Existence of nontrivial solutions for a system of fractional advection–dispersion equations

Abstract

In this paper, we investigate the existence of nontrivial solutions for a class of fractional advection–dispersion systems. The approach is based on the variational method by introducing a suitable fractional derivative Sobolev space. We take two examples to demonstrate the main results.

1 Introduction

Physical models containing fractional differential operators were extensively studied in recent years due to its capacity of simulating anomalous diffusion, i.e., diffusion which can not be accurately modeled by the usual advection–dispersion equation. A fractional advection–dispersion equation (ADE for short) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. Anomalous diffusion equations have been used in modeling turbulent flow [1,2,3], chaotic dynamics of classical conservative systems [4], and in contaminant transport of groundwater flow [5]. For more background information and applications on the fractional ADE, the reader is referred to [6,7,8,9,10].

Ervin and Loop [1] investigated the following fractional ADE:

$$\begin{aligned}&-\frac{\mathrm{d}}{{ d}t}(p{_0{\mathcal {D}}^{-\beta }_t}+(1-p){_t{\mathcal {D}}^{-\beta }_T})u'(t)+b(t)u'(t)+c(t)u(t) \nonumber \\&\quad =\nabla F(t,u(t)),\quad \mathrm{a.e.} \quad t\in [0,T] \end{aligned}$$

(1.1)

where ${_0{\mathcal {D}}_t}^{ {-\beta }}$ and ${_t{\mathcal {D}}_T}^{ {-\beta }}$ are the left and right Riemann–Liouville fractional integral operators respectively, with $0\le \beta <1$, $p\in [0,1]$ is a constant describing the skewness of the transport process, b, c, F are functions satisfying some suitable conditions. A special case of the fractional ADE describes symmetric transitions, where $p=\frac{1}{2}$ in (1.1). In this case,

$$\begin{aligned} p{_0{\mathcal {D}}_t}^{ {-\beta }}+(1-p){_t{\mathcal {D}}_T}^{ {-\beta }}=\frac{1}{2}{_0{\mathcal {D}}_t}^{ {-\beta }}+\frac{1}{2}{_t{\mathcal {D}}_T}^{ {-\beta }}. \end{aligned}$$

(1.2)

Another equation for a N-dimensional fractional ADE was given by Fix and Roop [7], and the equation may be written as

$$\begin{aligned} \frac{\partial \phi }{\partial t}=-\nabla \cdot (\mathbf{v }\phi )-\nabla \cdot (\nabla ^{-\beta }(-k\nabla \phi ))+f,\quad \mathrm{in}\quad \Omega , \end{aligned}$$

(1.3)

where $\phi (t,x)$ is the concentration of a solute at a point x in an arbitrary bounded connected set $\Omega \in {\mathbb {R}}^n$ at time t, $\mathbf{v }$ is the velocity of the fluid, k is the diffusion constant coefficient, $\mathbf{v }\phi$ and $-k\nabla \phi$ are the mass flux due to advection and dispersion respectively and f is a source term. The operator $\nabla ^{-\beta }$ with $0<\beta <1$ is a linear combination of the left and right Riemann–Liouville fractional integral operators, and its jth component is defined by

$$\begin{aligned} \nabla ^{-\beta }(-k\nabla \phi )_j=(p_{-\infty }{\mathcal {D}}^{-\beta }_{x_j}+(1-p){_{x_j}{\mathcal {D}}^{ -\beta }_{+\infty }})\left( -k\frac{\partial \phi }{\partial x_j}\right) ,\quad j=1,2,\ldots ,N, \end{aligned}$$

where $p\in [0,1]$ describes the skewness of the transport process, $_{-\infty }{\mathcal {D}}^{-\beta }_{x_j}$ and ${_{x_j}{\mathcal {D}}^{ -\beta }_{+\infty }}$ are the left and right Riemann–Liouville fractional integral operators, respectively. We take $p=\frac{1}{2}$ in (1.3), and get a special case of the fractional ADE (1.3) which describes symmetric transitions. In this case, the fractional order gradient operator $\nabla ^{-\beta }$ reduces to the following symmetric operator

$$\begin{aligned} (\nabla ^{-\beta })_j=\frac{1}{2} {_{-\infty }{\mathcal {D}}^{-\beta }_{x_j}}+\frac{1}{2} {_{x_j}{\mathcal {D}}^{-\beta }_{+\infty }},\quad j=1,2,\ldots ,N. \end{aligned}$$

Recently, many research results appeared for symmetric fractional ADE. By using the mountain pass theorem and Ekeland’s variational principle, Jiao and Zhou [11] established the existence of solution and nontrivial solution for the following symmetric fractional ADE, respectively,

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} ({ _{0}{\mathcal {D}}_t}^{-\beta }u')(t)+ \frac{1}{2} ({_t{\mathcal {D}}_T}^{-\beta }u')(t)\right) + \nabla F(t,u)=0, ~~ \text {a.e.} ~~ t\in [0,T],\\ \displaystyle u(0)=u(T)=0, \end{array}\right. \end{aligned}$$

(1.4)

where ${ _{0}{\mathcal {D}}_t}^{-\beta }$ and ${_t{\mathcal {D}}_T}^{-\beta }$ denote the left and right Riemann–Liouville fractional integrals of order $\beta$ with $0\le \beta <1$, respectively, $\nabla F(t,x)$ is the gradient of F at $x\in {\mathbb {R}}^n$. Teng et al. [12] proved the existence and multiplicity of solutions for a similar symmetric case for a class of nonsmooth fractional ADEs by using a variational method based on the nonsmooth critical point theory. Zhang et al. [13] studied the eigenvalue problem for the following symmetric fractional ADE:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} ({ _{0}{\mathcal {D}}_t}^{-\beta }u')(t)+ \frac{1}{2} ({_t{\mathcal {D}}_T}^{-\beta }u')(t)\right) + \lambda \nabla F(t,u)=0, ~~ \text {a.e.} ~~ t\in [0,T],\\ \displaystyle u(0)=u(T)=0, \end{array}\right. \end{aligned}$$

(1.5)

where $\lambda$ is a real nonnegative parameter. By using the three-critical-point theorem in [14, 15] respectively, several criteria for the existence of multiple nontrivial solutions for the eigenvalue problem (1.5) were established in [13]. For other research results about symmetric fractional ADE, we refer the reader to [16,17,18] for (1.4) and to [19,20,21,22,23] for the eigenvalue problem (1.5).

Motivated by the above works, in this paper, we will use the critical point theory to study the following symmetric fractional ADE system:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} ({ _{0}{\mathcal {D}}_t}^{-\beta _i}u_i')(t)+ \frac{1}{2} ({_t{\mathcal {D}}_T}^{-\beta _i}u_i')(t)\right) + F_{u_i}'(t,u_1(t),\ldots ,u_n(t))=0, ~~ \text {a.e.} ~~ t\in [0,T],\\ \displaystyle u_i(0)=u_i(T)=0 \end{array}\right. \quad \nonumber \\ \end{aligned}$$

(1.6)

for $1\le i\le n$, where $n\ge 1$, $T>0$, $0\le \beta _i<1$ for $1\le i\le n$, ${ _{0}{\mathcal {D}}_t}^{-\beta _i}$ and ${_t{\mathcal {D}}_T}^{-\beta _i}$ denote the left and right Riemann–Liouville fractional integrals of order $\beta _i$, respectively, $F: [0,T]\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is a given function. We will establish some conditions on F, which are easily to be verified, to guarantee the existence of a nontrivial solution for (1.6).

Obviously, if we take $\beta _i=\beta \in [0,1)$ for $1 \le i \le n$ in (1.6), then the fractional ADE (1.6) reduces to (1.4). If we take $\beta _i=0$ for $i=1,2,\ldots ,n$, then the fractional ADE (1.6) reduces to the classical second-order ADE of the following form

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle u''+ \nabla F(t,u(t))=0, ~~ \text {a.e.} ~~ t\in [0,T],\\ \displaystyle u(0)=u(T)=0, \end{array}\right. \end{aligned}$$

(1.7)

where $F: [0,T]\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is a given function satisfying some assumptions, $n\ge 1$, and $\nabla F (t, u)$ is the gradient of F at $u\in {\mathbb {R}}^n$. Many excellent results on the existence of solutions for (1.7) have been reported in [24, 25].

2 Preliminaries and the fractional derivative space

In this section, we firstly introduce some notations, definitions and preliminary results about fractional derivative which are to be used throughout this paper, then we define a suitable fractional derivative Sobolev space.

