Solvability of a Class of Fractional Advection–Dispersion Coupled Systems

Abstract

The purpose of this study is to provide some criteria for the existence and multiplicity of solutions for a class of fractional advection–dispersion coupled systems with nonlinear Sturm–Liouville conditions and instantaneous and non-instantaneous impulses. Specifically, the existence is derived through the Nehari manifold method, and the proof of multiplicity is based on Bonanno and Bisci’s critical point theorem, which does not require proof that the functional satisfies the Palais–Smale condition. Finally, to illustrate the obtained results, an example is provided.

1. Introduction

The fractional advection–dispersion equation is a mathematical model used to describe the transport and diffusion of substances in porous media or fluid flow systems. It is a generalization of the classical advection–dispersion equation and considers non-local transport and memory effects. The fractional advection–dispersion equation has a wide range of applications in many fields, such as fluid mechanics, environmental science, geology, and groundwater modeling. It is particularly useful in situations where the transport process exhibits non-local behavior, such as the transport of pollutants in heterogeneous media or fluid flow in fractured rock formations (see [1,2,3]). In recent years, there have also been some achievements in this area of research (see [4,5]); for example, [6] considered the following symmetric fractional advection–dispersion equations with Dirichlet boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\beta }(u^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\beta }(u^{\prime }(t))\right)+f(t,u(t))=0,a.e.t\in [0,T],\hfill \\ u(0)=u(T)=0,\hfill \end{array}\right.$$

where $0\le \beta <1$, ${}_{0}D_t^{-\beta }$ and ${}_{t}D_T^{-\beta }$ denoted the left and right Riemann–Liouville fractional integrals of order $\beta$, respectively. The authors proved the existence of at least one nonzero solution to the problem through Bonanno’s critical point theorem.

It is worth noting that in many practical applications, we need to consider the interactions between multiple systems, so it is particularly important to study the behavior and characteristics of coupled systems. The subsystems in a coupled system can interact and affect each other through physical fields, chemical reactions, information transfer, and other means. Fractional differential equation coupled systems have wide applications in many fields, such as control systems, signal processing, image processing, and biological systems (see [7,8,9,10,11,12]). Especially in [13], the following class of fractional order coupled boundary value problems was considered

$$\left\{\begin{array}{c}{}^{c}D^{\alpha }u(t)=f(t,u(t),v(t)),t\in [0,T],\hfill \\ {}^{c}D^{\beta }v(t)=g(t,u(t),v(t)),t\in [0,T],\hfill \\ (u+v)(0)=-(u+v)(T),{\int }_{\eta }^{\xi }(u-v)(s)\mathrm{d}s=A,0<\eta <\xi <T,\hfill \end{array}\right.$$

where $0<\alpha ,\beta \le 1$, ${}^{c}D^{\alpha }$ denoted the Caputo fractional derivative of order $\alpha$. Based on the fixed point theorem, the authors obtained the result that the system had at least one solution under the Lipschitz condition of the nonlinear term.

In addition, we would like to emphasize the value of impulses in practical applications. According to the duration of the impulse, it can be divided into an instantaneous impulse and a non-instantaneous impulse. The fractional instantaneous impulse differential equation is a mathematical model used to describe the behavior of the system under the action of instantaneous impulses. This equation combines the characteristics of instantaneous impulses and fractional differential equations and can be applied in various fields, such as image processing, biological systems, etc. (see [14,15,16]). And [17] considered the following class of fractional order implicit $(p,q)$-Laplacian systems with instantaneous impulses and Dirichlet boundary conditions

$$\left\{\begin{array}{c}{}_{t}D_T^{\alpha ;\varphi }{\mathsf{\Phi }}_p({}_0{}^{c}D_t^{\alpha ;\varphi }u(t))+{|u(t)|}^{p-2}u(t)+{}_{t}D_T^{\beta ;\varphi }{\mathsf{\Phi }}_q({}_0{}^{c}D_t^{\beta ;\varphi }u(t))+{|u(t)|}^{q-2}u(t)\hfill \\ =f(t,u(t),{}_{0}D_t^{\alpha ;\varphi }u(t),{}_{0}D_t^{\beta ;\varphi }u(t)),t\ne t_j,a.e.t\in [0,T],\hfill \\ \Delta {(}_t{D}_T^{\alpha -1;\varphi }{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha ;\varphi }u)+{}_{t}D_T^{\beta -1;\varphi }{\mathsf{\Phi }}_q{(}_0^c{D}_t^{\beta ;\varphi }u))(t_j)=I_j(u(t_j)),j=1,2,\cdots ,m,\hfill \\ u(0)=u(T)=0,\hfill \end{array}\right.$$

where $1<p\le q<\infty$, ${\displaystyle \frac{1}{p}}<\alpha \le 1$, ${\displaystyle \frac{1}{q}}<\beta \le 1$, ${}_{0}D_t^{\alpha ;\varphi }$, ${}_{t}D_T^{\alpha ;\varphi }$ and ${}_0{}^{c}D_t^{\alpha ;\varphi }$ denoted the left and right generalized Riemann–Liouville fractional derivatives and the left $\varphi$-Caputo fractional derivative of order $\alpha$, respectively. ${\mathsf{\Phi }}_p(s)={|s|}^{p-2}s$ $(s\ne 0)$, ${\mathsf{\Phi }}_p(0)=0$, $1<p<\infty$. The authors provided a criterion for the existence of at least one solution to the system through iterative techniques and Mountain Pass Lemma.

It should also be noted that in some cases, instantaneous impulses alone cannot accurately describe certain phenomena, and it is necessary to combine the non-instantaneous nature of non-instantaneous impulses, such as pharmacokinetics, control systems, signal processing, etc. (see [18,19,20,21,22]). For example, we were interested in the following fractional p-Laplacian differential equations with instantaneous and non-instantaneous impulsive boundary conditions in [23]

$$\left\{\begin{array}{c}{}_{t}D_T^{\alpha }{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(t))+{|u(t)|}^{p-2}u(t)=\lambda f_i(t,u(t)),t\in (s_i,t_{i+1}],i=0,1,\cdots ,m,\hfill \\ \Delta {(}_t{D}_T^{\alpha -1}{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(t_i)))=I_i(u(t_i)),i=1,2,\cdots ,m,\hfill \\ {}_{t}D_T^{\alpha -1}{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(t))={}_{t}D_T^{\alpha -1}{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(t_i^{+})),t\in (t_i,s_i],i=1,2,\cdots ,m,\hfill \\ {}_{t}D_T^{\alpha -1}{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(s_i^{-}))={}_{t}D_T^{\alpha -1}{\mathsf{\Phi }}_p{(}_0^c{D}_t^{\alpha }u(s_i^{+})),i=1,2,\cdots ,m,\hfill \\ u(0)=u(T)=0,\hfill \end{array}\right.$$

where $1<p<\infty$, ${\displaystyle \frac{1}{p}}<\alpha \le 1$, ${}_0{}^{c}D_t^{\alpha }$ and ${}_{t}D_T^{\alpha }$ denoted the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha$, respectively. ${\mathsf{\Phi }}_p(s)={|s|}^{p-2}s$ $(s\ne 0)$, ${\mathsf{\Phi }}_p(0)=0$, $1<p<\infty$. $\lambda >0$ was a parameter. The existence of an unbounded sequence of weak solutions was given by a recent critical point theory proposed by Bonanno and Bisci.

