考虑二阶Hamilton系统

$ \begin{equation} \label{euler} \left\{ \begin{array}{l l} \ddot{u}(t)=\nabla F(t, u(t)), \text{a}.\text{e}.t\in[0, T] \\u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{array} \right. \end{equation} $

(1.1)

其中 $T>0$, 设 $F:[0, T]\times\mathbb{R}^{N}\rightarrow\mathbb{R}$满足如下假设

(A)对每个 $x\in\mathbb{R}^{N}$, $F(t, x)$关于 $t$可测；对几乎所有的 $t\in[0, T]$, $F(t, x)$关于 $x$连续可微, 且存在 $a\in C(\mathbb{R}^{+}, \mathbb{R}^{+})$, $b\in L^{1}(0, T;\mathbb{R}^{+})$, 使得

$ |F(t, x)|\leq a(|x|)b(t), |\nabla F(t, x)|\leq a(|x|)b(t), $

对所有的 $x\in\mathbb{R}^{N}$和a.e. $t\in[0, T]$成立.

Mawhin和Willem在文[1]在非线性项有界, 即存在 $g\in L^{1}(0, T;\mathbb{R}^{+})$, 使得

$ |\nabla F(t, x)|\leq g(t), $

对所有 $x\in\mathbb{R}^{N}$和 ${\rm{a}}.{\rm{e}}.t \in [0,T]$成立时，得到了系统(1.1) 周期解的存在性定理.

文[2]假设非线性项是次线性增长的, 即存在 $f, g\in L^{1}(0, T;\mathbb{R}^{+})$, $\alpha\in[0, 1)$使得

$ |\nabla F(t, x)|\leq f(t)|x|^{\alpha}+g(t), $

对所有 $x\in\mathbb{R}^{N}$和 $\text{a}.\text{e}.t\in[0, T]$成立.

在具有线性增长非线性项, 即存在 $f, g\in L^{1}(0, T;\mathbb{R}^{+})$, 使得

$ \begin{equation}|\nabla F(t, x)|\leq f(t)|x|+g(t), \end{equation} $

(1.2)

对所有 $x\in\mathbb{R}^{N}$和 $\text{a}.\text{e}.t\in[0, T]$成立时, 文[3]中得到以下定理.

定理A^[3]  设 $F$满足(1.2)式, 且

$ \lim\limits_{|x|\rightarrow\infty}\frac{1}{|x|^{2}}\int_{0}^{T}F(t, x)\mathrm{d}t=+\infty. $

若 $\int_{0}^{T}f(t)\mathrm{d}t < \frac{12}{T}$, 则系统(1.1) 在Sobolev空间 $H_{T}^{1}$中至少有一个周期解.

文[4]将定理A中的强制性条件改进为下方有界的情形

$ \liminf\limits_{|x|\rightarrow\infty}\frac{1}{|x|^{2}}\int_{0}^{T}F(t, x)\mathrm{d}t>\frac{3T^{2}}{2\pi^{2}\left(12-T\int_{0}^{T}f(t)\mathrm{d}t\right)}\int_{0}^{T}f^{2}(t)\mathrm{d}t. $

当非线性项 $\nabla F(t, x)$线性增长时, 文[5-7]中分别在具有部分周期位势, 脉冲作用项, 单调性条件下得到了二阶Hamilton系统周期解的存在性定理.

设存在常数 $M_{0}>0$, $M_{1}>0$, $M_{2}>0$和非负函数 $\omega\in C([0, \infty), [0, \infty))$, 使得

(w₁) $\omega (s)\le \omega (t), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall s\le t, s, t\in [0, \infty ).$

(w₂) $\omega(s+t)\leq M_{0}(\omega(s)+\omega(t)), \ \ \ \ \ \ \ \ \forall s, t\in[0, \infty)$.

(w₃) $0\leq\omega(s)\leq M_{1}s^{p^{-}-1}+M_{2}, \ \ \ \ \ \ \ \ \ \ \forall s, t\in[0, \infty)$.

(w₄)当 $s\rightarrow\infty$时, $\omega(s)\rightarrow+\infty$.