Various definitions for the fractional derivative have been introduced over the past years [26]. In this paper, we focus on the Caputo fractional derivative and we refer the reader to [26, 27] for details.

For convenience, we denote

$$\begin{aligned} L^p([0,T],{\mathbb {R}})&=\left\{ u:[0,T]\rightarrow {\mathbb {R}}~{\big |}~\int _0^T|u(t)|^p\mathrm{d}t<+\infty \right\} ;\\ C([0,T],{\mathbb {R}})&=\{u:[0,T]\rightarrow {\mathbb {R}}~{\big |}~u(t) ~~\text {is continuous}\};\\ C^k([0,T],{\mathbb {R}})&=\{u:[0,T]\rightarrow {\mathbb {R}}~{\big |}~u^{(k)}(t)~~ \text {is continuous}\},~~ k=1,2,\ldots ;\\ C^\infty _0([0,T],{\mathbb {R}})&=\{u ~{\big |}u\in C^\infty ([0,T],{\mathbb {R}})~~\mathrm{with}~~u(0)=u(T)=0\};\\ \Vert u\Vert _{\infty }&=\max \limits _{t\in [0,T]}|u(t)|,\quad \Vert u\Vert _{L^p}=\left( \int _0^T|u(t)|^p\mathrm{d}t\right) ^{1/p}. \end{aligned}$$

Definition 2.1

(Left and right Riemann–Liouville fractional integrals [26]) Let g be a function defined on [a, b]. The left and right Riemann–Liouville fractional integrals of order $\gamma >0$ for function g, denoted by $(_a D_t^{-\gamma }g)$ and $(_t D_b^{-\gamma }g)$ respectively, are defined by

$$\begin{aligned} \left( {{_a{\mathcal {D}}_t}}^{ {-\gamma }}g\right) (t)= \frac{1}{\Gamma (\gamma )}\int _a^t{(t-s)^{\gamma -1}g(s)}\mathrm{d}s,~~ \gamma >0,~~ t\in [a,b] \end{aligned}$$

and

$$\begin{aligned} \left( {{_t{\mathcal {D}}_b}}^{-\gamma }g\right) (t)= \frac{1}{\Gamma (\gamma )}\int _t^b{(s-t)^{\gamma -1}g(s)}\mathrm{d}s,~~ \gamma >0,~~ t\in [a,b] \end{aligned}$$

provided that the right-hand sides are pointwise defined on [a, b], where $\Gamma$ is the “Gamma Function” defined by $\Gamma (\gamma )=\int _0^\infty t^{\gamma -1}e^{-t}\mathrm{d}t$, $\gamma >0$.

Definition 2.2

(Left and right Caputo fractional derivatives [26]) Let g be a function defined on [a, b], $\gamma \ge 0$ and $n \in \mathbb {N}$. We denote the left and right Riemann–Liouville fractional derivatives of order $\gamma \ge 0$ for function g by $(_a^c D_t^{\gamma }g)$ and $(_t^c{\mathcal {D}}_b^{\gamma }g)$ respectively.

• (i) If $\gamma \in (n-1,n)$, then

  $$\begin{aligned}&(_a^c{\mathcal {D}}_t^{\gamma }g)(t) \\&\quad =({{_a{\mathcal {D}}_t}}^{-(n-\gamma )}g^{(n)})(t) \\&\quad =\frac{1}{\Gamma (n-\gamma )} \left( \int _a^t{(t-s)^{n-\gamma -1}g^{(n)}(s)}\mathrm{d}s\right) ,~~ t\in [a,b] \end{aligned}$$

  and

  $$\begin{aligned} (_t^c{\mathcal {D}}_b^{\gamma }g)(t)= & {} (-1)^n({{_t{\mathcal {D}}_b}}^{-(n-\gamma )}g^{(n)})(t) \\= & {} \frac{(-1)^n }{\Gamma (n-\gamma )}\left( \int _t^b{(s-t)^{n-\gamma -1}g^{(n)}(s)}\mathrm{d}s\right) ,~~ t\in [a,b]. \end{aligned}$$

• (ii) If $\gamma =n$, then

  $$\begin{aligned} (_a^c\mathcal {D}_t^{n}g)(t) =g^{(n)}(t) \quad \text {and} \quad (_t^c\mathcal {D}_b^{n}g)(t) =(-1)^{n}g^{(n)}(t), ~~ t\in [a,b]. \end{aligned}$$

Remark 2.1

According to Definition 2.2, if $0< \gamma <1$, then

$$\begin{aligned} ({}_a^c{\mathcal {D}}_t^{\gamma }g)(t) =({{_a{\mathcal {D}}_t}}^{-(1-\gamma )}g')(t) =\frac{1}{\Gamma (1-\gamma )} \left( \int _a^t{(t-s)^{-\gamma }g'(s)}\mathrm{d}s\right) ,~~ t\in [a,b] \end{aligned}$$

and

$$\begin{aligned} (_t^c{\mathcal {D}}_b^{\gamma }g)(t) =-({{_t{\mathcal {D}}_b}}^{-(1-\gamma )}g')(t) =-\frac{1 }{\Gamma (1-\gamma )}\left( \int _t^b{(s-t)^{-\gamma }g'(s)}\mathrm{d}s\right) ,~~ t\in [a,b]. \end{aligned}$$

Property 2.1

[27] The left and right Riemann–Liouville fractional integral operators have the following property:

$$\begin{aligned} {{_a{\mathcal {D}}_t}}^{-\gamma _1}(_a {\mathcal {D}}_t^{-\gamma _2}f) ={_a{\mathcal {D}}_t}^{-\gamma _1-\gamma _2}f \quad \text {and} \quad _t {\mathcal {D}}_b^{-\gamma _1}({{_t{\mathcal {D}}_b}}^{-\gamma _2})={{_t{\mathcal {D}}_b}}^{-\gamma _1-\gamma _2}f,~~ \forall \gamma _1,\gamma _2>0 \end{aligned}$$

hold in all $t\in [a,b]$ for $f\in C([0,T],{\mathbb {R}})$.

Property 2.2

[27] Let $0<\alpha \le 1$ and $1\le p<\infty$. For any $f\in L^p([0,T],{\mathbb {R}})$, we have

$$\begin{aligned} \Vert {_a{\mathcal {D}}_{\xi }}^{-\alpha }f\Vert _{L^p([0,t])}\le \frac{t^\alpha }{\Gamma (\alpha _1)}\Vert f\Vert _{L^p([0,t])},~~ \xi \in [a,t], t\in [a,b]. \end{aligned}$$

Property 2.3

[27] The left and right Riemann–Liouville fractional integral operators have the following property:

$$\begin{aligned} \int _a^b\left( {{_a{\mathcal {D}}_t}}^{-\gamma }f\right) (t) g(t)\mathrm{d}t=\int _a^b\left( {{_t{\mathcal {D}}_b}}^{-\gamma }g\right) (t)f(t)\mathrm{d}t,~~ \gamma >0, \end{aligned}$$

provided that $f\in L^p[0,T],{\mathbb {R}})$, $g\in L^q[0,T],{\mathbb {R}})$ and $p\ge 1$, $q\ge 1$, $1/p+1/q\le 1+\gamma$ or $p\ne 1$, $q\ne 1$, $1/p+q/1= 1+\gamma$.

In order to establish a variational structure for (1.6), we must construct an appropriate function space. By Property 2.2, when $0<\alpha \le 1$, for any $f\in C^\infty ([0,T],{\mathbb {R}})$, we have $f\in L^p([0,T],{\mathbb {R}})$ and $(_0^c{\mathcal {D}}_t^{\alpha }f)\in L^p([0,T],{\mathbb {R}})$. Therefore, we now define the fractional derivative space $E^{\alpha }$ as the closure of ${C^\infty _0([0,T],{\mathbb {R}})}$ with respect to the norm $\Vert u\Vert ^{\alpha }=(\Vert u\Vert _{L^2}^2+\Vert _0^c{\mathcal {D}}_t^{\alpha }u\Vert _{L^2}^2 )^{1/2}$.