The Sturm–Liouville boundary condition is a type of boundary condition in boundary value problems, which is more general in form than the Dirichlet boundary condition. In the fields of physics and engineering, the Sturm–Liouville condition can be applied to solve problems such as wave equations, heat conduction equations, and fluid dynamics equations. In summary, the Sturm–Liouville condition is one of the important tools for solving Sturm–Liouville eigenvalue problems, and it has broad application prospects in fields such as mathematical physics, engineering technology, and scientific computing (see [24,25,26]). We investigated the following class of fractional advection–dispersion equation with Sturm–Liouville boundary conditions and impulsive conditions in [27]

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\beta }(u^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\beta }(u^{\prime }(t))\right)+\lambda f(t,u(t))=0,t\ne t_i,a.e.t\in [0,T],\hfill \\ a\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\beta }(u^{\prime }(0))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\beta }(u^{\prime }(0))\right)-bu(0)=A,\hfill \\ c\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\beta }(u^{\prime }(T))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\beta }(u^{\prime }(T))\right)+du(T)=B,\hfill \\ \Delta \left({}_{0}D_t^{-\beta }(u^{\prime }(t_i))+{}_{t}D_T^{-\beta }(u^{\prime }(t_i))\right)=I_i(u(t_i)),i=1,2,\cdots ,m,\hfill \end{array}\right.$$

where $0\le \beta <1$, $\lambda >0$ and a, b, c, $d>0$. A and B were constants. ${}_{0}D_t^{-\beta }$ and ${}_{t}D_T^{-\beta }$ denoted the left and right Riemann–Liouville fractional integrals of order $\beta$, respectively. When the Sturm–Liouville condition was non-homogeneous, under different conditions, the existence of at least two different solutions and infinitely many solutions to the equation were proved through the Mountain Pass Lemma and the Ricceri generalized variational principle, respectively. When the Sturm–Liouville condition was homogeneous, based on the symmetric Mountain Pass Lemma, we provided the existence results of infinite solutions to the equation.

Inspired by the above, this article attempts to explore the existence of solutions for the following fractional advection–dispersion coupled system with nonlinear Sturm–Liouville boundary conditions and instantaneous and non-instantaneous impulses using variational methods and critical point theory.

$$\left\{\begin{array}{c}-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\theta }(u^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\theta }(u^{\prime }(t))\right)+K(t)u(t)=\lambda F_{iu}(t,u(t),v(t))+\mu G_{iu}(t,u(t),v(t)),\hfill \\ t\in (s_i,t_{i+1}],i=0,1,\cdots ,m,\hfill \\ -{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\eta }(v^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\eta }(v^{\prime }(t))\right)+K(t)v(t)=\lambda F_{iv}(t,u(t),v(t))+\mu G_{iv}(t,u(t),v(t)),\hfill \\ t\in (s_i,t_{i+1}],i=0,1,\cdots ,m,\hfill \\ \Delta \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\theta }(u^{\prime }(t_i))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\theta }(u^{\prime }(t_i))\right)=I_i(u(t_i)),i=1,\cdots ,m,\hfill \\ \Delta \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\eta }(v^{\prime }(t_i))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\eta }(v^{\prime }(t_i))\right)=J_i(v(t_i)),i=1,\cdots ,m,\hfill \\ {}_{0}D_t^{-\theta }(u^{\prime }(t))+{}_{t}D_T^{-\theta }(u^{\prime }(t))={}_{0}D_t^{-\theta }(u^{\prime }(t_i^{+}))+{}_{t}D_T^{-\theta }(u^{\prime }(t_i^{+})),t\in (t_i,s_i],i=1,\cdots ,m,\hfill \\ {}_{0}D_t^{-\theta }(u^{\prime }(s_i^{-}))+{}_{t}D_T^{-\theta }(u^{\prime }(s_i^{-}))={}_{0}D_t^{-\theta }(u^{\prime }(s_i^{+}))+{}_{t}D_T^{-\theta }(u^{\prime }(s_i^{+})),i=1,\cdots ,m,\hfill \\ {}_{0}D_t^{-\eta }(v^{\prime }(t))+{}_{t}D_T^{-\eta }(v^{\prime }(t))={}_{0}D_t^{-\eta }(v^{\prime }(t_i^{+}))+{}_{t}D_T^{-\eta }(v^{\prime }(t_i^{+})),t\in (t_i,s_i],i=1,\cdots ,m,\hfill \\ {}_{0}D_t^{-\eta }(v^{\prime }(s_i^{-}))+{}_{t}D_T^{-\eta }(v^{\prime }(s_i^{-}))={}_{0}D_t^{-\eta }(v^{\prime }(s_i^{+}))+{}_{t}D_T^{-\eta }(v^{\prime }(s_i^{+})),i=1,\cdots ,m,\hfill \\ a\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\theta }(u^{\prime }(0))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\theta }(u^{\prime }(0))\right)-bh(u(0))=0,\hfill \\ c\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\theta }(u^{\prime }(T))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\theta }(u^{\prime }(T))\right)+dh(u(T))=0,\hfill \\ a\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\eta }(v^{\prime }(0))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\eta }(v^{\prime }(0))\right)-bh(v(0))=0,\hfill \\ c\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\eta }(v^{\prime }(T))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\eta }(v^{\prime }(T))\right)+dh(v(T))=0,\hfill \end{array}\right.$$

(1)

where $0\le \theta ,\eta <1$, $\lambda >0$ and $\mu >0$ are two parameters, $a,b,c,d>0$ are constants. ${}_{0}D_t^{-\gamma }$ and ${}_{t}D_T^{-\gamma }$ denote the left and right Riemann–Liouville fractional integrals of order $\gamma$, respectively. $0=s_0<t_1<s_1<t_2<s_2<\cdots <t_m<s_m<t_{m+1}=T$. The function $K\in C([0,T],\mathbb{R})$ and there exist $K_1$ and $K_2$, so that $0<K_1\le K(t)\le K_2$. $h\in C^1(\mathbb{R},\mathbb{R})$ and for $x\in \mathbb{R}$, there exists $h_0>0$ such that $|h(x)|\le h_0$. $F_i\in C^1((s_i,t_{i+1}]\times {\mathbb{R}}^2,\mathbb{R})$, $G_i\in C^1((s_i,t_{i+1}]\times {\mathbb{R}}^2,\mathbb{R})$, and $F_i(t,0,0)=G_i(t,0,0)=0$. The instantaneous impulses $I_i\in C^1(\mathbb{R},\mathbb{R})$ and $J_i\in C^1(\mathbb{R},\mathbb{R})$ start to change suddenly at the points $t_i$ and the non-instantaneous impulses continue during the finite intervals $(t_i,s_i]$, for $i=1,\cdots ,m$. In addition,

$$\begin{array}{cc}\hfill \Delta \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\gamma }(x^{\prime }(t_i))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\gamma }(x^{\prime }(t_i))\right)=& \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\gamma }(x^{\prime }(t_i^{+}))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\gamma }(x^{\prime }(t_i^{+}))\right)\hfill \\ & -\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-\gamma }(x^{\prime }(t_i^{-}))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\gamma }(x^{\prime }(t_i^{-}))\right),\hfill \end{array}$$

where

$$\begin{array}{c}{}_{0}D_t^{-\gamma }(x^{\prime }(s_i^{\pm }))+{}_{t}D_T^{-\gamma }(x^{\prime }(s_i^{\pm }))=\underset{t\to s_i^{\pm }}{lim}\left({}_{0}D_t^{-\gamma }(x^{\prime }(t))+{}_{t}D_T^{-\gamma }(x^{\prime }(t))\right),\hfill \\ {}_{0}D_t^{-\gamma }(x^{\prime }(t_i^{\pm }))+{}_{t}D_T^{-\gamma }(x^{\prime }(t_i^{\pm }))=\underset{t\to t_i^{\pm }}{lim}\left({}_{0}D_t^{-\gamma }(x^{\prime }(t))+{}_{t}D_T^{-\gamma }(x^{\prime }(t))\right).\hfill \end{array}$$

Concretely, we explore the existence of solutions to the boundary value problem (abbreviated as BVP) (1) by employing variational methods and critical point theory. After defining the function space and constructing the variational structure of BVP (1), it is proved that BVP (1) has at least one ground state solution and infinitely many solutions by the Nehari manifold method and critical point theory, respectively.

2. Preliminaries

This section undertakes a review of certain definitions and conclusions of fractional calculus.