受到文[8]和[9]的启发, 我们考虑用控制函数 $\omega(|x|)$替换线性增长条件(1.2) 中的 $|x|$, 并将上述结果推广到非自治 $p(t)$-拉普拉斯系统

$ \left\{ \begin{align} & \frac{d}{dt}(|\dot{u}(t){{|}^{p(t)-2}}\dot{u}(t))=\nabla F(t,u(t)),\text{a}.\text{e}.t\in [0,T] \\ & u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \\ \end{align} \right. $

(1.3)

其中 $p(t)\in C([0, T], \mathbb{R}^{+})$, $p(t)=p(t+T)$, 且

$ \begin{equation}1 < p^{-}:=\min\limits_{t\in[0, T]}p(t)\leq p^{+}:=\max\limits_{t\in[0, T]}p(t) < +\infty.\end{equation} $

(1.4)

临界点理论是研究微分方程和差分方程边值问题可解性的有效方法, 如文[10-12].非自治 $p(t)$-拉普拉斯系统来自于非线性弹性问题和流体力学, 该系统刻画了“逐点异性”的物理现象.近年来, 临界点理论已用于研究非自治 $p(t)$-拉普拉斯系统周期解的存在性, 参见文[13-21].

记 $p(t)\in C([0, T], \mathbb{R}^{+})$, 定义

$ L^{p(t)}([0, T], \mathbb{R}^{N})=\left\{u\in L^{1}([0, T], \mathbb{R}^{N});\int_{0}^{T}|u|^{p(t)}\mathrm{d}t < \infty \right\}. $

$ |u|_{p(t)}=\inf\left\{\lambda>0;\int_{0}^{T}|\frac{u}{\lambda}|^{p(t)}\mathrm{d}t\leq1\right\}. $

$ W_{T}^{1, p(t)}=\{u:[0, T]\rightarrow\mathbb{R}^{N}\mid u\in L^{p(t)}([0, T], \mathbb{R}^{N}), u(0)=u(T), \dot{u}\in L^{p(t)}(0, T;\mathbb{R}^{N})\}. $

当 $p^{-}>1$时, 空间 $W_{T}^{1, p(t)}$是自反的Banach空间, 其范数为

$ \|u\|=|u|_{p(t)}+|\dot{u}|_{p(t)}. $

记

$ \widetilde{W}_{T}^{1, p(t)}=\left\{u\in W_{T}^{1, p(t)}\mid\int_{0}^{T}u(t)\mathrm{d}t=0\right\}, $

则 $W_{T}^{1, p(t)}=\widetilde{W}_{T}^{1, p(t)}\oplus\mathbb{R}^{N}$.

引理2.1^[15]   $\forall\tilde{u}\in\widetilde{W}_{T}^{1, p(t)}$, 存在常数 $C_{0}>0$, $C_{1}>0$, $C_{2}>0$, 有

$ \begin{equation}\|\tilde{u}\|_{\infty}\leq2C_{0}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{1}, \end{equation} $

(2.1)

$ \begin{equation}\|\tilde{u}\|\leq C_{2}\left[\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+1\right], \end{equation} $

(2.2)

其中 $\|u\|_{\infty}=\max\limits_{0\leq t\leq T}|u(t)|, p^{-}=\min\limits_{0\leq t\leq T}p(t)$.

引理2.2^[15]   $\forall u\in W_{T}^{1, p(t)}$, $\bar{u}=\frac{1}{T}\int_{0}^{T}u(t)\mathrm{d}t$, 有

$ \|u\|\rightarrow\infty\Rightarrow\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t+|\bar{u}|\rightarrow+\infty. $

引理2.3^[16]  在Sobolev空间 $W_{T}^{1, p(t)}$上定义泛函 $\varphi$如下：

$ \varphi(u)=\int_{0}^{T}\frac{1}{p(t)}|\dot{u}(t) |^{p(t)}\mathrm{d}t+\int_{0}^{T}F(t, u(t))\mathrm{d}t,\ \ \ \ \ \ \forall u\in W_{T}^{1, p(t)}, $

则 $u\in W_{T}^{1, p(t)}$是问题(1.3) 的周期解当且仅当 $u$是泛函 $\varphi$的临界点, 且 $\varphi$连续可微，

$ \langle\varphi'(u), v\rangle=\int_{0}^{T}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t), \dot{v}(t)) \mathrm{d}t+\int_{0}^{T}(\nabla F(t, u(t)), v(t))\mathrm{d}t,\ \ \ \ \ \ \forall u, v\in W_{T}^{1, p(t)}. $

定义1  设 $X$为Banach空间, 若泛函 $\varphi\in C^{1}(X, \mathbb{R})$满足：对任何点列 $\{u_{n}\}\subset X$, 由 $\{\varphi(u_{n})\}$有界, $\varphi'(u_{n})\rightarrow 0$蕴含 $\{u_{n}\}$有收敛子列, 则称泛函 $\varphi$满足(PS)条件.

