具有多个参数的分数p-Laplace边值问题多解的存在性

The Existence of Multiple Solutions for the Valued p-Laplace Boundary Problem with Multiple Parameters

摘要: 分数阶微分方程是微分方程中重要的研究对象。带有p-Laplace算子的分数阶微分方程是分数阶微分方程的推广，也是一类重要的函数问题，因此研究带有p-Laplace算子的分数阶微分方程具有一定意义。对于分数阶p-Laplace微分方程解的存在性的研究已经相对比较成熟，但对于本文这类具有多个参数的分数p-Laplace边值问题的研究相对较少。本文使用临界点定理得到这类具有多个参数的分数p-Laplace微分方程的三个解的存在性。

Abstract: The research of the subordinate order differential is an important figure in the object division. The fractional differential equation with p-Laplace operator is an extension of the fractional differential equation, and it is also an important problem. Therefore, it is meaningful to study the fractional differential equation with p-Laplace operator. The research on the existence of solutions of fractional p-Laplace differential equations has been relatively mature, but there are relatively few researches on fractional p-Laplace boundary value problems with multiple parameters in this paper. In this paper, the critical point theorem is used to obtain the existence of three solutions of this type of fractional p-Laplace differential equation with multiple parameters.

1. 引言

分数阶p-Laplace微分方程是分数阶微分方程的推广。分数阶微分方程一般可以看作是应用分数阶微积分研究微分方程。分数阶微分方程可以描述物理、化学、生物等多个领域的数学模型。一些学者对分数阶微分方程的性质已经做了大量的研究。2020年，Bai等 [1] 研究了一类具有对流项的Caputo分数阶微分方程的格林函数。2020年，Bai等 [2] 研究了一类三点分数阶边值问题的解。2019年，Tian等 [3] 研究了带p-Laplace算子的分数阶微分方程边值问题的正解。2019年，Yue等 [4] 研究了具有振荡势的分数阶微分方程包含的无穷多个非负解。2019年，Jia等 [5] 研究了一类含导数和参数的分数阶微分方程的非局部问题。2020年，Wang等 [6] 研究了具有分数阶导数的混合p-Laplace边值问题解的存在唯一性。2020年，Kamache等 [7] 研究了具有两个控制参数的扰动非线性分数阶p-Laplace边值问题三个解的存在性。2020年，Kamache等 [8] 研究了一类新的分数阶p-Laplace边值问题弱解的存在性。

综上可见，分数阶p-Laplace微分方程的研究是一个重要的研究内容。在上述文献的基础上，本文运用临界点定理研究带多个参数的分数阶p-Laplace微分方程的边值问题，并得到该微分方程多个解的存在性

$\{\begin{array}{l}{}_{t}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)+\gamma {\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)\\ =\lambda F_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)+\mu G_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\\ u_i\left(a\right)=u_i\left(b\right)=0,1\le i\le n,t\in \left[a,b\right]\end{array}$ (1)

其中， ${\alpha }_i\in \left(0,1\right]$， ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$， $p>1$， ${}_{a}D{}_t^{{\alpha }_i}$ 和 ${}_{t}D{}_b^{{\alpha }_i}$ 是 ${\alpha }_i$ 阶的左右Riemann-Liouville分数导数且 ${\alpha }_i\in L^{\infty }\left(\left[a,b\right]\right)$， $\gamma ,\lambda ,\mu$ 是正参数，且 $F_{u_i},G_{u_i}$ 是 $F,G$ 关于 $u_i$ 的偏导数， $F_{u_i},G_{u_i}\in C\left(\left[a,b\right]\times R^n\right)$， ${\omega }_i\left(t\right)\in L^{\infty }\left(\left[a,b\right]\right)$ 且 ${\omega }_i^0\left(t\right)={\text{essinf}}_{\left[a,b\right]}{\omega }_i\left(t\right)>0$。

2. 预备知识

引理1 [9] 设u是定义在 $\left[a,b\right]$ 上的函数。其 $\alpha$ 阶的左右Riemann-Liouville分数导数定义如下

${}_{a}D{}_t^{\alpha }u\left(t\right):=\frac{{\text{d}}^n}{\text{d}{t}^n}{}_{a}D{}_t^{\alpha -n}u\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}\frac{{\text{d}}^n}{\text{d}{t}^n}{\displaystyle {\int }_a^t{\left(t-s\right)}^{n-a-1}u\left(s\right)\text{d}s},$