Definition 1

([28]). Let x be a function defined on $[0,T]$. The left and right Riemann–Liouville fractional integrals of order $0<\gamma \le 1$ for the function x denoted by ${}_{0}D_t^{-\gamma }x(t)$ and ${}_{t}D_T^{-\gamma }x(t)$, respectively, are defined by

$$\begin{array}{c}{}_{0}D_t^{-\gamma }x(t)={\displaystyle \frac{1}{\Gamma (\gamma )}}{\int }_0^t{(t-s)}^{\gamma -1}x(s)\mathrm{d}s,\hfill \\ {}_{t}D_T^{-\gamma }x(t)={\displaystyle \frac{1}{\Gamma (\gamma )}}{\int }_t^T{(s-t)}^{\gamma -1}x(s)\mathrm{d}s.\hfill \end{array}$$

Definition 2

([28]). Let x be a function defined on $[0,T]$. The left and right Riemann–Liouville fractional derivatives of order $0<\gamma \le 1$ for the function x denoted by ${}_{0}D_t^{\gamma }x(t)$ and ${}_{t}D_T^{\gamma }x(t)$, respectively, are defined by

$$\begin{array}{c}{}_{0}D_t^{\gamma }x(t)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{}_{0}D_t^{\gamma -1}x(t)={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\int }_0^t{(t-s)}^{-\gamma }x(s)\mathrm{d}s\right),\hfill \\ {}_{t}D_T^{\gamma }x(t)=-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{}_{t}D_T^{\gamma -1}x(t)={\displaystyle \frac{-1}{\Gamma (1-\gamma )}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\int }_t^T{(s-t)}^{-\gamma }x(s)\mathrm{d}s\right).\hfill \end{array}$$

Definition 3

([28]). Let $x\in AC([0,T],{\mathbb{R}}^N)$. Then the left and right Caputo fractional derivatives of order $0<\gamma \le 1$ for the function x denoted by ${}_0{}^{c}D_t^{\gamma }x(t)$ and ${}_t{}^{c}D_T^{\gamma }x(t)$, respectively, are defined by

$$\begin{array}{c}{}_0{}^{c}D_t^{\gamma }x(t)={}_0{}^{c}D_t^{\gamma -1}{x}^{\prime }(t)={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\int }_0^t{(t-s)}^{-\gamma }{x}^{\prime }(s)\mathrm{d}s,\hfill \\ {}_t{}^{c}D_T^{\gamma }x(t)=-{}_{t}D_T^{\gamma -1}{x}^{\prime }(t)={\displaystyle \frac{-1}{\Gamma (1-\gamma )}}{\int }_t^T{(s-t)}^{-\gamma }{x}^{\prime }(s)\mathrm{d}s.\hfill \end{array}$$

Property 1

([28]). Let x be continuous for a.e. $t\in [0,T]$ and ${\gamma }_1$, ${\gamma }_2>0$, the left and right Riemann–Liouville fractional integral operators have the following properties

$$\begin{array}{c}\hfill {}_{0}D_t^{-{\gamma }_1}{(}_0{D}_t^{-{\gamma }_2}x(t))={}_{0}D_t^{-{\gamma }_1-{\gamma }_2}x(t),{}_{t}D_T^{-{\gamma }_1}{(}_t{D}_T^{-{\gamma }_2}x(t))={}_{t}D_T^{-{\gamma }_1-{\gamma }_2}x(t).\end{array}$$

Property 2

([28]). If $x\in L^p([0,T],{\mathbb{R}}^N),y\in L^q([0,T],{\mathbb{R}}^N)$ and $p\ge 1$, $q\ge 1$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}\le 1+\gamma$ or $p\ne 1$, $q\ne 1$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1+\gamma$. Then

$${\int }_0^T{[}_0{D}_t^{-\gamma }x(t)]y(t)\mathrm{d}t={\int }_0^T{[}_t{D}_T^{-\gamma }y(t)]x(t)\mathrm{d}t.$$

Property 3

([28]). If $0<\gamma \le 1$ and $x\in AC([0,T],{\mathbb{R}}^N)$, then

$${}_{0}D_t^{-\gamma }{(}_0^c{D}_t^{\gamma }x(t))=x(t)-x(0),{}_{t}D_T^{-\gamma }{(}_t^c{D}_T^{\gamma }x(t))=x(t)-u(T).$$

Based on Definition 3 and Property 1, we can deduce that

$$\begin{array}{c}{\displaystyle \frac{1}{2}}{}_{0}D_t^{-\theta }(u^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\theta }(u^{\prime }(t))={\displaystyle \frac{1}{2}}{}_{0}D_t^{\alpha -1}{(}_0^c{D}_t^{\alpha }u(t))-{\displaystyle \frac{1}{2}}{}_{t}D_T^{\alpha -1}{(}_t^c{D}_T^{\alpha }u(t)),\hfill \\ {\displaystyle \frac{1}{2}}{}_{0}D_t^{-\eta }(v^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_T^{-\eta }(v^{\prime }(t))={\displaystyle \frac{1}{2}}{}_{0}D_t^{\beta -1}{(}_0^c{D}_t^{\beta }v(t))-{\displaystyle \frac{1}{2}}{}_{t}D_T^{\beta -1}{(}_t^c{D}_T^{\beta }v(t)),\hfill \end{array}$$

where $\alpha =1-{\displaystyle \frac{\theta }{2}}$, $\beta =1-{\displaystyle \frac{\eta }{2}}$ and ${\displaystyle \frac{1}{2}}<\alpha ,\beta \le 1$.

Let $L^p([0,T],\mathbb{R})(1\le p<\infty )$ and $C([0,T],\mathbb{R})$ be the p-Lebesgue space and continuous function space, respectively, with the norms

$$\begin{array}{c}{\parallel x\parallel }_{L^p}={\left({\int }_0^T{|x(t)|}^p\mathrm{d}t\right)}^{{\displaystyle \frac{1}{p}}},x\in L^p([0,T],\mathbb{R}),\hfill \\ {\parallel x\parallel }_{\infty }=\underset{t\in [0,T]}{max}|x(t)|,x\in C([0,T],\mathbb{R}).\hfill \end{array}$$

Definition 4.

Let ${\displaystyle \frac{1}{2}}<\gamma \le 1$ and $1\le p<\infty$. The fractional derivative space $E^{\gamma ,p}$ is defined as the closure of $C^{\infty }([0,T],\mathbb{R})$, that is, $E^{\gamma ,p}=\overline{C^{\infty }([0,T],\mathbb{R})}$ with the norm

$${\parallel x\parallel }_{\gamma ,p}={\left({\int }_0^T{|}_0^c{D}_t^{\gamma }{x(t)|}^p\mathrm{d}t+{\int }_0^T{|x(t)|}^p\mathrm{d}t\right)}^{{\displaystyle \frac{1}{p}}}.$$

Lemma 1

([29,30]). Let ${\displaystyle \frac{1}{2}}<\gamma \le 1$ and $1\le p<\infty$. For any $x\in L^p([0,T],\mathbb{R})$, we have

(1)

$${\parallel }_0{D}_{\xi }^{-\gamma }{x\parallel }_{L^p([0,t])}\le {\displaystyle \frac{t^{\gamma }}{\Gamma (\gamma +1)}}{\parallel x\parallel }_{L^p([0,t])}for\xi \in [0,t],t\in [0,T].$$

(2)

$${\parallel }_{\xi }{D}_T^{-\gamma }{x\parallel }_{L^p([t,T])}\le {\displaystyle \frac{{(T-t)}^{\gamma }}{\Gamma (\gamma +1)}}{\parallel x\parallel }_{L^p([t,T])}for\xi \in [t,T],t\in [0,T].$$

Lemma 2

([29]). If ${\displaystyle \frac{1}{2}}<\gamma \le 1$, then for any $x\in E^{\gamma ,2}$, we have

$$-cos\pi \gamma {\int }_0^T{|}_0^c{D}_t^{\gamma }{x(t)|}^2\mathrm{d}t\le -{\int }_0^T{(}_0^c{D}_t^{\gamma }x(t)){(}_t^c{D}_T^{\gamma }x(t))\mathrm{d}t\le -{\displaystyle \frac{1}{cos\pi \gamma }}{\int }_0^T{{|}_0^c{D}_t^{\gamma }x(t)|}^2\mathrm{d}t.$$

Based on Lemmas 1 and 2, similar to the proof of Lemma 4.7 in Reference [29], it can be concluded that the norm ${\parallel u\parallel }_{\gamma ,2}$ in $E^{\gamma ,2}$ is equivalent to

$${\parallel x\parallel }_{\gamma }={\left(-{\int }_0^T{(}_0^c{D}_t^{\gamma }x(t)){(}_t^c{D}_T^{\gamma }x(t))\mathrm{d}t+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t){(x(t))}^2\mathrm{d}t\right)}^{{\displaystyle \frac{1}{2}}}.$$

For any $u\in E^{\alpha ,2}$, $v\in E^{\beta ,2}$, define a space X to be $E^{\alpha ,2}\times E^{\beta ,2}$ with the norm

$${\parallel (u,v)\parallel }_X={\left({\parallel u\parallel }_{\alpha }^2+{\parallel v\parallel }_{\beta }^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Similar to Lemma 9 in [31], it can be proven that space X is a reflexive and separable Banach space. What’s more, it can derive the following lemma.