Property 2.4

[11] For $0<\alpha \le 1$,

• (i) $E^{\alpha }=\{u:[0,T]\rightarrow {\mathbb {R}}|u\in L^2([0,T],{\mathbb {R}}),({_{p}^c{\mathcal {D}}_t}^{\alpha }u)\in L^2([0,T],{\mathbb {R}}),u(0)=u(T)=0\}$;

• (ii) $E^{\alpha }$ is compactly embedded in $C([0,T],{{\mathbb {R}}})$ and is a reflexive and separable Banach space;

• (iii) $\Vert u\Vert _{L^2}\le \frac{T^{\alpha }}{\Gamma (\alpha +1)}\Vert _0^c{\mathcal {D}}_t^{\alpha }u\Vert _{L^2},~~ u\in E^{\alpha }$;

• (iv) $\Vert u\Vert _{\infty }\le \frac{\sqrt{2}T^{\alpha -1/2}}{\Gamma (\alpha )\sqrt{\alpha +1}}\Vert _0^c{\mathcal {D}}_t^{\alpha }u\Vert _{L^2},~~ u\in E^{\alpha }$.

Obviously, for $u\in E^{\alpha }$, if we define $\Vert u\Vert _{\alpha }=\Vert _0^c{\mathcal {D}}_t^{\alpha }u\Vert _{L^2}$, then by (iii) of Property 2.4, $\Vert u\Vert ^{\alpha }$ and $\Vert u\Vert _{\alpha }$ are two equivalent norms. Therefore, we will consider $E^{\alpha }$ with respect to the norm $\Vert u\Vert _{\alpha }$ for simplicity in the following.

Property 2.5

[11] if $1/2<\alpha \le 1$, then for any $u\in E^{\alpha }$, we have

$$\begin{aligned} |\cos (\pi \alpha )|\Vert u\Vert _\alpha ^2\le -\int _0^T({_{0}^c{\mathcal {D}}_t}^{\alpha }u)(t)({_{t}^c{\mathcal {D}}_T}^{\alpha }u) (t)\mathrm{d}t\le \frac{\Vert u\Vert _\alpha ^2}{|\cos (\pi \alpha )|} \end{aligned}$$

Let $E=E^{\alpha _1}\times E^{\alpha _2} \times \cdots \times E^{\alpha _n}$ endowed with norm $\Vert u\Vert _E=\Vert (u_1,u_2,\ldots ,u_n)\Vert _E=(\Vert u_1\Vert _{\alpha _1}^2+\Vert u_2\Vert _{\alpha _2}^2+\cdots +\Vert u_n\Vert _{\alpha _n}^2)^{1/2}$. Then, E is a reflexive and separable Banach space and compactly embedded in $(C([0,T], {{\mathbb {R}}}))^n$.

Definition 2.3

$u\in E$ is a solution of (1.6) on [0, T], if it satisfies (1.6); $u\in E$ is a nontrivial solution of (1.6) on [0, T], if it is a solution satisfying $\Vert u\Vert _E\ne 0$.

Definition 2.4

$u=(u_1,u_2,\ldots ,u_n)\in E$ is a weak solution of (1.6) if we have

$$\begin{aligned}&-\frac{1}{2} \int _0^T\sum \limits _{i=1}^{n} [ ({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i)(t)({_{t}^c{\mathcal {D}}_T}^{\alpha _i}v_i)(t)+({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i)(t)({_{0}^c{\mathcal {D}}_t}^{\alpha _i}v_i)(t)]\mathrm{d}t\\&\quad =\int _0^T\sum \limits _{i=1}^{n} F_{u_i}'(t,u_1(t),\ldots ,u_n(t))v_i(t)\mathrm{d}t \end{aligned}$$

for every $v=(v_1,v_2,\ldots ,v_n)\in E$.

In order to prove the equivalence between a weak solution and a solution of (1.6), we must rewrite (1.6). To do this, for each $1\le i\le n,$ let $\alpha _i=1-\frac{\beta _i}{2}$, then $\alpha _i\in (\frac{1}{2},1]$. According to Property 2.1 and Definition 2.2, the fractional ADE (1.6) can be transformed equivalently to the following system:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} { _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}\left( {_{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t)- \frac{1}{2} { _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)} \left( {_{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t)\right) \\ \qquad +\, F_{u_i}'(t,u_1(t),\ldots ,u_n(t))=0, \quad \text {a.e.}\quad t\in [0,T],\\ \displaystyle u_i(0)=u_i(T)=0 \end{array}\right. \end{aligned}$$

(2.1)

for $1\le i\le n.$

Lemma 2.1

If $u=(u_1,u_2,\ldots ,u_n)\in E$ is a weak solution of (1.6), then u must be a solution of (2.1).

Proof

Suppose $u=(u_1,u_2,\ldots ,u_n)\in E$ is a weak solution of (1.6). Define

$$\begin{aligned} w_i(t)=\int _0^tF_{u_i}'(s,u_1(s),\ldots ,u_n(s))\mathrm{d}s,\quad t\in [0,T],\quad 1\le i\le n. \end{aligned}$$

For any $v=(v_1,v_2,\ldots ,v_n)\in E$, noting that $v_i(T)=0$ for $1\le i\le n$, we obtain

$$\begin{aligned} \int _0^Tw_i(t)v_i'(t)\mathrm{d}t&=\int _0^T \left\{ v_i'(t)\int _0^tF_{u_i}'(s,u_1(s),\ldots ,u_n(s))\mathrm{d}s\right\} \mathrm{d}t\nonumber \\&=\int _0^T\left\{ \int _s^Tv_i'(t)\mathrm{d}s\mathrm{d}t\right\} F_{u_i}'(s,u_1(s),\ldots ,u_n(s))\mathrm{d}s\nonumber \\&=-\int _0^TF_{u_i}'(s,u_1(s),\ldots ,u_n(s))v_i(s)\mathrm{d}s,\quad 1\le i\le n. \end{aligned}$$

(2.2)

On the other hand, by Property 2.3, considering $u_i$, $v_i$$\in E^{\alpha _i}$, we have

$$\begin{aligned} \int _0^T\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t)\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}v_i\right) (t)\mathrm{d}t&=\int _0^T\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t) \left( -{ _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}v_i'\right) (t)\mathrm{d}t\nonumber \\&=-\int _0^T\left( { _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i)\right) (t)v_i'(t)\mathrm{d}t,\quad 1\le i\le n \end{aligned}$$

(2.3)

and

$$\begin{aligned} \int _0^T\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t)\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}v_i\right) (t)\mathrm{d}&=\int _0^T\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t) (-{ _{0}{\mathcal {D}}_t}^{(1-\alpha _i)}v_i')(t)\mathrm{d}t\nonumber \\&=\int _0^T\left( { _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i)\right) (t)v_i'(t)\mathrm{d}t,\quad 1\le i\le n. \end{aligned}$$

(2.4)

From (2.2) to (2.4), considering u is a weak solution of (1.6), we have

$$\begin{aligned}&\int _0^T \Big \{\frac{1}{2}\sum \limits _{i=1}^{n}[ ({ _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i))(t)-({ _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i))(t)] \nonumber \\&\quad +\sum \limits _{i=1}^{n}w_i(t)\}v_i'(t)\mathrm{d}t=0. \end{aligned}$$

(2.5)

Now, for any $v_i\in E^{\alpha _i}(1\le i\le n)$, we substitute $v=(0,\ldots ,v_i,\ldots ,0)\in E$ into (2.5) to obtain

$$\begin{aligned}&\int _0^T\Big \{\frac{1}{2}[ ({ _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i))(t)-({ _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i))(t)]+w_i(t)\Big \} \nonumber \\&\quad v_i'(t)\mathrm{d}t=0 \end{aligned}$$

(2.6)

for any $i=1,2,\ldots ,n$. The theory of Fourier series and (2.6) imply that

$$\begin{aligned}&\frac{1}{2}[ ({ _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i))(t)-({ _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i))(t)]+w_i(t)\equiv C_i, \\&\quad \mathrm{a.e.}\quad t\in [0,T], \end{aligned}$$

where $C_i\in {{\mathbb {R}}}$ is a constant. Hence, from $u_i\in E^{\alpha _i}$ and the definition of $w_i$, we have

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} { _{0}{\mathcal {D}}_t}^{-(1-\alpha _i)}\left( {_{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t)- \frac{1}{2} { _{t}{\mathcal {D}}_T}^{-(1-\alpha _i)}\left( {_{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t)\right) \\ \qquad +\, F_{u_i}'(t,u_1(t),\ldots ,u_n(t))=0, \quad \text {a.e.}\quad t\in [0,T],\\ \displaystyle u_i(0)=u_i(T)=0 \end{array}\right. \end{aligned}$$

for any $i=1,2,\ldots ,n$, which means that $u=(u_1(t),u_2(t),\ldots ,u_n(t))\in E$ is a solution of (2.1). $\square$