和

${}_{t}D{}_b^{\alpha }u\left(t\right):={\left(-1\right)}^n\frac{{\text{d}}^n}{\text{d}{t}^n}{}_{t}D{}_b^{\alpha -n}u\left(t\right)=\frac{{\left(-1\right)}^n}{\Gamma \left(n-\alpha \right)}\frac{{\text{d}}^n}{\text{d}{t}^n}{\displaystyle {\int }_t^b{\left(t-s\right)}^{n-a-1}u\left(s\right)\text{d}s},$

其中 $t\in \left[a,b\right]$，且右侧在 $\left[a,b\right]$ 上逐点定义， $n-1\le \alpha <n$， $n\in \mathbb{N}$。

定义1设 $0<{\alpha }_i\le 1$ 且 $1\le i\le n$， $1<p<\infty$。分数阶导数空间定义如下

$E_{{\alpha }_i}^p=\left\{u\left(t\right)\in L^p\left(\left[a,b\right],R\right)|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\in L^p\left(\left[a,b\right],R\right),u\left(a\right)=u\left(b\right)=0\right\},$

对于任意 $u\in E_{{\alpha }_i}^p$，定义如下的范数

${\Vert u\Vert }_{{\alpha }_i}={\left({\displaystyle {\int }_a^b\gamma {\left|u\left(t\right)\right|}^p\text{d}t}+{\displaystyle {\int }_a^b{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\text{d}t}\right)}^{1/p}.$ (2)

引理2 [10] 设 $0<{\alpha }_i\le 1$ 且 $1\le i\le n$， $1<p<\infty$。对于任意 $u\in E_{{\alpha }_i}^p$，有

${\Vert u_i\Vert }_{L^p}\le \frac{b^{{\alpha }_i}}{\Gamma \left({\alpha }_i+1\right)}{\Vert {}_{a}D{}_t^{{\alpha }_i}{u}_i\Vert }_{L^p}.$ (3)

而且，如果 ${\alpha }_i>p$ 和 $\frac{1}{p}+\frac{1}{q}=1$，有

${\Vert u_i\Vert }_{\infty }\le \frac{b^{{\alpha }_i-\frac{1}{p}}}{\Gamma \left({\alpha }_i\right)\Gamma {\left(\left({\alpha }_i-1\right)q+1\right)}^{1/q}}{\Vert {}_{a}D{}_t^{{\alpha }_i}{u}_i\Vert }_{L^p}.$ (4)

由引理2，可以得到

${\Vert u_i\Vert }_{L^p}\le \frac{b^{{\alpha }_i}}{\Gamma \left({\alpha }_i+1\right)}{\left({\displaystyle {\int }_a^b{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\text{d}t}\right)}^{1/p}$ (5)

其中 $0<{\alpha }_i\le 1$，且

${\Vert u_i\Vert }_{\infty }\le \frac{b^{\frac{{\alpha }_i-1}{p}}}{\Gamma \left({\alpha }_i\right){\left({\omega }_i^0\right)}^{1/p}\Gamma {\left(\left({\alpha }_i-1\right)q+1\right)}^{1/q}}{\left({\displaystyle {\int }_a^b{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\text{d}t}\right)}^{1/p}$ (6)

其中 ${\alpha }_i>p$ 和 $\frac{1}{p}+\frac{1}{q}=1$。

由(5)，范数(2)有如下等价范数

${\Vert u\Vert }_{{\alpha }_i}={\left({\displaystyle {\int }_a^b{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\text{d}t}\right)}^{1/p},\forall u\in E_{{\alpha }_i}^p,1\le i\le n.$ (7)

本文，令X是n个 $E_{{\alpha }_i}^p$ 空间的笛卡尔积且 $1\le i\le n$，也就是 $X=E_{{\alpha }_1}^p\times E_{{\alpha }_2}^p\times \cdots \times E_{{\alpha }_n}^p$，其范数定义如下

