \geq &\biggl(\frac{1}{p^{+}}-\frac{(4c_{1})^{p^{-}}r_{1}T}{p^{-}}\biggr) \int _{0} ^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\,dt-B(4c_{1})^{p^{-}}\biggl( \int _{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{p(t)}\biggr)^{\frac{ \beta }{p^{-}}} \\ & {} + \vert \overline{u} \vert ^{\beta }\biggl( \frac{1}{ \vert \overline{u} \vert ^{\beta }} \int _{0} ^{T}F_{2}(\overline{u}) \,dt-B\biggr)+c_{10} \end{aligned}$$

for all \(u\in W_{T}^{1,p(t)}\) and a constant \(c_{10}\), it follows from (iii), \(0<\beta <p^{-}\), \(0\leq r<\frac{p^{-}}{(4c_{1})^{p^{-}}Tp^{+}}\) and Lemma 2.3, \(\varphi (u)\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow +\infty \). Applying Theorem 1.1 and Corollary 1.1 in Mawhin–Willem [19], we complete the proof. □

Examples

In this section, we give some examples of \(F(t,x)\) and \(p(t)\) to illustrate our results.

Example 4.1

In system (1.1), let \(p(t)=3+\cos \omega t\), \(F(t,x)=F_{1}(t,x)+F_{2}(x)\), where

$$\begin{aligned}& T=\frac{2\pi }{\omega }, \quad\quad p^{+}=4,\quad\quad p^{-}=2,\quad\quad \lambda =2,\quad\quad \mu =3, \\& F_{1}(t,x)= \vert x \vert ^{2} \vert \sin \omega t \vert , \quad\quad F_{2}(x)=c(x)-\frac{1}{2}r \vert x \vert ^{2}, \end{aligned}$$

with

$$ r>0,\quad\quad c(x)=\frac{1}{2}r\bigl( \vert x_{1} \vert ^{3}+ \vert x_{2} \vert ^{2}+ \vert x_{3} \vert ^{2}+\cdots + \vert x _{N} \vert ^{2}\bigr). $$

By some computation, we can take \(0\leq r< \frac{1}{32c_{1}^{2}T}\) such that

$$ \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\geq -r \vert x-y \vert ^{2}, $$

and \(F_{1}(t,\cdot )\) is \((4,6)\)-subconvex, for all \(x,y\in R^{N}\) and a.e. \(t\in [0,T]\).

It is easy to verify

$$ \frac{1}{\mu p^{-}} \int _{0}^{T} F_{1}\bigl(t,\lambda p^{-}x\bigr)\,dt+ \int _{0} ^{T} F_{2}(x)\,dt \rightarrow +\infty $$

as \(\vert x \vert \rightarrow +\infty \).

Therefore all the conditions of Theorem 3.1 are satisfied, the problem (1.1) has at least one solution.

Example 4.2

In system (1.1), let \(p(t)=5+\cos \omega t\), \(F(t,x)=F_{1}(t,x)+F_{2}(x)\),

where

$$\begin{aligned}& \omega =\frac{2\pi }{T}, \quad\quad p^{+}=6, \quad\quad p^{-}=4, \quad\quad \gamma =\frac{3}{4}, \\& F_{1}(t,x)= \vert x \vert ^{\frac{7}{4}} \vert \sin \omega t \vert +\bigl(h(t),x\bigr), \quad\quad F_{2}(x)=c(x)- \frac{1}{4}r \vert x \vert ^{4}, \end{aligned}$$

with

$$ r>0, \quad\quad h(t)\in L^{1}\bigl([0,T],R^{+}\bigr), \quad\quad c(x)=\frac{1}{4}r\bigl( \vert x_{1} \vert ^{6}+ \vert x _{2} \vert ^{4}+ \vert x_{3} \vert ^{4}+\cdots + \vert x_{N} \vert ^{4}\bigr). $$

By some computation, we can take \(0\leq r\leq \frac{1}{32c_{1}^{2}T}\) such that

$$\begin{aligned}& \bigl(\nabla F_{2}(x)-\nabla F_{2}(y),x-y\bigr)\geq -r \vert x-y \vert ^{4}, \\& \bigl\vert \nabla F_{1}(t,x) \bigr\vert =\frac{7}{4} \vert x \vert ^{\frac{3}{4}} \vert \sin \omega t \vert + \bigl\vert h(t) \bigr\vert , \end{aligned}$$

for all \(x,y\in R^{N}\) and a.e. \(t\in [0,T]\).

On the other hand, we can verify that condition (iii) of Theorem 3.2 is satisfied. Therefore all the conditions of Theorem 3.2 hold, the problem (1.1) has at least one solution.

Example 4.3

In system (1.1), let \(p(t)=\frac{5}{2}+\cos \omega t\), \(F(t,x)=F_{1}(t,x)+F_{2}(x)\),

where

$$\begin{aligned}& \omega =\frac{2\pi }{T}, \quad\quad p^{+}= \frac{7}{2}, \quad\quad p^{-}=\frac{3}{2}, \quad \quad \beta =\frac{1}{2}, \\& F_{1}(t,x)= \vert x \vert ^{\frac{1}{2}}\biggl( \frac{1}{2}\sin \omega t+1\biggr)+\bigl(h(t),x\bigr) \geq - \vert x \vert ^{\frac{1}{2}}\frac{1}{2}\sin \omega t+\bigl(h(t),x\bigr), \\& F_{2}(x)=c(x)-\frac{2}{3}r \vert x \vert ^{\frac{3}{2}}, \end{aligned}$$

with

$$ r>0, \quad\quad h(t)\in L^{1}\bigl([0,T],R^{+}\bigr), \quad\quad c(x)=\frac{2}{3}r\bigl( \vert x_{1} \vert ^{2}+ \vert x _{2} \vert ^{\frac{3}{2}}+ \vert x_{3} \vert ^{\frac{3}{2}}+\cdots + \vert x_{N} \vert ^{ \frac{3}{2}}\bigr). $$

Similar to Example 4.1,we can see all the conditions of Theorem 3.3 are satisfied, the problem (1.1) has at least one solution.

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Acknowledgements

We would like to thank the referee for his/her valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper.

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The authors were supported by NNSF of China (No. 11501165); the Fundamental Research Funds for the Central Universities (2018B58614).

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Correspondence to Yuanfang Ru.

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Keywords

  • \(p(t)\)-Laplacian
  • Minimax principle
  • Least action principle
  • Periodic solution
  • Critical point