Lemma 3.

Let ${\parallel (u,v)\parallel }_{\infty }={\parallel u\parallel }_{\infty }+{\parallel v\parallel }_{\infty }$, for $(u,v)\in X$, there exists a constant $\Lambda >0$ such that

$${\parallel (u,v)\parallel }_{\infty }\le \Lambda {\parallel (u,v)\parallel }_X.$$

The proof of this Lemma is similar to Lemma 2.5 in [27], so it is omitted here.

Lemma 4

([30]). Assume that ${\displaystyle \frac{1}{2}}<\alpha ,\beta \le 1$ and the sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ weakly converges to x in $E^{\alpha ,2}$ or $E^{\beta ,2}$, that is, $x_n\rightharpoonup x$ in $E^{\alpha ,2}$ or $E^{\beta ,2}$. Then, $x_n\to x$ in $C([0,T],\mathbb{R})$, that is, $\parallel x_n{-x\parallel }_{\infty }\to 0$ as $n\to \infty$.

The following definition of the weak solution of BVP (1) can be given by simple calculations such as integration.

Definition 5.

A function $(u,v)\in X$ is called the weak solution of the BVP (1) if $(u,v)$ satisfying the following equation

$$\begin{array}{c}-{\displaystyle \frac{1}{2}}{\int }_0^T\left[{(}_0^c{D}_t^{\alpha }u(t)){(}_t^c{D}_T^{\alpha }\omega (t))+{(}_t^c{D}_T^{\alpha }u(t)){(}_0^c{D}_t^{\alpha }\omega (t))+{(}_0^c{D}_t^{\beta }v(t)){(}_t^c{D}_T^{\beta }\varpi (t))+{(}_t^c{D}_T^{\beta }v(t)){(}_0^c{D}_t^{\beta }\varpi (t))\right]\mathrm{d}t\hfill \\ +{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t)u(t)\omega (t)\mathrm{d}t+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t)v(t)\varpi (t)\mathrm{d}t+{\displaystyle \frac{b}{a}}h(u(0))\omega (0)+{\displaystyle \frac{d}{c}}h(u(T))\omega (T)+{\displaystyle \frac{b}{a}}h(v(0))\varpi (0)\hfill \\ +{\displaystyle \frac{d}{c}}h(v(T))\varpi (T)+{\displaystyle \sum _{i=1}^m}{I}_i(u(t_i))\omega (t_i)+{\displaystyle \sum _{i=1}^m}{J}_i(v(t_i))\varpi (t_i)\hfill \\ -\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[F_{iu}(t,u(t),v(t))\omega (t)+F_{iv}(t,u(t),v(t))\varpi (t)]\mathrm{d}t\hfill \\ -\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[G_{iu}(t,u(t),v(t))\omega (t)+G_{iv}(t,u(t),v(t))\varpi (t)]\mathrm{d}t=0,\forall (\omega ,\varpi )\in X.\hfill \end{array}$$

For each $(u,v)\in X$, we consider the functional $\mathsf{\Phi }:X\to \mathbb{R}$ as follows

$$\begin{array}{cc}\hfill \mathsf{\Phi }(u,v)=& -{\displaystyle \frac{1}{2}}{\int }_0^T{(}_0^c{D}_t^{\alpha }u(t)){(}_t^c{D}_T^{\alpha }u(t))\mathrm{d}t+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t){(u(t))}^2\mathrm{d}t\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^T{(}_0^c{D}_t^{\beta }v(t)){(}_t^c{D}_T^{\beta }v(t))\mathrm{d}t+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t){(v(t))}^2\mathrm{d}t\hfill \\ & +{\displaystyle \frac{b}{a}}H(u(0))+{\displaystyle \frac{d}{c}}H(u(T))+{\displaystyle \frac{b}{a}}H(v(0))+{\displaystyle \frac{d}{c}}H(v(T))\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{u(t_i)}{I}_i(s)\mathrm{d}s+{\displaystyle \sum _{i=1}^m}{\int }_0^{v(t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u(t),v(t))\mathrm{d}t\hfill \\ & -\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u(t),v(t))\mathrm{d}t\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel (u,v)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(u(0))+H(v(0))]+{\displaystyle \frac{d}{c}}[H(u(T))+H(v(T))]+{\displaystyle \sum _{i=1}^m}{\int }_0^{u(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{v(t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u(t),v(t))\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u(t),v(t))\mathrm{d}t,\hfill \end{array}$$

where $H(x)={\int }_0^{x}h(s)\mathrm{d}s$, $F_i(t,x,y)={\int }_0^x{F}_{is}(t,s,y)\mathrm{d}s+{\int }_0^y{F}_{is}(t,x,s)\mathrm{d}s$ and $G_i(t,x,y)={\int }_0^x{G}_{is}(t,s,y)\mathrm{d}s+{\int }_0^y{G}_{is}(t,x,s)\mathrm{d}s$. It is easy to find that for $(u,v)\in X$

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}^{\prime }(u,v)(\omega ,\varpi )=& -{\displaystyle \frac{1}{2}}{\int }_0^T[{(}_0^c{D}_t^{\alpha }u(t)){(}_t^c{D}_T^{\alpha }\omega (t))+{(}_t^c{D}_T^{\alpha }u(t)){(}_0^c{D}_t^{\alpha }\omega (t))]\mathrm{d}t+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t)u(t)\omega (t)\mathrm{d}t\hfill \\ & -{\displaystyle \frac{1}{2}}{\int }_0^T[{(}_0^c{D}_t^{\beta }v(t)){(}_t^c{D}_T^{\beta }\varpi (t))+{(}_t^c{D}_T^{\beta }v(t)){(}_0^c{D}_t^{\beta }\varpi (t))]\mathrm{d}t+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}K(t)v(t)\varpi (t)\mathrm{d}t\hfill \\ & +{\displaystyle \frac{b}{a}}h(u(0))\omega (0)+{\displaystyle \frac{d}{c}}h(u(T))\omega (T)+{\displaystyle \frac{b}{a}}h(v(0))\varpi (0)+{\displaystyle \frac{d}{c}}h(v(T))\varpi (T)\hfill \\ & +{\displaystyle \sum _{i=1}^m}{I}_i(u(t_i))\omega (t_i)+{\displaystyle \sum _{i=1}^m}{J}_i(v(t_i))\varpi (t_i)-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[(F_{iu}(t,u(t),v(t))\omega (t)+\hfill \\ & F_{iv}(t,u(t),v(t)))\varpi (t)]\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[(G_{iu}(t,u(t),v(t))\omega (t)+G_{iv}(t,u(t),v(t)))\varpi (t)]\mathrm{d}t.\hfill \end{array}$$

According to the existing knowledge, if $(u,v)\in X$ is a solution of ${\mathsf{\Phi }}^{\prime }(u,v)(\omega ,\varpi )=0$ for all $(\omega ,\varpi )\in X$, then $(u,v)$ is a weak solution of the BVP (1).