$\Vert u\Vert ={\displaystyle {\sum }_{i=1}^n{\Vert u_i\Vert }_{E_{{\alpha }_i}^p}},u=\left(u_1,u_2,\cdots ,u_n\right)\in X,$

其中 ${\Vert u_i\Vert }_{E_{{\alpha }_i}^p}$ 在(7)中定义。显然，X是紧嵌入在 $C{\left(\left[a,b\right],R\right)}^n$ 中。

现在，在X上定义如下泛函：

$\begin{array}{c}T\left(u\right)=\frac{1}{p}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\gamma {\left|u_i\left(t\right)\right|}^p\text{d}t+{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\right)\text{d}t}}\\ -\lambda {\displaystyle {\int }_a^{b}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}-\mu {\displaystyle {\int }_a^{b}G\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t},\end{array}$ (8)

$\phi \left(u\right)=\frac{1}{p}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\gamma {\left|u_i\left(t\right)\right|}^p\text{d}t+{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}u\left(t\right)\right|}^p\right)\text{d}t}},$ (9)

$\Phi \left(u\right)={\displaystyle {\int }_a^{b}G\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t},$ (10)

$\omega \left(u\right)={\displaystyle {\int }_a^{b}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t},$ (11)

其中 $u=\left(u_1,u_2,\cdots ,u_n\right)\in X$， ${\Upsilon }_X$ 表示所有泛函 $\phi$ 的类且 ${\rm T}\left(u\right)=\phi \left(u\right)-\mu \Phi \left(u\right)-\lambda \omega \left(u\right)$。

显然，Τ是一个Gâteaux可微泛函且它在点 $u\in X$ 的Gâteaux导数定义如下

$\begin{array}{c}\langle T^{\prime }\left(u\right),v\rangle ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\text{d}t}}+\gamma {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)v_i\left(t\right)\text{d}t}}\\ -\lambda {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{F}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}}-\mu {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{G}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}}\end{array}$ (12)

其中 $u=\left(u_1,u_2,\cdots ,u_n\right)\in X$， $v=\left(v_1,v_2,\cdots ,v_n\right)\in X$。类似地，得到

$\langle {\phi }^{\prime }\left(u\right),v\rangle ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\text{d}t}}+\gamma {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)v_i\left(t\right)\text{d}t}},$ (13)

$\langle {\Phi }^{\prime }\left(u\right),v\rangle ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{G}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}},$ (14)

$\langle {\omega }^{\prime }\left(u\right),v\rangle ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{F}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}}.$ (15)

引理3 [11] 设 $0<{\alpha }_i\le 1$ 和 $1<p<\infty$，分数阶导数空间X是一个自反可分的Banach空间。

引理4 [12] [13] 设 $0<\alpha \le 1$ 且 $u,v\in L^p\left(\left[a,b\right],R\right)$， $1<p<\infty$，则有

${\displaystyle {\int }_a^{b}v\left(t\right){}_{t}D{}_b^{\alpha }u\left(t\right)\text{d}t}={\displaystyle {\int }_a^{b}u\left(t\right){}_{a}D{}_t^{\alpha }v\left(t\right)\text{d}t}.$

定义2 如果函数u使得 ${}_{a}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)\in C\left[a,b\right]$ 且满足问题(1)的方程和边界条件，则 $u\in X$ 是问题(1)的经典解。

定义3如果函数 $u\in X$ 满足 $\langle T^{\prime }\left(u\right),v\rangle =0,v\in X$，则 $u\in X$ 是问题(1)的弱解。

引理5如果 $u\in X$ 是问题(1)的弱解，则 $u\in X$ 是问题(1)的经典解。

证明：如果 $u\in X$ 是问题(1)的弱解，由定义3，有 $\langle T^{\prime }\left(u\right),v\rangle =0,v\in X$，即

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i{(t)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\text{d}t}}+\gamma {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)v_i\left(t\right)\text{d}t}}\\ -\lambda {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{F}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}}-\mu {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{G}_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)v_i\left(t\right)\text{d}t}}=0\end{array}$ (16)

由引理4可知

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\text{d}t}}\\ ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{}_{t}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)v_i\left(t\right)\text{d}t}}\end{array}$ (17)