Lemma 2.2

Suppose $s_i>0$ and $c_i\ge 0$ are constants for $1\le i\le n$ with $\sum \nolimits _{i=1}^nc_i^2>n$. Then

$$\begin{aligned} \sum \limits _{i=1}^n c_i^{s_i}\le n\left( \sum \limits _{i=1}^n c_i^{2}\right) ^{s_0/2}, \end{aligned}$$

where $s_0=\max \nolimits _{1\le j \le n}s_j.$

Proof

Without loss of generality, suppose $c_1>1$, then $c_1^{s_0}>1$. For any $i\in \{1,2,\ldots ,n\}$, if $c_i\ge 1$, then

$$\begin{aligned} c_i^{s_i}\le c_i^{s_0}\le \sum \limits _{i=1}^n c_i^{s_0};\end{aligned}$$

(2.7)

if $c_i< 1$, then

$$\begin{aligned} c_i^{s_i}< 1 < c_1^{s_0}\le \sum \limits _{i=1}^n c_i^{s_0}. \end{aligned}$$

(2.8)

It follows from (2.7) and (2.8) that

$$\begin{aligned} c_i^{s_i}\le \sum \limits _{i=1}^n c_i^{s_0},\quad 1\le i \le n, \end{aligned}$$

and thus we have

$$\begin{aligned} \sum \limits _{i=1}^n c_i^{s_i}\le n\sum \limits _{i=1}^n c_i^{s_0}\le n\left( \sum \limits _{i=1}^n c_i^{2}\right) ^{s_0/2}. \end{aligned}$$

$\square$

3 Existence of nontrivial solutions

We study the existence of nontrivial solutions for the fractional ADE (1.6) in this section. Our tool is a critical point theorem which was developed by Bonanno and D’Aguì [28].

Lemma 3.1

[28] Let X be a reflexive real Banach space, $\varphi :X\rightarrow {{\mathbb {R}}}$ be a sequentially weakly lower semicontinuous functional, and $\psi :X\rightarrow {{\mathbb {R}}}$ be a sequentially weakly upper semicontinuous functional such that $\varphi -\psi$ is coercive. Assume that there exist a sequentially weakly continuous function $I:X\rightarrow {{\mathbb {R}}}$ and $r\in (\inf _X(\varphi +I), \sup _X(\psi +I))$ such that

$$\begin{aligned} \rho (I,r):=\sup _{(\varphi +I)(y)>r}\frac{(\psi +I)(y)-\sup _{(\varphi +I)(x)\le r}(\psi +I)(x)}{(\varphi +I)(y)-r}>1. \end{aligned}$$

Then the restriction of the function $\varphi -\psi$ to $(\varphi +I)^{-1}(r,+\infty )$ has a global minimum.

Theorem 3.1

Let $n\ge 1$, $T>0$, $\frac{1}{2}\le \alpha _i<1$ for $1\le i\le n$, $F(\cdot ,u_1,u_2,\ldots ,u_n): [0,T]\times {{\mathbb {R}}}^n\rightarrow {{\mathbb {R}}}$ is measurable with respect to $t\in [0,T]$ for every $(u_1,u_2,\ldots ,u_n) \in {{\mathbb {R}}}^n$, and $F(t,\cdot ,\ldots ,\cdot )$ is continuously differentiable with respect to $(u_1,u_2,\ldots ,u_n) \in {{\mathbb {R}}}^n$ for a.e. $t \in [0, T ]$. Assume that

• (H1) $F(t,0,\ldots ,0)=0$ for any $t\in [0,T]$;

• (H2) There exists $\omega =(\omega _1,\omega _2,\ldots ,\omega _n)\in E$ such that

  $$\begin{aligned} 0<\Vert \omega \Vert _E^2<2\min \limits _{1\le i\le n}|\cos \pi \alpha _i|\int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t; \end{aligned}$$

• (H3) There exist $c_i \in [0,|\cos (\pi \alpha _i)|\Gamma ^2(\alpha _i+1)/2T^{2\alpha _i})$, $b_i(t)\in L^{2/(2-s_i)}([0,T],{{\mathbb {R}}})$ and $s_i\in (0,2)$ for $1\le i\le n$ and $k(t)\in L^1([0,T],{{\mathbb {R}}}^+)$ such that

  $$\begin{aligned}&F(t,u_1,u_2,\ldots ,u_n)\le \sum \limits _{i=1}^nc_i|u_i|^2+\sum \limits _{i=1}^nb_i(t)|u_i|^{s_i}+k(t), \\&\quad (t,u_1,u_2,\ldots ,u_n)\in [0,T]\times {{\mathbb {R}}}^n. \end{aligned}$$

Then (1.6) has at least one nontrivial solution $u^*\in E$.

Proof

In order to apply Lemma 3.1 to the system (2.1), we introduce the functionals $\varphi$, $\psi$ and I for $u\in E$ as follows:

$$\begin{aligned} \varphi (u)&=-\frac{1}{2}\int _0^T\sum \limits _{i=1}^n\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t)\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t)\mathrm{d}t;\\ \psi (u)&=\int _0^TF(t,u_1(t),\ldots ,u_n(t))\mathrm{d}t;\\ I(u)&=\sum \limits _{i=1}^n\Vert u_i\Vert _{\infty }^2. \end{aligned}$$

Since E is compactly embedded in $(C([0,T],{{\mathbb {R}}}))^n$, it is well known that $\varphi$ is a sequentially weakly lower semicontinuous function, $\psi$ is a sequentially weakly upper semicontinuous function, and I is a sequentially weakly continuous function. Moreover, both $\varphi$ and $\psi$ are Gâteaux differentiable functions whose Gâteaux derivatives at the point $u=(u_1,u_2,\ldots ,u_n)\in E$ are the functions $\varphi '(u)\in E^*$ and $\psi '(u)\in E^*$ respectively, given by

$$\begin{aligned} \varphi '(u)(v)=-\frac{1}{2} \int _0^T\sum \limits _{i=1}^{n}[ ({ _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i)(t)({_{t}^c{\mathcal {D}}_T}^{\alpha _i}v_i)(t)+({ _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i)(t)({_{0}^c{\mathcal {D}}_t}^{\alpha _i}v_i)(t)]\mathrm{d}t\qquad \end{aligned}$$

(3.1)

and

$$\begin{aligned} \psi '(u)(v)=\int _0^T\sum \limits _{i=1}^{n} F_{u_i}'(t,u_1(t),\ldots ,u_n(t))v_i(t)\mathrm{d}t \end{aligned}$$

(3.2)

for every $v=(v_1,v_2,\ldots ,v_n)\in E$.

Obviously, from (3.1), (3.2) and Definition 2.4, we get that a critical point $u^*\in E$ of $\varphi -\psi$ must be a weak solution of (1.6). In the following, we will apply Lemma 3.1 to prove the existence of a critical point for $\varphi -\psi$.

For any $u\in E$, from Property 2.5, (H3), the Hölder inequality and (iii) of Property 2.4, we get

$$\begin{aligned}&\varphi (u)-\psi (u)\nonumber \\&\quad =-\frac{1}{2}\int _0^T\sum \limits _{i=1}^n\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i\right) (t) \left( {_{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i\right) (t)\mathrm{d}t-\int _0^TF(t,u_1(t),\ldots ,u_n(t))\mathrm{d}t \nonumber \\&\quad \ge \sum \limits _{i=1}^n\frac{|\cos \pi \alpha _i|}{2}\Vert u_i\Vert _{\alpha _i}^2-\sum \limits _{i=1}^nc_i\int _0^T|u_i(t)|^2\mathrm{d}t-\sum \limits _{i=1}^n\int _0^Tb_i(t)|u_i(t)|^{s_i}\mathrm{d}t-\int _0^Tk(t)\mathrm{d} t \nonumber \\&\quad \ge \sum \limits _{i=1}^n\frac{|\cos \pi \alpha _i|}{2}\Vert u_i\Vert _{\alpha _i}^2-\sum \limits _{i=1}^nc_i\Vert u_i\Vert _{L^2}^2-\sum \limits _{i=1}^n\Vert b_i\Vert _{L^{2/(2-s_i)}}\Vert u_i\Vert ^{s_i}_{L^2}-\int _0^Tk(t)\mathrm{d} t \nonumber \\&\quad \ge \sum \limits _{i=1}^n\frac{|\cos \pi \alpha _i|}{2}\Vert u_i\Vert _{\alpha _i}^2-\sum \limits _{i=1}^n\frac{c_i T^{2\alpha _i}}{\Gamma ^2(\alpha _i+1)}\Vert u_i\Vert _{\alpha _i}^2-\sum \limits _{i=1}^n\frac{T^{\alpha _is_i}\Vert b_i\Vert _{L^{2/(2-s_i)}}}{\Gamma ^{s_i}(\alpha _i+1)}\Vert u_i\Vert _{\alpha _i}^{s_i} \nonumber \\&\quad -\int _0^Tk(t)\mathrm{d} t \nonumber \\&\quad =\sum \limits _{i=1}^n\left( \frac{|\cos \pi \alpha _i|}{2}-\frac{c_i T^{2\alpha _i}}{\Gamma ^2(\alpha _i+1)}\right) \Vert u_i\Vert _{\alpha _i}^2-\sum \limits _{i=1}^n\frac{T^{\alpha _is_i}\Vert b_i\Vert _{L^{2/(2-s_i)}}}{\Gamma ^{s_i}(\alpha _i+1)}\Vert u_i\Vert _{\alpha _i}^{s_i}-\int _0^Tk(t)\mathrm{d} t \nonumber \\&\quad \ge {\mathcal {M}}\sum \limits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^2-{\mathcal {N}}\sum \limits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^{s_i}-\int _0^Tk(t)\mathrm{d} t, \end{aligned}$$