Lemma 5

([32]). Let X be a reflexive real Banach space, $\mathsf{\Psi }:X\to \mathbb{R}$ be a continuous, coercive, sequentially weakly lower semi-continuous and Gateaux differentiable functional, and $\mathsf{{\rm Y}}:X\to \mathbb{R}$ be a sequentially weakly upper semi-continuous and Gateaux differentiable functional. For all $r>{inf}_X\mathsf{\Psi }$, put

$$\phi (r)=\underset{(x,y)\in {\mathsf{\Psi }}^{-1}((-\infty ,r))}{inf}{\displaystyle \frac{\left({sup}_{(x,y)\in ({\mathsf{\Psi }}^{-1}(-\infty ,r))}\mathsf{{\rm Y}}(x,y)\right)-\mathsf{{\rm Y}}(x,y)}{r-\mathsf{\Psi }(x,y)}}$$

and

$${\phi }^{+}=\underset{r\to \infty }{lim\; inf}\phi (r),{\phi }^{-}=\underset{r\to {({inf}_X\mathsf{\Psi })}^{+}}{lim\; inf}\phi (r),$$

then, the following conclusions hold:

(1)

If ${\phi }^{+}<\infty$, then for each $\lambda \in (0,{\displaystyle \frac{1}{{\phi }^{+}}})$, the following alternative holds: either the functional $\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$ has a global minimum, or there exists a sequence ${\left\{(x_n,y_n)\right\}}_{n\in \mathbb{N}}$ of critical points (local minima) of $\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$ such that ${lim}_{n\to \infty }\mathsf{\Psi }(x_n,y_n)=\infty$.

(2)

If ${\phi }^{-}<\infty$, then for each $\lambda \in (0,{\displaystyle \frac{1}{{\phi }^{-}}})$, the following alternative holds: either there exists a global minimum of Ψ which is a local minimum of $\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$, or there exists a sequence ${\left\{(x_n,y_n)\right\}}_{n\in \mathbb{N}}$ of pairwise distinct critical points (local minima) of $\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$ with ${lim}_{n\to \infty }\mathsf{\Psi }(x_n,y_n)={inf}_X\mathsf{\Psi }$, which weakly converges to a global minimum of Ψ.

3. Main Results

In this section, we will present the main results of this article. Firstly, we provide the following hypotheses.

• (H₁) $h^{\prime }(x)x^2<h(x)x$. For $i=1,\cdots ,m$, $I_i^{\prime }(x)x^2<I_i(x)x$, $J_i^{\prime }(x)x^2<J_i(x)x$.
• (H₂) For $i=1,\cdots ,m$, there exist $l>0$, $0<\sigma <1$ and $\delta >0$ such that

  $$I_i(x)\ge -{l|x|}^{\sigma },J_i(x)\ge -{l|x|}^{\sigma },for|x|\le \delta .$$
• (H₃) For $i=1,\cdots ,m$, there exist $k>0$ and $0<\iota <1$, such that

  $$|I_i(x)|\le k{x}^{\iota },|J_i(x)|\le k{x}^{\iota },forx>0.$$
• (H₄) For $i=0,1,\cdots ,m$,

  $$\begin{array}{c}{F}_{ix}(t,x,y)x+F_{iy}(t,x,y)y<{\displaystyle \frac{\partial F_i(t,x,y)}{\partial x}}{x}^2+{\displaystyle \frac{\partial F_i(t,x,y)}{\partial x}}xy+{\displaystyle \frac{\partial F_i(t,x,y)}{\partial y}}xy+{\displaystyle \frac{\partial F_i(t,x,y)}{\partial y}}{y}^2,\hfill \\ G_{ix}(t,x,y)x+G_{iy}(t,x,y)y<{\displaystyle \frac{\partial G_i(t,x,y)}{\partial x}}{x}^2+{\displaystyle \frac{\partial G_i(t,x,y)}{\partial x}}xy+{\displaystyle \frac{\partial G_i(t,x,y)}{\partial y}}xy+{\displaystyle \frac{\partial G_i(t,x,y)}{\partial y}}{y}^2.\hfill \end{array}$$
• (H₅) For $i=0,1,\cdots ,m$,

  $$\begin{array}{c}\underset{|x|\to 0}{lim}{\displaystyle \frac{F_{ix}(t,x,y)}{|x|}}=0,\underset{|y|\to 0}{lim}{\displaystyle \frac{F_{iy}(t,x,y)}{|y|}}=0,\hfill \\ \underset{|x|\to 0}{lim}{\displaystyle \frac{G_{ix}(t,x,y)}{|x|}}=0,\underset{|y|\to 0}{lim}{\displaystyle \frac{G_{iy}(t,x,y)}{|y|}}=0\hfill \end{array}$$

  uniformly for $t\in (s_i,t_{i+1}]$.
• (H₆) For $i=0,1,\cdots ,m$, there exists $\varsigma >2$ such that

  $$\underset{|x|\to \infty ,|y|\to \infty }{lim}{\displaystyle \frac{F_i(t,x,y)}{{|x|}^{\varsigma }+{|y|}^{\varsigma }}}=\infty ,\underset{|x|\to \infty ,|y|\to \infty }{lim}{\displaystyle \frac{G_i(t,x,y)}{{|x|}^{\varsigma }+{|y|}^{\varsigma }}}=\infty .$$
• (H₇) For $i=0,1,\cdots ,m$, there exist $\varsigma >2$, $\varphi$, W and $Z>0$, for $t\in (s_i,t_{i+1}]$, $|x|\ge \varphi$ and $|y|\ge \varphi$, such that

  $$\begin{array}{c}{F}_{ix}(t,x(t),y(t))x(t)+F_{iy}(t,x(t),y(t))y(t)-\varsigma F_i(t,x(t),y(t))\ge -{W(|x|}^2+{|y|}^2)-Z,\hfill \\ G_{ix}(t,x(t),y(t))x(t)+G_{iy}(t,x(t),y(t))y(t)-\varsigma G_i(t,x(t),y(t))\ge -{W(|x|}^2+{|y|}^2)-Z.\hfill \end{array}$$

Remark 1.

Typically, the superquadratic Ambrosetti–Rabinowitz condition is employed to guarantee the boundedness of ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}$; however, this article substantiates it through $(H_6)$ and $(H_7)$. The Ambrosetti–Rabinowitz condition encompasses $(H_6)$ and $(H_7)$, so these assumptions are weaker than the Ambrosetti–Rabinowitz condition.

Let $\mathcal{N}=\{(u,v)\in X\setminus \left\{(0,0)\right\}:{\mathsf{\Phi }}^{\prime }(u,v)(u,v)=0\}$. Assume that $(H_1)$ and $(H_4)$ hold, if $(u,v)\in \mathcal{N}$ is the critical point of ${\mathsf{\Phi }|}_{\mathcal{N}}$, then ${\mathsf{\Phi }}^{\prime }(u,v)=0$.

Lemma 6.

Suppose $(H_1)$–$(H_6)$ hold. For each $(u,v)\in X\setminus \{(0,0)\}$, there is a unique $\varrho >0$ such that $(\varrho u,\varrho v)\in \mathcal{N}$. And $\mathsf{\Phi }(\varrho u,\varrho v)={max}_{\varrho \ge 0}\mathsf{\Phi }(\varrho u,\varrho v)>0$.

Proof. 