将(17)代入(16)，得

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({}_{t}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{(t)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)+\gamma {\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)\right)v_i\left(t\right)\text{d}t}}\\ -{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\lambda F_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)+\mu G_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\right)v_i\left(t\right)\text{d}t}}=0\end{array}$ (18)

由dubois-Reymond定理和(18)，得

$\begin{array}{l}{}_{t}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)+\gamma {\left|u_i\left(t\right)\right|}^{p-2}{u}_i\left(t\right)\\ =\lambda F_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)+\mu G_{u_i}\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right),t\in \left[a,b\right].\end{array}$

则满足问题(1)的方程。因为 $F_{u_i}$ 和 $G_{u_i}$ 是连续的，则 ${}_{t}D{}_b^{{\alpha }_i}\left(\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)\right)\in C\left[a,b\right]$。又因为 $u\in X$，有 $u_i\left(a\right)=u_i\left(b\right)=0$，则满足问题(1)的边界条件，所以当 $u\in X$ 是问题(1)的弱解时，则 $u\in X$ 是问题(1)的经典解。

引理6 ( [14], Theorem 26. A(d))设 $A:X\to X^{*}$ 是实自反可分Banach空间X上的一个单调，强制，半连续算子，假设 $\left\{w_1,w_1,\cdots \right\}$ 是X上得一组基。则满足下面的结论：

如果A是严格单调的，则逆算子 $A^{-1}:X^{*}\to X$ 存在。且逆算子是严格单调，半连续和有界的。如果A是一致单调的，则逆算子 $A^{-1}$ 是连续的。如果A是强单调的，则逆算子 $A^{-1}$ 是Lipschitz连续的。

定理1 [15] 设X是可分自反的实Banach空间， $\phi :X\to R$ 是强制且弱序列下半连续的C¹泛函， $\phi$ 在X的每个有界子集上有界且属于 ${\Upsilon }_X$， ${\phi }^{\prime }$ 在 $X^{*}$ 上有连续逆， $\omega :X\to R$ 是有紧导数的C¹泛函。假设存在 $\phi$ 的严格局部最小值 $u_0$ 有 $\phi \left(u_0\right)=\omega \left(u_0\right)=0$，令

${\rho }_1=\max \left\{0,\lim {\sup }_{\Vert u\Vert \to +\infty }\frac{\omega \left(u\right)}{\phi \left(u\right)},\lim {\sup }_{u\to u_0}\frac{\omega \left(u\right)}{\phi \left(u\right)}\right\},{\rho }_2={\sup }_{u\in {\phi }^{-1}\left(\left(0,+\infty \right)\right)}\frac{\omega \left(u\right)}{\phi \left(u\right)},$

假设 ${\rho }_1<{\rho }_2$，那么对于每一个紧区间 $\left[{\theta }_1,{\theta }_2\right]\subset \left(\frac{1}{{\rho }_2},\frac{1}{{\rho }_1}\right)$ (有 $\frac{1}{0}=+\infty ,\frac{1}{+\infty }=0$ )，存在 $R>0$ 满足：对于任何 $\lambda \in \left[{\theta }_1,{\theta }_2\right]$ 和具有紧导数的C¹泛函 $\varphi :X\to R$，存在 $\xi >0$，使得对于任何 $\mu \in \left[0,\xi \right]$，等式 ${\phi }^{\prime }\left(u\right)-\mu {\varphi }^{\prime }\left(u\right)-\lambda {\omega }^{\prime }\left(u\right)=0$ 在X中至少有三个范数小于R的解。

3. 主要结果

引理7 泛函 $\phi$ 是强制，弱序列下半连续的且在X的每个有界子集上有界且属于 ${\Upsilon }_X$， ${\phi }^{\prime }$ 在 $X^{*}$ 上有连续逆。