(3.3)

where

$$\begin{aligned} {\mathcal {M}}:=\min \limits _{1\le i\le n}\left( \frac{|\cos \pi \alpha _i|}{2}-\frac{c_i T^{2\alpha _i}}{\Gamma ^2(\alpha _i+1)}\right) >0\quad \mathrm{and }\quad {\mathcal {N}}:=\max \limits _{1\le i\le n}\frac{T^{\alpha _is_i}\Vert b_i\Vert _{L^{2/(2-s_i)}}}{\Gamma ^{s_i}(\alpha _i+1)}. \end{aligned}$$

From Lemma 2.2, when $\sum \nolimits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^2>n$, we have

$$\begin{aligned} \sum \limits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^{s_i}\le n\left( \sum \limits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^{2}\right) ^{s_0/2}, \end{aligned}$$

(3.4)

where $s_0=\max \nolimits _{1\le i \le n}s_i \in (0,2)$. Hence, when $\sum \nolimits _{i=1}^n\Vert u_i\Vert _{\alpha _i}^2>n$, we substitute (3.4) into (3.3) to obtain

$$\begin{aligned} \varphi (u)-\psi (u)\ge {\mathcal {M}}\Vert u\Vert _E^2-n{\mathcal {N}}\Vert u\Vert _E^{s_0}-\int _0^Tk(t)\mathrm{d} t. \end{aligned}$$

Thus, by ${\mathcal {M}}>0$ and $s_0 \in (0,2)$, we have

$$\begin{aligned} \lim \limits _{\Vert u\Vert _E\rightarrow +\infty }(\varphi (u)-\psi (u))=+\infty , \end{aligned}$$

which means $\varphi (u)-\psi (u)$ is coercive.

Next, we will prove $\rho (I,r)>1$ for some $r\in (\inf _E(\varphi +I), \sup _E(\psi +I))$.

Firstly, by Property 2.5 and (H2), we have

$$\begin{aligned} (\psi -\varphi )(\omega )&=\int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t \nonumber \\&- \left( -\frac{1}{2}\int _0^T\sum \limits _{i=1}^n \left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}\omega _i\right) (t) \left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}\omega _i\right) (t)\mathrm{d}t\right) \nonumber \\&\ge \int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t-\left( \frac{1}{2}\sum \limits _{i=1}^n\frac{\Vert \omega _i\Vert _{\alpha _i}^2}{|\cos (\pi \alpha _i)|}\right) \nonumber \\&\ge \int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t-\left( \frac{\Vert \omega \Vert _E^2}{2\min \limits _{1\le i \le n}|\cos (\pi \alpha _i)|}\right) >0 \end{aligned}$$

(3.5)

and

$$\begin{aligned} (\varphi +I)(\omega )&=-\frac{1}{2}\int _0^T\sum \limits _{i=1}^n\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}\omega _i\right) (t)\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}\omega _i\right) (t)\mathrm{d}t+\sum \limits _{i=1}^n\Vert u_i\Vert _{\infty }^2\nonumber \\&\ge \frac{1}{2}\sum \limits _{i=1}^n{|\cos (\pi \alpha _i)|}{\Vert \omega _i\Vert _{\alpha _i}^2}+\sum \limits _{i=1}^n\Vert u_i\Vert _{\infty }^2\nonumber \\&\ge {\frac{1}{2}\min \limits _{1\le i \le n}|\cos (\pi \alpha _i)|}{\Vert \omega \Vert _E^2}+\sum \limits _{i=1}^n\Vert u_i\Vert _{\infty }^2>0. \end{aligned}$$

(3.6)

From (H1), (3.5) and (3.6), we have

$$\begin{aligned}&\lim \limits _{r\rightarrow 0}\frac{(\psi +I)(\omega )-\int _0^T\max \limits _{\sum \limits _{i=1}^n|\xi _i|^2\le \frac{r}{1+c_0}}F(t,\xi _1,\ldots ,\xi _n)\mathrm{d}t- \frac{r}{1+c_0}}{(\varphi +I)(\omega )-r} =\frac{(\psi +I)(\omega )}{(\varphi +I)(\omega )}\nonumber \\&\quad =1+\frac{(\psi -\varphi )(\omega )}{(\varphi +I)(\omega )}>1, \end{aligned}$$

(3.7)

where $c_0=\min \nolimits _{1\le i \le n}\frac{|\cos (\pi \alpha _i)|\Gamma ^2(\alpha _i)(\alpha _i+1)}{4T^{2\alpha _i-1}}$.

Secondly, from $(\varphi +I)(\omega )>0$, combining (3.7), we may choose a constant $r_0\in {{\mathbb {R}}}$ satisfying

$$\begin{aligned} 0<r_0<(\varphi +I)(\omega ) \end{aligned}$$

(3.8)

and

$$\begin{aligned} \frac{(\psi +I)(\omega )-\int _0^T\max \nolimits _{\sum \limits _{i=1}^n|\xi _i|^2\le \frac{r_0}{1+c_0}}F(t,\xi _1,\ldots ,\xi _n)\mathrm{d}t- \frac{r}{1+c_0}}{(\varphi +I)(\omega )-r_0}>1. \end{aligned}$$

(3.9)

For $x\in \{x {\big |}x\in E,(\varphi +I)(x)\le r_0\}$, by Property 2.5 and (iii) of Property 2.4, we have

$$\begin{aligned} r_0\ge (\varphi +I)(x)&=-\frac{1}{2}\int _0^T\sum \limits _{i=1}^n\left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}x_i\right) (t)\left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}x_i\right) (t)\mathrm{d}t+\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\nonumber \\&\ge \frac{1}{2}\sum \limits _{i=1}^n|\cos (\pi \alpha _i)|\Vert x_i\Vert _{\alpha _i}^2+\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\nonumber \\&\ge \frac{1}{2}\sum \limits _{i=1}^n|\cos (\pi \alpha _i)|\frac{\Gamma ^2(\alpha _i)((\alpha _i-1)/2+1)}{T^{2\alpha _i-1}}\Vert x_i\Vert _{\infty }^2+\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\nonumber \\&\ge (1+c_0)\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2. \end{aligned}$$

(3.10)

By (3.10), we conclude

$$\begin{aligned} \{x \big |x\in E,(\varphi +I)(x)\le r_0\}\subseteq \left\{ x \big |x\in E,\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\le \frac{r_0}{1+c_0}\right\} . \end{aligned}$$

Then,

$$\begin{aligned} \sup _{(\varphi +I)(x)\le r_0}(\psi +I)(x)&=\sup _{(\varphi +I)(x)\le r_0} \left\{ \int _0^TF(t,x_1(t),\ldots ,x_n(t))\mathrm{d}t+\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\right\} \nonumber \\&\le \sup _{(\varphi +I)(x)\le r_0}\int _0^TF(t,x_1(t),\ldots ,x_n(t))\mathrm{d}t+\sup _{(\varphi +I)(x)\le r_0}\sum \limits _{i=1}^n\Vert x_i\Vert _{\infty }^2\nonumber \\&\le \int _0^T\max \limits _{\sum \limits _{i=1}^n|\xi _i|^2\le \frac{r}{1+c_0}}F(t,\xi _1,\ldots ,\xi _n)\mathrm{d}t+ \frac{r_0}{1+c_0}. \end{aligned}$$

(3.11)