引理2.4^[1]  (极小作用原理)若泛函 $\varphi:X\mapsto\mathbb{R}$弱下半连续, 且 $\varphi$在自反的Banach空间 $X$中强制, 即当 $\|u\|\rightarrow\infty$时, 有 $\varphi(u)\rightarrow+\infty, $则泛函 $\varphi$在空间 $X$中有极小值.

引理2.5^[1]  (鞍点定理)设 $E$是Hilbert空间, $E=E_{1}\oplus E_{2}$, 其中 $E_{2}\neq\{0\}$是有限维子空间。若 $\varphi\in C^{1}(X, \mathbb{R})$满足(PS)条件和以下两个条件

(ⅰ)存在 $e\in B_{\rho}\cap E_{2}$和常数 $\omega>\sigma$, 使得 $\varphi|_{ e+E_{1}}\geq\omega$,

(ⅱ)存在常数 $\sigma$和 $\rho$, 使得 $\varphi|_{\partial B_{\rho}\cap E_{2}}\leq\sigma$,

则 $\varphi$有临界值 $c\geq\omega$且

$ c=\inf\limits_{h\in\Gamma}\max\limits_{x\in B_{\rho}\cap E_{1}}\varphi(h(x)), $

其中 $\Gamma=\{h\in C(\bar{B}_{\rho}\cap E_{1}, E):h\mid_{\partial B_{\rho}\cap E_{1}}=\texttt{id}_{\partial B_{\rho}\cap E_{1}}\}, $ $\texttt{id}$表示恒等算子, $B_{\rho}$是 $E$中以0为中心半径为 $r$的开球, $\partial B_{\rho}$表示 $B_{\rho}$的边界.

定理3.1  设 $\omega\in C([0, \infty), [0, \infty))$, 满足( $\omega_{1}$)-( $\omega_{4}$).设存在 $f, g\in L^{1}(0, T;\mathbb{R}^{+})$, 使得

$ \begin{equation}|\nabla F(t, x)|\leq f(t)\omega(|x|)+g(t), \end{equation} $

(3.1)

对所有 $x\in\mathbb{R}^{N}$和 $\text{a}.\text{e}.t\in[0, T]$成立.且

$ \begin{equation}\liminf\limits_{|x|\rightarrow\infty} \frac{1}{\omega^{q^{+}}(|x|)}\int_{0}^{T}F(t, x)\mathrm{d}t>\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}, \end{equation} $

(3.2)

其中 $\frac{1}{p^{-}}+\frac{1}{q^{+}}=1$, 若

$ \begin{equation}\int_{0}^{T}f(t)\mathrm{d}t < \frac{1}{(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}p^{+}}, \end{equation} $

(3.3)

则问题(1.3) 在Sobolev空间 $W_{T}^{1, p(t)}$中至少有一个周期解.

注  定理3.1推广与改进了定理A和文献[8]中定理1.5.首先, 定理3.1中将对应结果推广到了非自治 $p(t)$-拉普拉斯系统; 另一方面, 易见式(3.2) 中极限是下方有界的.

取 $p(t)\equiv 2$, 则 $p^{-}=p^{+}=2$, 令

$ F(t, x)=\left(\frac{3}{5}T-t\right)\ln^{2}(1+|x|^{2})+\beta(t)\ln(1+|x|^{2}), \omega(|x|)=\ln(1+|x|^{2}), $

其中 $\beta(t)\in L^{1}(0, T;\mathbb{R}^{+})$, 则 $F$满足定理3.1的条件, 但不满足定理A和文[8]中定理1.5.