证明：由(7)和(9)，得 $\phi \left(u\right)\ge \frac{1}{p}{\displaystyle {\int }_a^b{\displaystyle {\sum }_{i=1}^n\left({\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|}^p\right)\text{d}t}}=\frac{1}{p}{u}^p$，即当 $\Vert u\Vert \to \infty$，有 $\phi \left(u\right)\to \infty$，因此，泛函 $\phi$ 是强制的。假设M是X上的有界子集，即在X的一个子集上 $\Vert u\Vert \le M$，由(9)，有 $\left|\phi \left(u\right)\right|\le \frac{1}{p}{\Vert u\Vert }^p\le \frac{M^p}{p}$，因此，泛函 $\phi$ 在X的每个有界子集上有界。由于 ${\Vert u\Vert }_{{\alpha }_i}^p$ 的弱序列下半连续性，得泛函 $\phi =\frac{1}{p}{\displaystyle {\int }_a^b{\displaystyle {\sum }_{i=1}^n\left(\gamma {\left|u_i\left(t\right)\right|}^p\text{d}t+{\omega }_i\left(t\right){\left|{}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|}^p\right)\text{d}t}}$ 是弱序列下半连续的且属于 ${\Upsilon }_X$。

下面证明 ${\phi }^{\prime }$ 在 $X^{*}$ 上有连续逆。由(13)，有

$\begin{array}{l}\langle {\phi }^{\prime }\left(u\right)-{\phi }^{\prime }\left(v\right),u-v\rangle \\ ={\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i{\left(t\right)}^{p-2}}}}\left({\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)-{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\right){}_{a}D{}_t^{{\alpha }_i}\left(u_i\left(t\right)-v_i\left(t\right)\right)\text{d}t\\ +\gamma {\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({\varphi }_p\left(u_i\left(t\right)\right)-{\varphi }_p\left(v_i\left(t\right)\right)\right)\left(u_i\left(t\right)-v_i\left(t\right)\right)\text{d}t}}.\end{array}$ (19)

引入文献 [16] 中的不等式，即

$\left({\left|s_1\right|}^{p-2}{s}_1-{\left|s_2\right|}^{p-2}{s}_2\right)\left(s_1-s_2\right)\ge \{\begin{array}{l}{\left|s_1-s_2\right|}^p,p\ge 2\\ \frac{{\left|s_1-s_2\right|}^2}{{\left(\left|s_1\right|+\left|s_2\right|\right)}^{2-p}},1<p\le 2,\end{array}$

得到

$\begin{array}{l}\left({\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)-{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\right)\left({}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\\ \ge \{\begin{array}{l}\frac{1}{{\omega }_i\left(t\right)}{\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|}^p,p\ge 2\\ \frac{1}{{\omega }_i\left(t\right)}\frac{{\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|}^2}{{\left(\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|+\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|\right)}^{2-p}},1<p\le 2,\end{array}\end{array}$ (20)

当 $1<p\le 2$ 时，由(20)得

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|}^p\text{d}t}}\\ \le {\left({\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i\left(t\right)}}}\frac{{\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|}^2}{{\left(\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|+\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|\right)}^{2-p}}\text{d}t\right)}^{\frac{p}{2}}\\ {\left({\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i{\left(t\right)}^{\frac{p}{2-p}}}}\left(\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|+\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|\right)\text{d}t\right)}^{\frac{2-p}{2}},\end{array}$ (21)

即

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{1}{{\omega }_i\left(t\right)}\frac{{\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|}^2}{{\left(\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right|+\left|{\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right|\right)}^{2-p}}\text{d}t}}\\ \ge \frac{2^{p-2}{\left({\omega }_1^0\right)}^{\frac{2\left(p-1\right)}{p}}}{\widetilde{{\omega }_1^0}}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\Vert u_i-v_i\Vert }_{{\alpha }_i}^2{\left({\Vert u_i\Vert }_{{\alpha }_i}^p+{\Vert v_i\Vert }_{{\alpha }_i}^p\right)}^{\frac{p-2}{p}}}\end{array}$ (22)

因此，由(19)和(21)得

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)-{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\right)\left({}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)}}\\ \ge \frac{2^{p-2}{\left({\omega }_1^0\right)}^{\frac{2\left(p-1\right)}{p}}}{\widetilde{{\omega }_1^0}}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\Vert u_i-v_i\Vert }_{{\alpha }_i}^2{\left({\Vert u_i\Vert }_{{\alpha }_i}^p+{\Vert v_i\Vert }_{{\alpha }_i}^p\right)}^{\frac{p-2}{p}}}>0\end{array}$ (23)