Therefore, by (3.8), (3.11) and (3.9), we get

$$\begin{aligned} \rho (I,r_0)&=\sup _{(\varphi +I)(y)>r_0}\frac{(\psi +I)(y)-\sup _{(\varphi +I)(x)\le r_0}(\psi +I)(x)}{(\varphi +I)(y)-r_0}\nonumber \\&\ge \frac{(\psi +I)(\omega )-\sup _{(\varphi +I)(x)\le r_0}(\psi +I)(x)}{(\varphi +I)(\omega )-r_0}\nonumber \\&\ge \frac{(\psi +I)(\omega )-\int _0^T\max \nolimits _{\sum \limits _{i=1}^n|\xi _i|^2\le \frac{r}{1+c_0}}F(t,\xi _1,\ldots ,\xi _n)\mathrm{d}t- \frac{r_0}{1+c_0}}{(\varphi +I)(\omega )-r_0}>1. \end{aligned}$$

(3.12)

So Lemma 3.1 guarantees that $\varphi -\psi$ has a critical point $u^*=(u_1^*,u_2^*,\ldots ,u_n^*) \in E$ such that $(\varphi +I)(u^*)>r_0$. By Property 2.5 and (iv) of Property 2.4, we get

$$\begin{aligned} r_0<(\varphi +I)(u^*)&=-\frac{1}{2}\int _0^T\sum \limits _{i=1}^n \left( { _{0}^c{\mathcal {D}}_t}^{\alpha _i}u_i^*\right) (t) \left( { _{t}^c{\mathcal {D}}_T}^{\alpha _i}u_i^*\right) (t)\mathrm{d}t+\sum \limits _{i=1}^n\Vert u_i^*\Vert _{\infty }^2\\&\le \frac{1}{2}\sum \limits _{i=1}^n\frac{\Vert u_i^*\Vert _{\alpha _i}^2}{|\cos (\pi \alpha _i)|} +\sum \limits _{i=1}^n\frac{2T^{2\alpha _i-1}}{\Gamma ^2(\alpha _i)(\alpha _i+1)}\Vert u_i^*\Vert _{\alpha _i}^2\\&=\sum \limits _{i=1}^n \left( \frac{1}{2|\cos (\pi \alpha _i)|}+\frac{2T^{2\alpha _i-1}}{\Gamma ^2(\alpha _i)(\alpha _i+1)}\right) \Vert u_i^*\Vert _{\alpha _i}^2 \\&\le \Vert u^*\Vert _E^2 \max \limits _{1\le i \le n} \left( \frac{1}{2|\cos (\pi \alpha _i)|} +\frac{2T^{2\alpha _i-1}}{\Gamma ^2(\alpha _i)(\alpha _i+1)}\right) , \end{aligned}$$

which means that

$$\begin{aligned} \Vert u^*\Vert _E>r_0^{\frac{1}{2}}/\max \limits _{1\le i \le n}\left( \frac{1}{2|\cos (\pi \alpha _i)|}+\frac{2T^{2\alpha _i-1}}{\Gamma ^2(\alpha _i)(\alpha _i+1)}\right) ^{\frac{1}{2}}, \end{aligned}$$

and thus $u^*\in E$ is a nontrivial solution of (1.6). $\square$

Now we deduce a particular but verifiable consequence of Theorem 3.1 where the test function $\omega$ is specified. For convenience, put

$$\begin{aligned} {\mathcal {B}}(\alpha _i, T)=&\frac{16}{T^2}\int _0^{T}t^{2(1-\alpha _i)}\mathrm{d}t+\frac{16}{T^2}\int ^{T}_{T/4}\left( \left( t-\frac{T}{4}\right) ^{2(1-\alpha _i)}-2\left( t^2-\frac{T}{4}t\right) ^{1-\alpha _i}\right) \mathrm{d}t \\&+\frac{16}{T^2}\int ^{T}_{3T/4}\left( t-\frac{3T}{4})^{2(1-\alpha _i)}-2\left( t^2-\frac{3T}{4}t\right) ^{1-\alpha _i}\right. \\&\left. +2\left( t^2-Tt+\frac{3 T^2}{16}\right) ^{1-\alpha _i}\right) \mathrm{d}t \end{aligned}$$

for $1\le i \le n$.

Corollary 3.1

Let F be as that defined in Theorem 3.1, and both (H1) and (H3) of Theorem 3.1 hold. Assume that

• (H4) There exist $d_i>0$ for $1\le i\le n$ such that

  • (i) $F(t,\xi _1,\ldots ,\xi _n)\ge 0$ for all $(t,\xi _1,\ldots ,\xi _n)\in ([0,\frac{T}{4})\bigcup [\frac{3T}{4},T])\times [0,d_1\Gamma (2-\alpha _1)]\times [0,d_2\Gamma (2-\alpha _2)]\times \cdots \times [0,d_n\Gamma (2-\alpha _n)]$;

  • (ii) $2\min \nolimits _{1\le i\le n}|\cos \pi \alpha _i|\int _{T/4}^{3T/4}F(t,\Gamma (2-\alpha _1)d_1,\Gamma (2-\alpha _2)d_2,\ldots ,\Gamma (2-\alpha _n)d_n)>\sum \nolimits _{i=1}^n d_i^2{\mathcal {B}}(\alpha _i, T)$.

Then (1.6) has at least one nontrivial solution $u^*\in E$.

Proof

We only need to show that (H2) of Theorem 3.1 are fulfilled by choosing $\omega =(\omega _1(t),\omega _2(t),\ldots ,\omega _n(t))$ with

$$\begin{aligned} \omega _i(t)=\left\{ \begin{array}{lll} \displaystyle \frac{4d_i\Gamma (2-\alpha _i)}{T}t,&{}t\in \left[ 0,\frac{T}{4}\right] , \\ \displaystyle d_i\Gamma (2-\alpha _i),&{}t\in \left[ \frac{T}{4},\frac{3T}{4}\right] ,\\ \displaystyle \frac{4d_i\Gamma (2-\alpha _i)}{T}(T-t),&{}t\in \left[ \frac{3T}{4},T\right] \end{array}\right. \end{aligned}$$

(3.13)

for $1\le i \le n$.

We calculate directly that

$$\begin{aligned} \left( {{_a^c{\mathcal {D}}_t}}^{\alpha _i}\omega _i\right) (t)=\frac{4d_i}{T}\left\{ \begin{array}{lll} \displaystyle t^{1-\alpha _i},&{}t\in \left[ 0,\frac{T}{4}\right] ,\\ \displaystyle t^{1-\alpha _i}-\left( t-\frac{T}{4}\right) ^{1-\alpha _i},&{} t\in \left[ \frac{T}{4},\frac{3T}{4}\right] , \\ \displaystyle t^{1-\alpha _i}-\left( t-\frac{T}{4}\right) ^{1-\alpha _i}-\left( t-\frac{3T}{4}\right) ^{1-\alpha _i}, &{}t\in \left[ \frac{3T}{4},T\right] \end{array}\right. \end{aligned}$$

for $1\le i \le n$.

Obviously, $\omega _i\in L^2([0,T],{\mathbb {R}})$ and $({_{0}^c{\mathcal {D}}_t}^{\alpha }\omega _i)\in L^2([0,T],{\mathbb {R}})$ for $1\le i \le n$. Noting $\omega _i(0)=\omega _i(T)=0$ for $1\le i \le n$, we conclude $\omega _i\in E^{\alpha _i}$, and thus $\omega =(\omega _1,\omega _2,\ldots ,\omega _n) \in E$.