证  由条件( $\omega_{1}$)-( $\omega_{3}$), 式(3.1), (2.1), 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \ \ \ \ \left|\int_{0}^{T}F(t,u(t))\mathrm{d}t-\int_{0}^{T}F(t,\bar{u})\mathrm{d}t\right|\\ &&=\left|\int_{0}^{T}\int_{0}^{1}(\nabla F(t,\bar{u}+s\tilde{u}(t)), \tilde{u}(t))\mathrm{d}s\mathrm{d}t\right|\\ &&\leq\int_{0}^{T}\int_{0}^{1}f(t)\omega(| \bar{u}+s\tilde{u}(t)|)| \tilde{u}(t)|\mathrm{d}s\mathrm{d}t+\int_{0}^{T}\int_{0}^{1}g(t)|\tilde{u}(t) |\mathrm{d}s\mathrm{d}t\\ &&\leq\int_{0}^{T}\int_{0}^{1}f(t)M_{0}[\omega(|\bar{u}|)+\omega(|\tilde{u}(t)|)]| \tilde{u}(t)|\mathrm{d}s\mathrm{d}t+\int_{0}^{T}\int_{0}^{1}g(t)|\tilde{u}(t) |\mathrm{d}s\mathrm{d}t\\ &&\leq M_{0}\int_{0}^{T}\int_{0}^{1}f(t)[\omega(|\bar{u}|)+M_{1}|\tilde{u}(t)|^{p^{-}-1}+M_{2}]| \tilde{u}(t)|\mathrm{d}s\mathrm{d}t+\int_{0}^{T}\int_{0}^{1}g(t)|\tilde{u}(t) |\mathrm{d}s\mathrm{d}t\\ &&\leq\omega(|\bar{u}|)\|\tilde{u} \|_{\infty}M_{0}\int_{0}^{T}f(t)\mathrm{d}t+\|\tilde{u} \|_{\infty}^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\\ &&\ \ \ \ \ \ +\|\tilde{u} \|_{\infty}\left[M_{0}M_{2}\int_{0}^{T}f(t)\mathrm{d}t+\int_{0}^{T}g(t)\mathrm{d}t\right]\\ &&\leq \omega(|\bar{u}|)\left[2C_{0}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{1}\right]M_{0}\int_{0}^{T}f(t)\mathrm{d}t\\ &&\ \ \ \ \ \ +\left[2C_{0}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{1}\right]^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\\ &&\ \ \ \ \ \ +\left[2C_{0}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{1}\right]\left[M_{0}M_{2}\int_{0}^{T}f(t)\mathrm{d}t+\int_{0}^{T}g(t)\mathrm{d}t\right]\\ &&\leq(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t~\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t+C_{3}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{4}\omega(|\bar{u}|)+C_{5}\\ &&\ \ \ \ \ \ +2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t~\omega(|\bar{u}|)\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}. \end{eqnarray*} $

(3.4)

利用Young不等式, 及 $\frac{1}{p^{-}}+\frac{1}{q^{+}}=1$, 有

$ \begin{eqnarray*} && \ \ \ \ \ \ 2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t\omega(|\bar{u}|)\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}\\ &&\leq\frac{1}{p^{-}}\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ +\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}|). \end{eqnarray*} $

(3.5)

由式(3.4) 和(3.5), 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \varphi(u)=\int_{0}^{T}\frac{1}{p(t)}|\dot{u}(t) |^{p(t)}\mathrm{d}t+\int_{0}^{T}F(t, u(t))\mathrm{d}t\\ &&\geq\frac{1}{p^{+}}\int_{0}^{T}|\dot{u}(t) |^{p(t)}\mathrm{d}t+\left[\int_{0}^{T}F(t, u(t))\mathrm{d}t-\int_{0}^{T}F(t, \bar{u})\mathrm{d}t\right]+\int_{0}^{T}F(t, \bar{u})\mathrm{d}t\\ &&\geq\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}(t) |^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ -\frac{1}{p^{-}}\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ +\left[\frac{1}{\omega^{q^{+}}(|\bar{u}|)}\int_{0}^{T}F(t, \bar{u})\mathrm{d}t-\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\right]\omega^{q^{+}}(|\bar{u}|)\\ &&\ \ \ \ \ \ -C_{3}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}-C_{4}\omega(|\bar{u}|)-C_{5}\\ &&=\frac{1}{p^{+}}\left(1-\frac{1}{p^{-}}\right)\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t-C_{3}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}-C_{4}\omega(|\bar{u}|)-C_{5}\\ &&\ \ \ \ \ \ +\left[\frac{1}{\omega^{q^{+}}(|\bar{u}|)}\int_{0}^{T}F(t, \bar{u})\mathrm{d}t-\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\right]\omega^{q^{+}}(|\bar{u}|). \end{eqnarray*} $