Focusing on $(H_5)$, we find that for $i=0,1,\cdots ,m$ and $ϵ>0$, there exists $\delta >0$, and for $|u|\le \delta$, $|v|\le \delta$, we have

$$F_i(t,u,v)\le {ϵ(|u|}^2+{|v|}^2),G_i(t,u,v)\le {ϵ(|u|}^2+{|v|}^2).$$

Then, let $\rho ={\displaystyle \frac{\delta }{\Lambda }}$, for $(u,v)\in X\setminus \{(0,0)\}$ with ${\parallel (u,v)\parallel }_X=\rho$, so there is ${\parallel (u,v)\parallel }_{\infty }\le \delta$; then

$$\begin{array}{cc}\hfill \mathsf{\Phi }(u,v)=& {\displaystyle \frac{1}{2}}{\parallel (u,v)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(u(0))+H(v(0))]+{\displaystyle \frac{d}{c}}[H(u(T))+H(v(T))]+{\displaystyle \sum _{i=1}^m}{\int }_0^{u(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{v(t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u(t),v(t))\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u(t),v(t))\mathrm{d}t\hfill \\ \hfill \ge & {\displaystyle \frac{1}{2}}{\parallel (u,v)\parallel }_X^2-({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0{\parallel (u,v)\parallel }_{\infty }-{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{l}{\sigma +1}}\left(|u(t_i){|}^{\sigma +1}+{|v(t_i)|}^{\sigma +1}\right)\hfill \\ & -(\lambda +\mu )ϵ{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{(|u(t)|}^2+{|v(t)|}^2)\mathrm{d}t\hfill \\ \hfill \ge & {\displaystyle \frac{1}{2}}{\parallel (u,v)\parallel }_X^2-({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0\Lambda \parallel (u,v){\parallel }_X-{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{l{\Lambda }^{\sigma +1}}{\sigma +1}}\parallel (u,v){\parallel }_X^{\sigma +1}-(\lambda +\mu )ϵT{\Lambda }^2{\parallel (u,v)\parallel }_X^2.\hfill \end{array}$$

Since $1<\sigma +1<2$, we prove that $\mathsf{\Phi }(u,v)>0$ provided $ϵ<{\displaystyle \frac{1}{2(\lambda +\mu )T{\Lambda }^2}}$ and $\rho$ is large enough. That is, there are $\rho >0$ and $\overline{\sigma }>0$, such that $\mathsf{\Phi }(u,v)\ge \overline{\sigma }$ for ${\parallel (u,v)\parallel }_X=\rho$. Furthermore, when ${0<\parallel (u,v)\parallel }_X<\rho$, there is $\mathsf{\Phi }(u,v)>0$.

Next, according to $(H_6)$, for $i=0,1,\cdots ,m$ and $(t,u,v)\in (s_i,t_{i+1}]\times {\mathbb{R}}^2$, there exist $C_1>0$ and $C_2>0$ such that

$$F_i(t,u,v)\ge C_1{(|u|}^{\varsigma }+{|v|}^{\varsigma })-C_2,G_i(t,u,v)\ge C_1{(|u|}^{\varsigma }+{|v|}^{\varsigma })-C_2.$$

Hence, we have

$$\begin{array}{cc}\hfill \mathsf{\Phi }(\zeta u,\zeta v)=& {\displaystyle \frac{{\zeta }^2}{2}}{\parallel (u,v)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(\zeta u(0))+H(\zeta v(0))]+{\displaystyle \frac{d}{c}}[H(\zeta u(T))+H(\zeta v(T))]+{\displaystyle \sum _{i=1}^m}{\int }_0^{\zeta u(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{\zeta v(t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,\zeta u(t),\zeta v(t))\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,\zeta u(t),\zeta v(t))\mathrm{d}t\hfill \\ \hfill \le & {\displaystyle \frac{{\zeta }^2}{2}}{\parallel (u,v)\parallel }_X^2+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0\zeta \parallel (u,v){\parallel }_{\infty }+{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k{\zeta }^{\iota +1}}{\iota +1}}(|u(t_i){|}^{\iota +1}+|v(t_i){|}^{\iota +1})\hfill \\ & -(\lambda +\mu ){\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[C_1{(|\zeta u(t)|}^{\varsigma }+{|\zeta v(t)|}^{\varsigma })-C_2]\mathrm{d}t\hfill \\ \hfill \le & -(\lambda +\mu ){\zeta }^{\varsigma }{C}_1{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{(|u(t)|}^{\varsigma }+{|v(t)|}^{\varsigma })\mathrm{d}t+{\displaystyle \frac{{\zeta }^2}{2}}{\parallel (u,v)\parallel }_X^2+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0\Lambda \zeta {\parallel (u,v)\parallel }_X\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k_i{(\Lambda \zeta )}^{\iota +1}}{\iota +1}}{\parallel (u,v)\parallel }_X^{\iota +1}+(\lambda +\mu )C_{2}T.\hfill \end{array}$$

By $\varsigma >2$ and $1<\iota +1<2$, we get $\mathsf{\Phi }(\zeta u,\zeta v)\to -\infty$ as $\zeta \to \infty$.

For $\varrho >0$, let $g_{u,v}(\varrho )=\mathsf{\Phi }(\varrho u,\varrho v)$. The above proves that $g_{u,v}$ has at least one maximum point $\varrho$, and the following shows that $\varrho$ is unique. Suppose that $\varrho$ is the critical point of $g_{u,v}$; then

$$\begin{array}{cc}\hfill g_{u,v}^{\prime }(\varrho )={\mathsf{\Phi }}^{\prime }(\varrho u,\varrho v)(u,v)=& {\varrho \parallel (u,v)\parallel }_X^2+{\displaystyle \frac{b}{a}}[h(\varrho u(0))u(0)+h(\varrho v(0))v(0)]+{\displaystyle \frac{d}{c}}[h(\varrho u(T))u(T)+h(\varrho v(T))v(T)]\hfill \\ & +{\displaystyle \sum _{i=1}^m}{I}_i(\varrho u(t_i))u(t_i)+{\displaystyle \sum _{i=1}^m}{J}_i(\varrho v(t_i))v(t_i)-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[F_{iu}(t,\varrho u(t),\varrho v(t))u(t)\hfill \\ & +F_{iv}(t,\varrho u(t),\varrho v(t))v(t)]\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[G_{iu}(t,\varrho u(t),\varrho v(t))u(t)\hfill \\ & +G_{iv}(t,\varrho u(t),\varrho v(t))v(t)]\mathrm{d}t=0.\hfill \end{array}$$

By simple calculation, from $(H_1)$ and $(H_4)$, we get

$$\begin{array}{cc}\hfill g_{u,v}^{″}(\varrho )=& {\displaystyle \frac{b}{a{\varrho }^2}}[h^{\prime }(\varrho u(0)){(\varrho u(0))}^2-h(\varrho u(0))\varrho u(0)+h^{\prime }(\varrho v(0)){(\varrho v(0))}^2-h(\varrho v(0))\varrho v(0)]\hfill \\ & +{\displaystyle \frac{d}{c{\varrho }^2}}[h^{\prime }(\varrho u(T)){(\varrho u(T))}^2-h(\varrho u(T))\varrho u(T)+h^{\prime }(\varrho v(T)){(\varrho v(T))}^2-h(\varrho v(T))\varrho v(T)]\hfill \\ & +{\displaystyle \frac{1}{{\varrho }^2}}{\displaystyle \sum _{i=1}^m}[I_i^{\prime }(\varrho u(t_i)){(\varrho u(t_i))}^2-I_i(\varrho u(t_i))\varrho u(t_i)]+{\displaystyle \frac{1}{{\varrho }^2}}{\displaystyle \sum _{i=1}^m}[J_i^{\prime }(\varrho v(t_i)){(\varrho v(t_i))}^2-J_i(\varrho v(t_i))\varrho v(t_i)]\hfill \\ & -{\displaystyle \frac{\lambda }{{\varrho }^2}}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[{\displaystyle \frac{\partial F_{iu}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho u)}}{(\varrho u(t))}^2+{\displaystyle \frac{\partial F_{iu}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho v)}}{\varrho }^{2}u(t)v(t)\hfill \\ & +{\displaystyle \frac{\partial F_{iv}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho u)}}{\varrho }^{2}u(t)v(t)+{\displaystyle \frac{\partial F_{iv}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho v)}}{(\varrho v(t))}^2\hfill \\ & -F_{iu}(t,\varrho u(t),\varrho v(t))\varrho u(t)-F_{iv}(t,\varrho u(t),\varrho v(t))\varrho v(t)]\mathrm{d}t\hfill \\ & -{\displaystyle \frac{\mu }{{\varrho }^2}}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[{\displaystyle \frac{\partial G_{iu}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho u)}}{(\varrho u(t))}^2+{\displaystyle \frac{\partial G_{iu}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho v)}}{\varrho }^{2}u(t)v(t)\hfill \\ & +{\displaystyle \frac{\partial G_{iv}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho u)}}{\varrho }^{2}u(t)v(t)+{\displaystyle \frac{\partial G_{iv}(t,\varrho u(t),\varrho v(t))}{\partial (\varrho v)}}{(\varrho v(t))}^2\hfill \\ & -G_{iu}(t,\varrho u(t),\varrho v(t))\varrho u(t)-G_{iv}(t,\varrho u(t),\varrho v(t))\varrho v(t)]\mathrm{d}t\hfill \\ \hfill <& 0,\hfill \end{array}$$

which means that if $\varrho$ is a critical point of $g_{u,v}$, then $\varrho$ is a strictly local maximum of $g_{u,v}$ and the critical point is unique. □

Lemma 7.