Furthermore, we have

$$\begin{aligned}&\Vert \omega _i\Vert _{\alpha _i}^2=\int _0^T| \left( {{_a^c{\mathcal {D}}_t}}^{\alpha _i}\omega _i\right) (t)|^2\mathrm{d}t\\&\quad =\int _0^{T/4}+\int _{T/4}^{3T/4}+\int _{3T/4}^T|\left( {{_a^c{\mathcal {D}}_t}}^{\alpha _i}\omega _i\right) (t)|^2\mathrm{d}t\\&\quad =\frac{16d_i^2}{T^2}\int _0^{T}t^{2(1-\alpha _i)}\mathrm{d}t +\frac{16d_i^2}{T^2}\int ^{T}_{T/4}\left( \left( t-\frac{T}{4}\right) ^{2(1-\alpha _i)}-2\left( t^2-\frac{T}{4}t\right) ^{1-\alpha _i}\right) \mathrm{d}t\\&\quad \quad +\frac{16d_i^2}{T^2}\int ^{T}_{3T/4} \left. \left( t-\frac{3T}{4}\right) ^{2(1-\alpha _i)}-2\left( t^2-\frac{3T}{4}t\right) ^{1-\alpha _i}+2\left( t^2-Tt+\frac{3 T^2}{16}\right) ^{1-\alpha _i}\right) \mathrm{d}t\\&\quad =d_i^2{\mathcal {B}}(\alpha _i, T). \end{aligned}$$

On the other hand, according to (3.13), we get $0\le \omega _i(t)\le d_i\Gamma (2-\alpha _i)\,(i=1,2,\ldots ,n)$ for all $t\in [0,T]$. Then condition (i) of (H4) ensures that

$$\begin{aligned} \int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t&=\int _0^{T/4}+\int _{T/4}^{3T/4}+\int _{3T/4}^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t\nonumber \\&\ge \int _{T/4}^{3T/4}F(t,d_1\Gamma (2-\alpha _1),\ldots ,d_n\Gamma (2-\alpha _n))\mathrm{d}t. \end{aligned}$$

(3.14)

Condition (ii) of (H4) and (3.14) ensure that

$$\begin{aligned}&2\min \limits _{1\le i\le n}|\cos \pi \alpha _i|\int _0^TF(t,\omega _1(t),\ldots ,\omega _n(t))\mathrm{d}t\\&\quad \ge 2\min \limits _{1\le i\le n}|\cos \pi \alpha _i|\int _{T/4}^{3T/4}F(t,d_1\Gamma (2-\alpha _1),\ldots ,d_n\Gamma (2-\alpha _n))\mathrm{d}t\\&\quad >\sum \limits _{i=1}^n d_i^2{\mathcal {B}}(\alpha _i, T)=\sum \limits _{i=1}^n\Vert \omega _i\Vert _{\alpha _i}^2=\Vert \omega \Vert _E^2, \end{aligned}$$

which means that $\omega$ satisfies (H2) of Theorem 3.1. $\square$

Remark 3.1

Other candidates for the test function $\omega$ in (3.13) can take other forms. For example,

$$\begin{aligned} \overline{\omega _i}(t)=\left\{ \begin{array}{lll} \displaystyle \frac{16d_i\Gamma (1-\alpha _i)}{T^2} \left( \frac{T}{2}-t\right) t,&{}t\in \left[ 0,\frac{T}{4}\right] , \\ \displaystyle d_i\Gamma (1-\alpha _i),&{}t\in \left[ \frac{T}{4},\frac{3T}{4}\right] , \\ \displaystyle \frac{16d_i\Gamma (1-\alpha _i)}{T^2}\left( \frac{T}{2}-t\right) (t-T),&{}t\in \left[ \frac{3T}{4},T\right] \\ \end{array}\right. \end{aligned}$$

(3.15)

for $1\le i \le n$.

In this case,

$$\begin{aligned}&\left( {{_a^c{\mathcal {D}}_t}}^{\alpha _i}\overline{\omega _i}\right) (t) =\frac{32d_i}{T^2}\\&\quad \times \left\{ \begin{array}{lll} \displaystyle \frac{t^{2-\alpha _i}}{2-\alpha _i}+\frac{(T-4t)t^{1-\alpha _i}}{4(1-\alpha _i)},&{} t\in \left[ 0,\frac{T}{4}\right] ,\\ \displaystyle \frac{t^{2-\alpha _i}}{2-\alpha _i}-\frac{(4t-T)t^{1-\alpha _i}}{4(1-\alpha _i)}+\frac{(4t-T)^{2-\alpha _i})}{(1-\alpha _i)(2-\alpha _i)4^{2-\alpha _i}}, &{}t\in \left[ \frac{T}{4},\frac{3T}{4}\right] ,\\ \displaystyle \frac{t^{2-\alpha _i}}{2-\alpha _i}-\frac{(4t-T)t^{1-\alpha _i}}{4(1-\alpha _i)}+\frac{(4t-T)^{2-\alpha _i}-(4t-3T)^{2-\alpha _i}}{(1-\alpha _i)(2-\alpha _i)4^{2-\alpha _i}}, &{}t\in \left[ \frac{3T}{4},T\right] \end{array}\right. \end{aligned}$$

and

$$\begin{aligned}&\overline{{\mathcal {B}}}(\alpha _i, T) =\frac{32^2d_i^2}{T^4}\int _0^{T} \left( \frac{t^{2(2-\alpha _i)}}{(2-\alpha _i)^2}+\frac{(T/4-t)^2t^{2(2-\alpha _i)}}{(2-\alpha _i)^2}+\frac{2(T/4-t)t^{3-2\alpha _i}}{(1-\alpha _i)(2-\alpha _i)}\right) \mathrm{d}t \\&\quad +\frac{32^2d_i^2}{T^4}\int ^{T}_{T/4}\left( \frac{(t-T/4)^{2(2-\alpha _i)}}{(1-\alpha _i)^2(2-\alpha _i)^2} +\frac{2(t^2-T/4t)^{2-\alpha _i}}{(1-\alpha _i)(2-\alpha _i)^2 } -\frac{2(t-T/4)^{3-\alpha _i}t^{1-\alpha _i}}{(1-\alpha _i)^2(2-\alpha _i)}\right) \mathrm{d}t\\&\quad +\frac{32^2d_i^2}{T^4}\int ^{T}_{3T/4} \left( \frac{(t-3/4T)^{2(2-\alpha _i)}}{(1-\alpha _i)^2(2-\alpha _i)^2 } -\frac{2(t^2-3/4Tt)^{2-\alpha _i}}{(1-\alpha _i)(2-\alpha _i)^2 }\right. \\&\quad \left. +\frac{2(t-T/4)(t-3/4T)^{2-\alpha _i}t^{1-\alpha _i}}{(1-\alpha _i)^2(2-\alpha _i) }-\frac{2(t^2-Tt+3/16T^2)^{2-\alpha _i}}{(1-\alpha _i)^2(2-\alpha _i)^2 }\right) \mathrm{d}t \end{aligned}$$

for $1\le i \le n$.

Remark 3.2

If we take $\beta _i=\beta \in [0,1)$ for $1 \le i \le n$ in (1.6), then (1.6) reduces to (1.4). The result about (1.4) in [11, Theorem 5.1] is that (1.4) has at least one solution $u^*\in E$ if

$$\begin{aligned} |F(t,u)|\le \overline{a}|u|^2+\overline{b}(t)|u|^{s}+\overline{c}(t), \end{aligned}$$

(3.16)

where $\overline{a}\in [0,|\cos (\pi \alpha |\Gamma ^2(\alpha +1/2T^{2\alpha })$, $s\in (0,2)$, $\overline{b}\in L^{2/(2-s)}([0,T],{{\mathbb {R}}})$, and $\overline{c}\in L^1([0,T],{{\mathbb {R}}})$.

The restriction on F in (3.16) is similar to the (H3) in Theorem 3.1. Obviously, comparing with the fact that (1.6) has a nontrivial solution $u^*\in E$ satisfying

$$\begin{aligned} \Vert u^*\Vert _E>r_0^{\frac{1}{2}}/\max \limits _{1\le i \le n}\left( \frac{1}{2|\cos (\pi \alpha _i)|}+\frac{2T^{2\alpha _i-1}}{\Gamma ^2(\alpha _i)(\alpha _i+1)}\right) ^{\frac{1}{2}} \end{aligned}$$

in our paper, it can not be ruled out that $u^*$ is a zero solution for (1.4) in [11].

Example 3.1

Consider the following fractional advection-dispersion equations:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} \left( { _{0}{\mathcal {D}}_t}^{-0.5}u_1'\right) (t)+ \frac{1}{2} \left( {_t{\mathcal {D}}_T}^{-0.5}u_1'\right) (t)\right) + F_{u_1}(t, u_1,u_2)=0, ~~ \text {a.e.} ~~ t\in [0,1],\\ \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} \left( { _{0}{\mathcal {D}}_t}^{-0.4}u_2'\right) (t) + \frac{1}{2} \left( {_t{\mathcal {D}}_T}^{-0.4}u_2'\right) (t)\right) +F_{u_2}(t, u_1,u_2)=0, ~~ \text {a.e.} ~~ t\in [0,1],\\ \displaystyle u_1(0)=u_1(1)=0,\quad u_2(0)=u_2(1)=0, \end{array}\right. \nonumber \\ \end{aligned}$$

(3.17)

where $F:[0,1]\times {{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}$ is the function defined by

$$\begin{aligned} F(t,u_1,u_2)= |1-2t|\left\{ \frac{1}{4}\left( u_1^2+u_2^2\right) \sin \sqrt{u_1^2+u_2^2}+2\root 4 \of {{u^2_1+u_2^2}}e^{-\left( u^2_1+u_2^2\right) }\right\} . \end{aligned}$$

(3.18)

Comparing (3.17) with (1.6), we have $T=1$, $n=2$, $\beta _1=0.5$ and $\beta _2=0.4$, which lead to $\alpha _1=0.75$ and $\alpha _2=0.8$ in (2.1).