当 $p\ge 2$ 时，由(20)得

$\begin{array}{l}{\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)-{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\right)\left({}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)}}\\ \ge {\left({\omega }_1^0\right)}^{p-2}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\Vert u_i-v_i\Vert }_{{\alpha }_i}^2}>0\end{array}$ (24)

由(23)和(24)，得

${\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)\right)-{\varphi }_p\left({\omega }_i\left(t\right){}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)\right)\left({}_{a}D{}_t^{{\alpha }_i}{u}_i\left(t\right)-{}_{a}D{}_t^{{\alpha }_i}{v}_i\left(t\right)\right)}}>0$ (25)

其中 $1<p<\infty$。

类似地，当 $p\ge 1$ 时，可得

${\displaystyle {\int }_a^b{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left({\varphi }_p\left(u_i\left(t\right)\right)-{\varphi }_p\left(v_i\left(t\right)\right)\right)\left(u_i\left(t\right)-v_i\left(t\right)\right)\text{d}t}}>0$ (26)

由(25)和(26)，得

$\langle {\phi }^{\prime }\left(u\right)-{\phi }^{\prime }\left(v\right),u-v\rangle >0$ (27)

即 ${\phi }^{\prime }$ 是严格单调算子，由引理3和引理6，得 ${\left({\phi }^{\prime }\right)}^{-1}$ 在 $X^{*}$ 上是连续的。

引理8 泛函 $\varphi$ 和 $\omega$ 在X上是连续Gâteaux可微的，且 ${\varphi }^{\prime }$ 和 ${\omega }^{\prime }$ 是紧的。

证明：对于 $u_n\subset X$，设在X中 $u_n\rightharpoonup u$，即当 $n\to \infty$，在 $\left[a,b\right]$ 上 $u_n$ 一致收敛到u。因此有

$\underset{n\to \infty }{\lim }\inf \varphi \left(u_n\right)\le {\displaystyle {\int }_a^b\underset{n\to \infty }{\lim }\inf G\left(t,u_n\left(t\right)\right)\text{d}t}={\displaystyle {\int }_a^{b}G\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}=\Phi \left(u\right),$

其中 $u=\left(u_1,u_2,\cdots ,u_n\right)\in X$，则 $\Phi$ 是弱序列下半连续的。对于任意 $t\in \left[a,b\right]$，G关于u和v是连续可微的，基于勒贝格控制收敛定理， ${\varphi }^{\prime }\left(u_n\right)$ 强收敛到 ${\varphi }^{\prime }\left(u\right)$，即 ${\varphi }^{\prime }$ 在X上是强连续的，则 ${\varphi }^{\prime }$ 是紧算子，且(14)为Gâteaux导数 ${\varphi }^{\prime }\in X^{*}$ 在点 $u\in X$ 的泛函，同理可得泛函 $\omega$ 在X上是连续Gâteaux可微的且 ${\omega }^{\prime }$ 是紧的。

定理2 假设存在一个非负常数 $\eta$ 和一个函数 $\varpi =\left(u_{11},u_{21},\cdots ,u_{n1}\right)\in X$，使得满足下面的条件

(i) $\max \left\{\underset{\left(u_1,u_2,\cdots ,u_n\right)\to \left(0,0,\cdots ,0\right)}{\lim \sup }\frac{F\left(t,u_1,u_2,\cdots ,u_n\right)}{{\left|u_1\right|}^p+{\left|u_2\right|}^p+\cdots +{\left|u_n\right|}^p},\underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{F\left(t,u_1,u_2,\cdots ,u_n\right)}{{\left|u_1\right|}^p+{\left|u_2\right|}^p+\cdots +{\left|u_n\right|}^p}\right\}\le \eta$，

(ii) $\frac{{\displaystyle {\int }_a^{b}F\left(t,u_{11}\left(t\right),u_{21}\left(t\right),\cdots ,u_{n1}\left(t\right)\right)\text{d}t}}{{\Vert u_{11}\Vert }^p+{\Vert u_{21}\Vert }^p+\cdots +{\Vert u_{n1}\Vert }^p}>TM\eta$，