由引理1.2, $\|u\|\rightarrow\infty\Rightarrow\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t+|\bar{u}|\rightarrow+\infty$, 由式(3.2) 和 $(\omega_{4})$, 并注意到 $p^{-}>1$, 当 $\|u\|\rightarrow\infty$时, $\varphi(u)\rightarrow+\infty$.注意到当 $p^{-}>1$时, 空间 $W_{T}^{1, p(t)}$是自反的Banach空间, 泛函 $\varphi$弱下半连续^[20], 由极小作用原理可知, 泛函 $\varphi$至少有一个临界点, 从而得到问题(1.3) 至少有一个周期解.

定理3.2  设非负函数 $\omega$满足( $\omega_{1}$)-( $\omega_{4}$), $F$满足(3.1) 和(3.3), 且

$ \limsup\limits_{|x|\rightarrow\infty} \frac{1}{\omega^{q^{+}}(|x|)}\int_{0}^{T}F(t, x)\mathrm{d}t < -K, $

(3.6)

其中

$ K=\frac{1}{q^{+}}\left[1+\frac{\frac{1}{p^{-}}\left(1+\frac{1}{p^{+}}\right) }{\frac{1}{2}\left(1-\frac{1}{p^{+}p^{-}}\right) }\right]\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}, $

(3.7)

则问题(1.3) 在Sobolev空间 $W_{T}^{1, p(t)}$中至少有一个周期解.

注  取 $p(t)\equiv\sin\frac{2\pi t}{T}+5$, 则 $p^{-}=4$, $q^{+}=\frac{4}{3}$, 令

$ F(t, x)=\left(\frac{2}{5}T-t\right)|x|^{4}+T^{3}|x|^{2}, \omega(|x|)=|x|^{3}, $

则 $F$满足定理3.2中的条件, 但不满足文[13-21]中定理.

证  我们将利用鞍点定理来证明定理3.2, 对 $\forall u\in W_{T}^{1, p(t)}$, 设 $\bar{u}=\frac{1}{T}\int_{0}^{T}u(t)\mathrm{d}t, u(t)=\tilde{u}(t)+\bar{u}.$

第1步  证明泛函 $\varphi$满足(PS)条件, 即任何点列 $\{u_{n}\}\subset W_{T}^{1, p(t)}$, 由 $ \{\varphi(u_{n})\}$有界, $\varphi'(u_{n})\rightarrow 0$, $(n\rightarrow\infty)$, 可推得 $\{u_{n}\}$有收敛子列.首先证明 $\{u_{n}\}$在 $W_{T}^{1, p(t)}$有界.

类似于(3.4) 的证明, 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \left|\int_{0}^{T}(\nabla F(t,u(t)),\tilde{u}(t))\mathrm{d}t\right|\\ &&\leq(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t~\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ +C_{3}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{4}~\omega(|\bar{u}|)+C_{5}\\ &&\ \ \ \ \ \ +2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t~\omega(|\bar{u}|)\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}. \end{eqnarray*} $

(3.8)

由式(3.5), (3.8), 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \|\tilde{u}_{n}\|\geq\langle\varphi'(u_{n}),\tilde{u}_{n}\rangle\\ &&=\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t+\int_{0}^{T}(\nabla F(t,u_{n}(t)),\tilde{u}_{n}(t))\mathrm{d}t\\ &&\geq\left[1-(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ -\frac{1}{p^{-}}\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}p^{-}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ -\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}_{n}|)\\ &&\ \ \ \ \ \ -C_{3}\left(\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}-C_{4}~\omega(|\bar{u}_{n}|)-C_{5}\\ &&=\left(1-\frac{1}{p^{+}p^{-}}\right)\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t-C_{3}\left(\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}-C_{4}~\omega(|\bar{u}_{n}|)-C_{5}\\ &&\ \ \ \ \ \ -\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}_{n}|). \end{eqnarray*} $