Suppose $(H_1)$–$(H_7)$ hold. Then there is $(u,v)\in \mathcal{N}$ such that $\mathsf{\Phi }(u,v)=\overline{k}={inf}_{\mathcal{N}}\mathsf{\Phi }\ge {inf}_{\partial B_{\rho }}\mathsf{\Phi }\ge \overline{\sigma }>0$.

Proof. 

According to Lemma 4 and the Control Convergence Theorem, $\mathsf{\Phi }$ and ${\mathsf{\Phi }}^{\prime }$ are weakly lower semi-continuous. Next, we show that ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}$ is bounded in X. Let ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}\subset \mathcal{N}$ be the minimization sequence of $\mathsf{\Phi }$; then $\mathsf{\Phi }(u_n,v_n)=\overline{k}+o(1)$, ${\mathsf{\Phi }}^{\prime }(u_n,v_n)(u_n,v_n)=0$.

Assuming that $\parallel (u_n,v_n){\parallel }_X\to \infty$ as $n\to \infty$, let $(U_n,V_n)={\displaystyle \frac{(u_n,v_n)}{\parallel (u_n,v_n){\parallel }_X}}$, so that $\parallel (U_n,V_n){\parallel }_X=1$. By Lemma 4, passing to a subsequence, we can suppose that $(U_n,V_n)\rightharpoonup (U,V)$ in X; then, $(U_n,V_n)\to (U,V)$ in $C([0,T],{\mathbb{R}}^2)$. Furthermore,

$$\begin{array}{cc}& \lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u_n(t),v_n(t))\mathrm{d}t+\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u_n(t),v_n(t))\mathrm{d}t\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel (u_n,v_n)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(u_n(0))+H(v_n(0))]+{\displaystyle \frac{d}{c}}[H(u_n(T))+H(v_n(T))]+{\displaystyle \sum _{i=1}^m}{\int }_0^{u_n(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{v_n(t_i)}{J}_i(s)\mathrm{d}s-\mathsf{\Phi }(u_n,v_n)\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}\parallel (u_n,v_n){\parallel }_X^2+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0\Lambda \parallel (u_n,v_n){\parallel }_X+{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k{\Lambda }^{\iota +1}}{\iota +1}}{\parallel (u_n,v_n)\parallel }_X^{\iota +1}+C_3,\hfill \end{array}$$

where $C_3>0$, which implies that

$$\begin{array}{cc}& \lambda {\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{F}_i(t,u_n(t),v_n(t))\mathrm{d}t}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}+\mu {\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{G}_i(t,u_n(t),v_n(t))\mathrm{d}t}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\parallel (u_n,v_n){\parallel }_X^2}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0\Lambda {\displaystyle \frac{\parallel (u_n,v_n){\parallel }_X}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}+{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k{\Lambda }^{\iota +1}}{\iota +1}}{\displaystyle \frac{\parallel (u_n,v_n){\parallel }_X^{\iota +1}}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}+{\displaystyle \frac{C_3}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}\hfill \\ \hfill \to & 0\hfill \end{array}$$

(2)

as $n\to \infty$. Turning our attention to the continuities of $F_{iu}$, $F_{iv}$, $G_{iu}$, $G_{iv}$ $(i=0,1,\cdots ,m)$, combine $(H_7)$ and we obtain for $i=0,1,\cdots ,m$, $t\in (s_i,t_{i+1}]$ and $(u,v)\in {\mathbb{R}}^2$,

$$\begin{array}{c}{F}_{iu}(t,u,v)u+F_{iv}(t,u,v)v-\varsigma F_i(t,u,v)\ge -W(u^2+v^2)-Z,\hfill \\ G_{iu}(t,u,v)u+G_{iv}(t,u,v)v-\varsigma G_i(t,u,v)\ge -W(u^2+v^2)-Z.\hfill \end{array}$$

Further, it can be concluded that

$$\begin{array}{cc}\hfill \overline{k}+o(1)=& \mathsf{\Phi }(u_n,v_n)={\displaystyle \frac{1}{2}}{\parallel (u_n,v_n)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(u_n(0))+H(v_n(0))]+{\displaystyle \frac{d}{c}}[H(u_n(T))+H(v_n(T))]\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{u_n(t_i)}{I}_i(s)\mathrm{d}s+{\displaystyle \sum _{i=1}^m}{\int }_0^{v_n(t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u_n(t),v_n(t))\mathrm{d}t\hfill \\ & -\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u_n(t),v_n(t))\mathrm{d}t\hfill \\ \hfill \ge & {\displaystyle \frac{1}{2}}{\parallel (u_n,v_n)\parallel }_X^2+{\displaystyle \frac{b}{a}}[H(u_n(0))+H(v_n(0))]+{\displaystyle \frac{d}{c}}[H(u_n(T))+H(v_n(T))]+{\displaystyle \sum _{i=1}^m}{\int }_0^{u_n(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{v_n(t_i)}{J}_i(s)\mathrm{d}s-{\displaystyle \frac{(\lambda +\mu )}{\varsigma }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}W(|u_n{(t)|}^2+|v_n(t){|}^2)\mathrm{d}t-{\displaystyle \frac{\lambda +\mu }{\varsigma }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}Z\mathrm{d}t\hfill \\ & -{\displaystyle \frac{\lambda }{\varsigma }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[F_{i{u}_n}(t,u_n(t),v_n(t))u_n(t)+F_{i{v}_n}(t,u_n(t),v_n(t))v_n(t)]\mathrm{d}t\hfill \\ & -{\displaystyle \frac{\mu }{\varsigma }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}[G_{i{u}_n}(t,u_n(t),v_n(t))u_n(t)+G_{i{v}_n}(t,u_n(t),v_n(t))v_n(t)]\mathrm{d}t\hfill \\ \hfill \ge & ({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\varsigma }})\parallel (u_n,v_n){\parallel }_X^2+{\displaystyle \frac{1}{\varsigma }}{\mathsf{\Phi }}^{\prime }(u_n,v_n)(u_n,v_n)-({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0(|u_n|+|v_n|)-({\displaystyle \frac{b}{\varsigma a}}+{\displaystyle \frac{d}{\varsigma c}})h_0(|u_n|+|v_n|)\hfill \\ & -{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k}{\iota +1}}(|u_n{|}^{\iota +1}+|v_n{|}^{\iota +1})-{\displaystyle \frac{1}{\varsigma }}{\displaystyle \sum _{i=1}^m}k(|u_n{|}^{\iota +1}+|v_n{|}^{\iota +1})-{\displaystyle \frac{(\lambda +\mu )WT}{\varsigma }}{\parallel (u_n,v_n)\parallel }_{\infty }^2\hfill \\ & -{\displaystyle \frac{(\lambda +\mu )ZT}{\varsigma }},\hfill \end{array}$$

which means that there exists $C_4>0$ such that

$$\underset{n\to \infty }{lim}{\parallel (U_n,V_n)\parallel }_{\infty }^2=\underset{n\to \infty }{lim}{\displaystyle \frac{\parallel (u_n,v_n){\parallel }_{\infty }^2}{\parallel (u_n,v_n){\parallel }_X^2}}\ge C_4>0.$$