则对于每个紧区间 $\left[{\theta }_1,{\theta }_2\right]\subset \left(\frac{1}{{\rho }_2},\frac{1}{{\rho }_1}\right)$，存在 $N>0$ 满足：对于任何 $\lambda \in \left[{\theta }_1,{\theta }_2\right]$，存在 $\xi >0$，使得对于任何 $\mu \in \left[0,\xi \right]$，问题(1)在X中至少有三个范数小于N的解。

证明：下面将使用定理1去证明问题(1)在分数阶导数空间 $X=E_{{\alpha }_1}^p\times E_{{\alpha }_2}^p\times \cdots \times E_{{\alpha }_n}^p$ 以及其范数 $u={\displaystyle {\sum }_{i=1}^n{\Vert u_i\Vert }_{E_{{\alpha }_i}^p}}$， $u=\left(u_1,u_2,\cdots ,u_n\right)\in X$ 中至少有三个范数小于N的解。由引理3，得分数阶导数空间X是一个自反可分的Banach空间。再由引理7和引理8，得泛函 $\phi$ 是强制，弱序列下半连续的且在X的每个有界子集上有界， ${\phi }^{\prime }$ 在 $X^{*}$ 上有连续逆。且泛函 $\varphi$ 和 $\omega$ 在X上是连续Gâteaux可微的，且 ${\varphi }^{\prime }$ 和 ${\omega }^{\prime }$ 是紧的。

存在 $\phi$ 的严格局部最小值 $u_0=\left(u_{01},u_{02},\cdots ,u_{0n}\right)=\left(0,0,\cdots ,0\right)\in X$，有 $\phi \left(u_0\right)=\omega \left(u_0\right)=0$。

由(i)得，存在 ${\epsilon }_1,{\epsilon }_2>0$，有

$F\left(t,u_1,u_2,\cdots ,u_n\right)\le \eta \left({\left|u_1\right|}^p+{\left|u_2\right|}^p+\cdots +{\left|u_n\right|}^p\right),$ (28)

其中 $t\in \left[a,b\right]$， $\left|\left(u_1,u_2,\cdots ,u_n\right)\right|\in \left(0,{\epsilon }_1\right)\cup \left({\epsilon }_2,\infty \right)$。

由于F的连续性，存在 $r>0$， $\sigma >p$，有

$F\left(t,u_1,u_2,\cdots ,u_n\right)\le \eta \left({\left|u_1\right|}^p+{\left|u_2\right|}^p+\cdots +{\left|u_n\right|}^p\right)+r\left({\left|u_1\right|}^{\sigma }+{\left|u_2\right|}^{\sigma }+\cdots +{\left|u_n\right|}^{\sigma }\right)$ (29)

其中 $t\in \left[a,b\right]$， $\left|\left(u_1,u_2,\cdots ,u_n\right)\right|\in R$。

由(29)， $u=\left(u_1,u_2,\cdots ,u_n\right)\in X$ 和引理2，有

$\begin{array}{c}\omega \left(u\right)={\displaystyle {\int }_a^{b}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}\\ \le \eta {\displaystyle {\int }_a^b\left({\left|u_1\right|}^p+{\left|u_2\right|}^p+\cdots +{\left|u_n\right|}^p\right)\text{d}t}+r{\displaystyle {\int }_a^b\left({\left|u_1\right|}^{\sigma }+{\left|u_2\right|}^{\sigma }+\cdots +{\left|u_n\right|}^{\sigma }\right)}\\ \le \eta TM\left({\Vert u_1\Vert }_{{\alpha }_1}^p+{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +{\Vert u_n\Vert }_{{\alpha }_n}^p\right)+Tr\xi \left({\Vert u_1\Vert }_{{\alpha }_1}^{\sigma }+{\Vert u_2\Vert }_{{\alpha }_2}^{\sigma }+\cdots +{\Vert u_n\Vert }_{{\alpha }_n}^{\sigma }\right)\end{array}$

其中 $M=\max \left\{\frac{b^{p{\alpha }_i-1}}{{\left(\Gamma \left({\alpha }_i\right)\right)}^p{w}_i^0{\left(\left({\alpha }_i-1\right)q+1\right)}^{\frac{p}{q}}}|1\le i\le n\right\}$ 且 $\xi =\max \left\{{\left(\frac{b^{{\alpha }_i-\frac{1}{p}}}{\Gamma \left({\alpha }_i\right)\sqrt[\frac{1}{p}]{w_i^0{\left(\left({\alpha }_i-1\right)q+1\right)}^{\frac{p}{q}}}}\right)}^{\sigma }|1\le i\le n\right\}$。