(3.9)

另一方面, 由式(2.1), 可得

$ \|\tilde{u}_{n}\|\leq C_{2}\left[\left(\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+1\right]. $

(3.10)

由式(3.9), (3.10), 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}_{n}|)+C_{4}~\omega(|\bar{u}_{n}|)\\ &&\geq\left(1-\frac{1}{p^{+}p^{-}}\right)\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t-C_{6}\left(\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}-C_{7}\\ &&\geq\frac{1}{2}\left(1-\frac{1}{p^{+}p^{-}}\right)\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t-C_{8}, \end{eqnarray*} $

(3.11)

其中 $C_{8}=-\min\limits_{s\in[0, +\infty)}\left\{\frac{1}{2}\left(1-\frac{1}{p^{+}p^{-}}\right)s^{p^{-}}-C_{6}s-C_{7}\right\}>0$.由式(3.9), 有

$ \begin{eqnarray*} &&\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\leq\frac{1}{\frac{1}{2}\left(1-\frac{1}{p^{+}p^{-}}\right)}\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}_{n}|)\\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +C_{9}~\omega(|\bar{u}_{n}|)+C_{10}. \end{eqnarray*} $

(3.12)

由式(3.4), (3.5), (3.12), 有

$ \begin{eqnarray*} &&\ \ \ \ \ \ \varphi(u_{n})=\int_{0}^{T}\frac{1}{p(t)}|\dot{u}_{n}(t) |^{p(t)}\mathrm{d}t+\int_{0}^{T}F(t,u_{n}(t))\mathrm{d}t\\ &&\leq\frac{1}{p^{-}}\int_{0}^{T}|\dot{u}_{n}(t) |^{p(t)}\mathrm{d}t+\left[\int_{0}^{T}F(t,u_{n}(t))\mathrm{d}t-\int_{0}^{T}F(t,\bar{u}_{n})\mathrm{d}t\right]+\int_{0}^{T}F(t,\bar{u}_{n})\mathrm{d}t\\ &&\leq\left[\frac{1}{p^{-}}+(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}_{n}(t) |^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ +\frac{1}{p^{-}}\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ +\int_{0}^{T}F(t,\bar{u}_{n})\mathrm{d}t+\frac{1}{q^{+}}\left(\frac{2M_{0}C_{0}\int_{0}^{T}f(t)\mathrm{d}t}{\sqrt[p^{-}]{\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}p^{-}\int_{0}^{T}f(t)\mathrm{d}t}}\right)^{q^{+}}\omega^{q^{+}}(|\bar{u}_{n}|)\\ &&\ \ \ \ \ \ +C_{3}\left(\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{4}\omega(|\bar{u}_{n}|)+C_{5}\\ &&\leq\left[\frac{1}{\omega^{q^{+}}(|\bar{u}_{n}|)}\int_{0}^{T}F(t,\bar{u}_{n})\mathrm{d}t+K\right]\omega^{q^{+}}(|\bar{u}_{n}|)+C_{11}~\omega(|\bar{u}_{n}|)\\ &&\ \ \ \ \ \ +C_{12}~\omega^{\frac{q^{+}}{p^{-}}}(|\bar{u}_{n}|)+C_{13}~\omega^{\frac{1}{p^{-}}}(|\bar{u}_{n}|)+C_{14}, \end{eqnarray*} $

(3.13)

其中 $K$为式(3.7) 中定义的正常数.反设 $\{u_{n}\}$在 $W_{T}^{1, p(t)}$中无界, 当 $n\rightarrow\infty$时, $\| u_{n}\|\rightarrow\infty$.

由引理2.2, 当 $n\rightarrow\infty$时, $\|u_{n}\|\rightarrow\infty\Rightarrow\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t+|\bar{u}_{n}|\rightarrow+\infty.$

当 $n\rightarrow\infty$时, $| \bar{u}_{n}|\rightarrow+\infty$, 由( $\omega_{4}$), 有 $\omega(| \bar{u}_{n}|)\rightarrow+\infty$.由式(3.6), 式(3.13), 并注意到 $p^{-}>1$, 当 $n\rightarrow\infty$时, $\varphi(u_{n})\rightarrow-\infty$.