Obviously, both (H1) and (H3) of Theorem 3.1 are satisfied since $F(t,0,0)=0$ and

$$\begin{aligned} F(t,u_1,u_2)\le c_1u_1^2+ c_2u_2^2+ b_1(t)\sqrt{|u_1|}+b_2(t)\sqrt{|u_2|}, \end{aligned}$$

where $c_1 =\frac{1}{4}\in [0,\frac{|\cos (\pi \alpha _1)|\Gamma ^2(\alpha _1+1)}{2})\approx [0,0.2986)$, $c_2 =\frac{1}{4}\in [0,\frac{|\cos (\pi \alpha _2)|\Gamma ^2(\alpha _2+1)}{2})\approx [0,0.3509)$ and $b_1(t)=b_2(t)= 2 |1-2t|\in L^2([0,1],{\mathbb {R}})$.

Next, we will show that (H2) of Theorem 3.1 is also satisfied.

We choose

$$\begin{aligned} \omega _1(t)=\Gamma (1.25)t(1-t)\quad \mathrm{and}\quad \omega _2(t)=\Gamma (1.2)t(1-t),\quad t\in [0,1], \end{aligned}$$

then, one has $\omega =(\omega _1,\omega _2)\in E=E_{0.75}\times E_{0.8}$. We then have

$$\begin{aligned} ({ _{0}^c{\mathcal {D}}_t}^{0.75}\omega _1)(t)=t^{0.25}-\frac{8}{5}t^{1.25}\quad \mathrm{and }\quad ({ _{0}^c{\mathcal {D}}_t}^{0.8}\omega _1)(t)=t^{0.2}-\frac{5}{3}t^{1.2}, \end{aligned}$$

and thus we have $\Vert \omega _1\Vert _{0.75}^2\approx 0.1181$ and $\Vert \omega _2\Vert _{0.8}^2\approx 0.1424$. Hence

$$\begin{aligned} \Vert \omega \Vert _E^2=\Vert \omega _1\Vert _{0.75}^2+\Vert \omega _2\Vert _{0.8}^2\approx 0.2605. \end{aligned}$$

(3.19)

Moreover,

$$\begin{aligned}&2\min \limits _{1\le i\le 2}|\cos \pi \alpha _i|\int _0^1F(t,\omega _1(t),\omega _2(t))\mathrm{d}t\nonumber \\&\quad =2\cos \frac{\pi }{4}\int _0^1|1-2t| \frac{2 \root 4 \of {\Gamma ^2(1.25)+\Gamma ^2(1.2)}\sqrt{t(1-t)}}{e^{(\Gamma ^2(1.25)+\Gamma ^2(1.2))t^2(1-t)^2}}\mathrm{d}t\nonumber \\&\qquad +2\cos \frac{\pi }{4}\int _0^1|1-2t|\frac{\Gamma ^2(1.25)+\Gamma ^2(1.2)}{4}t^2(1-t)^2\sin t(1-t)\mathrm{d}t\nonumber \\&\quad \approx 0.5123. \end{aligned}$$

(3.20)

Hence, we conclude that (H2) of Theorem 3.1 is satisfied from (3.19) and (3.20).

Therefore, from Theorem 3.1, the fractional ADE (3.17) has a nontrivial solution $u^*=(u_1^*,u_2^*)\in E_{0.75}\times E_{0.8}$.

Example 3.2

Consider the following fractional advection-dispersion equations:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} \left( { _{0}{\mathcal {D}}_t}^{-0.6}u_1'\right) (t)+ \frac{1}{2} \left( {_t{\mathcal {D}}_T}^{-0.6}u_1'\right) (t)\right) + F_{u_1}(t, u_1,u_2)=0, ~~ \text {a.e.} ~~ t\in [0,1],\\ \displaystyle \frac{\text {d}}{\text {d}t}\left( \frac{1}{2} \left( { _{0}{\mathcal {D}}_t}^{-0.8}u_2'\right) (t)+ \frac{1}{2} \left( {_t{\mathcal {D}}_T}^{-0.8}u_2'\right) (t)\right) +F_{u_2}(t, u_1,u_2)=0, ~~ \text {a.e.} ~~ t\in [0,1],\\ \displaystyle u_1(0)=u_1(1)=0,\quad u_2(0)=u_2(1)=0, \end{array}\right. \nonumber \\ \end{aligned}$$

(3.21)

where $F:[0,1]\times {{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}$ is the function defined by

$$\begin{aligned} F(t,u_1,u_2)= \frac{ |1-2t|}{10}\ln \left( 1+u_1^2+u_2^2\right) +8\left( u^2_1+u_2^2\right) ^{\frac{3}{4}}(t-t^2)^{\frac{1}{2}}. \end{aligned}$$

(3.22)

Comparing (3.21) with (1.6), we have $T=1$, $n=2$, $\beta _1=0.6$ and $\beta _2=0.8$, which lead to $\alpha _1=0.7$ and $\alpha _2=0.6$ in (2.1).

Obviously, both (H1) and (H3) of Theorem 3.1 are satisfied since $F(t,0,0)=0$ and

$$\begin{aligned} F(t,u_1,u_2)\le c_1u_1^2+ c_2u_2^2+ b_1(t)|u_1|^{\frac{3}{2}}+b_2(t)|u_2|^{\frac{3}{2}}, \end{aligned}$$

where $c_1 =\frac{1}{10}\in [0,\frac{|\cos (\pi \alpha _1)|\Gamma ^2(\alpha _1+1)}{2})\approx [0,0.2425)$, $c_2 =\frac{1}{10}\in [0,\frac{|\cos (\pi \alpha _2)|\Gamma ^2(\alpha _2+1)}{2})\approx [0,0.1233)$ and $b_1(t)=b_2(t)= 8(t-t^2)^{\frac{1}{2}}\in L^2([0,1],{\mathbb {R}})$.

Next, we will show that (H2) of Theorem 3.1 is also satisfied.

We choose

$$\begin{aligned} \omega _1(t)=\Gamma (1.3)t(1-t)\quad \mathrm{and}\quad \omega _2(t)=\Gamma (1.4)t(1-t),\quad t\in [0,1], \end{aligned}$$

then, one has $\omega =(\omega _1,\omega _2)\in E=E_{0.7}\times E_{0.6}$. We then have

$$\begin{aligned} ({ _{0}^c{\mathcal {D}}_t}^{0.7}\omega _1)(t)=t^{0.3}-\frac{20}{13}t^{1.3}\quad \mathrm{and }\quad ({ _{0}^c{\mathcal {D}}_t}^{0.6}\omega _1)(t)=t^{0.4}-\frac{10}{7}t^{1.4}, \end{aligned}$$

and thus we have $\Vert \omega _1\Vert _{0.7}^2\approx 0.0989$ and $\Vert \omega _2\Vert _{0.6}^2\approx 0.0723$. Hence

$$\begin{aligned} \Vert \omega \Vert _E^2=\Vert \omega _1\Vert _{0.7}^2+\Vert \omega _2\Vert _{0.6}^2\approx 0.1712. \end{aligned}$$

(3.23)

Moreover,

$$\begin{aligned}&2\min \limits _{1\le i\le 2}|\cos \pi \alpha _i|\int _0^1F(t,\omega _1(t),\omega _2(t))\mathrm{d}t\nonumber \\&\quad =2\cos \frac{2\pi }{5}\int _0^1\frac{ |1-2t|}{10}\ln [1+(\Gamma ^2(1.3)+\Gamma ^2(1.4))(t-t^2)^2]\mathrm{d} t\nonumber \\&\qquad + 16\cos \frac{2\pi }{5}\int _0^1[\Gamma ^2(1.3)+\Gamma ^2(1.4)]^{\frac{3}{4}}(t-t^2)^2\mathrm{d}t\nonumber \\&\quad \approx 0.2346. \end{aligned}$$

(3.24)

Hence, we conclude that (H2) of Theorem 3.1 is satisfied from (3.23) and (3.24).

Therefore, from Theorem 3.1, the fractional ADE (3.21) has a nontrivial solution $u^*=(u_1^*,u_2^*)\in E_{0.7}\times E_{0.6}$.