由 $\sigma >p$，有

$\begin{array}{l}\underset{\left(u_1,u_2,\cdots ,u_n\right)\to \left(0,0,\cdots ,0\right)}{\lim \sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)}\\ \le \underset{\left(u_1,u_2,\cdots ,u_n\right)\to \left(0,0,\cdots ,0\right)}{\lim \sup }\frac{\eta TM\left({\Vert u_1\Vert }_{{\alpha }_1}^p+{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +{\Vert u_n\Vert }_{{\alpha }_n}^p\right)}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ +\underset{\left(u_1,u_2,\cdots ,u_n\right)\to \left(0,0,\cdots ,0\right)}{\lim \sup }\frac{Tr\xi \left({\Vert u_1\Vert }_{{\alpha }_1}^{\sigma }+{\Vert u_2\Vert }_{{\alpha }_2}^{\sigma }+\cdots +{\Vert u_n\Vert }_{{\alpha }_n}^{\sigma }\right)}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ \le p\eta TM\end{array}$ (30)

由(28)，有

$\begin{array}{l}\underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)}\\ \le \underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{{\displaystyle {\int }_{\Vert \left(u_1,u_2,\cdots ,u_n\right)\Vert \le {\epsilon }_2}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ +\underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{{\displaystyle {\int }_{\Vert \left(u_1,u_2,\cdots ,u_n\right)\Vert >{\epsilon }_2}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ \le \underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{{\displaystyle {\int }_{\Vert \left(u_1,u_2,\cdots ,u_n\right)\Vert >{\epsilon }_2}F\left(t,u_1\left(t\right),\cdots ,u_n\left(t\right)\right)\text{d}t}}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ \le \underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{\eta TM\left({\Vert u_1\Vert }_{{\alpha }_1}^p+{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +{\Vert u_n\Vert }_{{\alpha }_n}^p\right)}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}\\ \le p\eta TM\end{array}$ (31)

由(30)和(31)，有

${\rho }_1=\max \left\{0,\underset{\left(u_1,u_2,\cdots ,u_n\right)\to \left(0,0,\cdots ,0\right)}{\lim \sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)},\underset{\left(u_1,u_2,\cdots ,u_n\right)\to +\infty }{\lim \sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)}\right\}\le p\eta TM$

由(ii)，有

$\begin{array}{c}{\rho }_2=\underset{\left(u_1,u_2,\cdots ,u_n\right)\in {\phi }^{-1}\left(\left(0,+\infty \right)\right)}{\sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)}\\ =\underset{\left(u_1,u_2,\cdots ,u_n\right)\in X/\left\{\left(0,0,\cdots ,0\right)\right\}}{\sup }\frac{\omega \left(u_1,u_2,\cdots ,u_n\right)}{\phi \left(u_1,u_2,\cdots ,u_n\right)}\\ \ge \frac{{\displaystyle {\int }_a^{b}F\left(t,u_{11}\left(t\right),\cdots ,u_{n1}\left(t\right)\right)\text{d}t}}{\frac{1}{p}{\Vert u_1\Vert }_{{\alpha }_1}^p+\frac{1}{p}{\Vert u_2\Vert }_{{\alpha }_2}^p+\cdots +\frac{1}{p}{\Vert u_n\Vert }_{{\alpha }_n}^p}>p\eta TM\ge {\rho }_1.\end{array}$

则对于每个紧区间 $\left[{\theta }_1,{\theta }_2\right]\subset \left(\frac{1}{{\rho }_2},\frac{1}{{\rho }_1}\right)$，存在 $N>0$ 满足：对于任何 $\lambda \in \left[{\theta }_1,{\theta }_2\right]$，存在 $\xi >0$，使得对于任何 $\mu \in \left[0,\xi \right]$，问题(1)在X中至少有三个范数小于N的解。

基金项目

2021研究生专业课程建设项目，2021北京邮电大学“高新课程”建设项目。