当 $n\rightarrow\infty$时, $\int_{0}^{T}|\dot{u}_{n}(t)|^{p(t)}\mathrm{d}t\rightarrow+\infty$, 式(3.12), 有 $\omega(| \bar{u}_{n}|)\rightarrow+\infty$.由式(3.6), 式(3.13), 并注意到 $p^{-}>1$, 当 $n\rightarrow\infty$时, $\varphi(u_{n})\rightarrow-\infty$.

这与 $ \{\varphi(u_{n})\}$有界矛盾！故 $\{u_{n}\}$在 $W_{T}^{1, p(t)}$中有界.注意到当 $p^{-}>1$时, $W_{T}^{1, p(t)}$紧嵌入 $C([0, T];\mathbb{R}^{N})$和 $W_{T}^{1, p(t)}$的一致凸性, 类似于文献[19]中定理3.2的证明, $\{u_{n}\}$在 $W_{T}^{1, p(t)}$中有收敛子列, 故泛函 $\varphi$满足(PS)条件.

第2步  取 $E_{1}=\widetilde{W}_{T}^{1, p(t)}$, $E_{2}=\mathbb{R}^{N}$, 则 $W_{T}^{1, p(t)}=\widetilde{W}_{T}^{1, p(t)}\oplus\mathbb{R}^{N}$.我们证明鞍点定理的环绕条件成立.对 $u\in E_{1}$, 类似于(3.4) 的证明, 有

$ \begin{eqnarray*} &&\left|\int_{0}^{T}F(t, u(t))\mathrm{d}t-\int_{0}^{T}F(t, 0)\mathrm{d}t\right|\leq (4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\\ &&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +C_{15}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}+C_{16}. \end{eqnarray*} $

(3.14)

由式(3.14), 有

$ \begin{eqnarray*} &&\varphi(u)=\int_{0}^{T}\frac{1}{p(t)}|\dot{u}(t) |^{p(t)}\mathrm{d}t+\left[\int_{0}^{T}F(t, u(t))\mathrm{d}t-\int_{0}^{T}F(t, 0)\mathrm{d}t\right]+\int_{0}^{T}F(t, 0)\mathrm{d}t\\ &&\ \ \ \ \geq\left[\frac{1}{p^{+}}-(4C_{0})^{p^{-}}M_{0}M_{1}\int_{0}^{T}f(t)\mathrm{d}t\right]\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t-C_{15}\left(\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\right)^{\frac{1}{p^{-}}}\\ &&\ \ \ \ \ \ -C_{16}+\int_{0}^{T}F(t, 0)\mathrm{d}t. \end{eqnarray*} $

对 $u\in E_{1}=\widetilde{W}_{T}^{1, p(t)}$, 由引理2.2, $\|u\|\rightarrow\infty\Rightarrow\int_{0}^{T}|\dot{u}(t)|^{p(t)}\mathrm{d}t\rightarrow+\infty$.由(3.3) 式知,

$ \frac{1}{p^{+}}-(4C_{0})^{p^{-}}K_{0}K_{1}\int_{0}^{T}f(t)\mathrm{d}t>0, $

故当 $\|u\|\rightarrow +\infty$时, 有 $\varphi(u)\rightarrow+\infty$.对 $u\in E_{1}=\widetilde{W}_{T}^{1, p(t)}$成立.显然存在常数 $\eta$, 使得 $\varphi(u)\geq\eta$.

另一方面, 对 $y\in E_{2}=\mathbb{R}^{N}$, 由式(3.6), 对 $\forall\varepsilon >0$, 当 $|y|$充分大时, 有

$ \begin{eqnarray*} &&\varphi(y)=\int_{0}^{T}F(t, y)\mathrm{d}t\leq\left(-K+\varepsilon\right)\omega^{q^{+}}(|y|), \end{eqnarray*} $

令 $\varepsilon$充分小, 当 $|y|\rightarrow +\infty$时, $\omega(|y|)\rightarrow+\infty$, 则 $\varphi(y)\rightarrow-\infty$.因此存在正常数 $\rho$, 使得 $\varphi|_{\partial B_{\rho}\cap E_{2}}\leq\eta-1=\sigma$.