Then we know that $(U,V)\ne (0,0)$. Let

$${\mathsf{\Omega }}_1=\{t\in (s_i,t_{i+1}]:(U,V)\ne (0,0)\},{\mathsf{\Omega }}_2=(s_i,t_{i+1}]\setminus {\mathsf{\Omega }}_1,i=0,1,\cdots ,m.$$

Based on $(H_5)$ and $(H_6)$, for $i=0,1,\cdots ,m$, $t\in (s_i,t_{i+1}]$ and $(u,v)\in {\mathbb{R}}^2$, there exist $\overline{W}>0$ and $\overline{Z}>0$, such that

$$F_i(t,u,v)\ge -\overline{W}(u^2+v^2)-\overline{Z},G_i(t,u,v)\ge -\overline{W}(u^2+v^2)-\overline{Z}.$$

Combining $(H_6)$ and Fatou’s Lemma, for $t\in (s_i,t_{i+1}]$, we have

$$\begin{array}{cc}& \underset{n\to \infty }{lim\; inf}{\int }_{s_i}^{t_{i+1}}\left[{\displaystyle \frac{F_i(t,u_n(t),v_n(t))}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}+{\displaystyle \frac{G_i(t,u_n(t),v_n(t))}{\parallel (u_n,v_n){\parallel }_X^{\varsigma }}}\right]\mathrm{d}t\hfill \\ \hfill =& \underset{n\to \infty }{lim\; inf}[{\int }_{{\mathsf{\Omega }}_1}{\displaystyle \frac{F_i(t,u_n(t),v_n(t))}{|(u_n(t),v_n(t)){|}^{\varsigma }}}|(U_n(t),V_n(t)){|}^{\varsigma }\mathrm{d}t+{\int }_{{\mathsf{\Omega }}_1}{\displaystyle \frac{G_i(t,u_n(t),v_n(t))}{|(u_n(t),v_n(t)){|}^{\varsigma }}}{|(U_n(t),V_n(t))|}^{\varsigma }\mathrm{d}t\hfill \\ & +{\int }_{{\mathsf{\Omega }}_2}{\displaystyle \frac{F_i(t,u_n(t),v_n(t))}{|(u_n(t),v_n(t)){|}^{\varsigma }}}|(U_n(t),V_n(t)){|}^{\varsigma }\mathrm{d}t+{\int }_{{\mathsf{\Omega }}_2}{\displaystyle \frac{G_i(t,u_n(t),v_n(t))}{|(u_n(t),v_n(t)){|}^{\varsigma }}}{|(U_n(t),V_n(t))|}^{\varsigma }\mathrm{d}t]\hfill \\ \hfill =& \infty ,\hfill \end{array}$$

which is a contradiction with (2). In other words, ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}$ is bounded in X.

From $(u_n,v_n)\rightharpoonup (u,v)$ in X, it can be inferred that $(u_n,v_n)\to (u,v)$ in $C([0,T]\times {\mathbb{R}}^2,\mathbb{R})$. And further, we get $\parallel (u_n,v_n)-(u,v){\parallel }_X^2\to 0$ as $n\to \infty$, then $(u_n,v_n)\to (u,v)$ in X.

For ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}\subset \mathcal{N}$, one has

$${\mathsf{\Phi }}^{\prime }(u,v)(u,v)\le {\underline{lim}}_{n\to \infty }{\mathsf{\Phi }}^{\prime }(u_n,v_n)(u_n,v_n)=0.$$

We claim that $(u,v)\ne (0,0)$. If not, suppose $(u,v)=(0,0)$, and then $(u_n,v_n)\to (0,0)$ in $C([0,T]\times {\mathbb{R}}^2,\mathbb{R})$. And because ${\mathsf{\Phi }}^{\prime }(u_n,v_n)=0$, this means $\parallel (u_n,v_n){\parallel }_X\to 0$. This contradicts ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}\subset \mathcal{N}$. Based on the above discussion, we see that there exists a unique $\varrho >0$, so that $(\varrho u,\varrho v)\in \mathcal{N}$ and

$$\overline{k}\le \mathsf{\Phi }(\varrho u,\varrho v)\le {\underline{lim}}_{n\to \infty }\mathsf{\Phi }(\varrho u_n,\varrho v_n)\le \underset{n\to \infty }{lim}\mathsf{\Phi }(\varrho u_n,\varrho v_n).$$

On the other hand, for all $(u_n,v_n)\in \mathcal{N}$, $g_{u_n,v_n}^{″}(\varrho )<0$ and $\varrho =1$ is the global maximum of $g_{u_n,v_n}$, then $\mathsf{\Phi }(\varrho u_n,\varrho v_n)\le \mathsf{\Phi }(u_n,v_n)$, and

$$\overline{k}\le \mathsf{\Phi }(\varrho u,\varrho v)\le \underset{n\to \infty }{lim}\mathsf{\Phi }(u_n,v_n)=\overline{k}.$$

Thus, there is $(\varrho u,\varrho v)\in \mathcal{N}$ such that $\mathsf{\Phi }(\varrho u,\varrho v)=\overline{k}$. □

Theorem 1.

Suppose $(H_1)$–$(H_7)$ hold. Then the BVP (1) admits at least one nontrivial ground state solution.

Proof. 

Lemma 7 implies that $(u,v)$ is the nonzero critical point of ${\mathsf{\Phi }|}_{\mathcal{N}}$, and we know that ${\mathsf{\Phi }}^{\prime }(u,v)=0$, then $(u,v)$ is the nontrivial ground state solution of BVP (1). □

Now, we introduce several notations. Let $\mathsf{\Psi }(u,v)={\displaystyle \frac{1}{2}}{\parallel (u,v)\parallel }_X^2$ and

$$\begin{array}{cc}\hfill \mathsf{{\rm Y}}(u,v)=& -{\displaystyle \frac{b}{\lambda a}}[H(u(0))+H(v(0))]-{\displaystyle \frac{d}{\lambda c}}[H(u(T))+H(v(T))]-{\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{i=1}^m}{\int }_0^{u(t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & -{\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{i=1}^m}{\int }_0^{v(t_i)}{J}_i(s)\mathrm{d}s+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,u(t),v(t))\mathrm{d}t+{\displaystyle \frac{\mu }{\lambda }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,u(t),v(t))\mathrm{d}t.\hfill \end{array}$$

For $(t,u,v)\in (s_i,t_{i+1}]\times {\mathbb{R}}^2$ $(i=0,1,\cdots ,m)$, define

$$\begin{array}{c}A=\underset{|\xi |\to \infty ,|\overline{\xi }|\to \infty }{lim\; inf}{\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{max}_{|u|\le \xi ,|v|\le \overline{\xi }}(F_i(t,u,v)+\frac{\mu }{\lambda }{G}_i(t,u,v))\mathrm{d}t}{{\xi }^2+{\overline{\xi }}^2}},\hfill \\ B=\underset{|\xi |\to \infty ,|\overline{\xi }|\to \infty }{lim\; sup}{\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}(F_i(t,\xi ,\overline{\xi })+\frac{\mu }{\lambda }{G}_i(t,\xi ,\overline{\xi }))\mathrm{d}t}{{\xi }^2+{\overline{\xi }}^2}},\hfill \end{array}$$

and let ${\lambda }_1={\displaystyle \frac{K_{2}T}{2B}}$, ${\lambda }_2={\displaystyle \frac{K_2}{2A{\Lambda }^2}}$. Then, we obtain the following result.

Theorem 2.

Assume that $(H_3)$ hold and $A<{\displaystyle \frac{B}{T{\Lambda }^2}}$. Then for $\lambda \in ({\lambda }_1,{\lambda }_2)$, the BVP (1) possesses an unbounded sequence of solutions in X.

Proof. 

Take $\lambda \in ({\lambda }_1,{\lambda }_2)$, and let ${\left\{(a_n,b_n)\right\}}_{n\in \mathbb{N}}$ be a real sequence such that ${lim}_{n\to \infty }{a}_n=\infty$, ${lim}_{n\to \infty }{b}_n=\infty$; then we have

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{max}_{|u|\le a_n,|v|\le b_n}[F_i(t,u,v)+\frac{\mu }{\lambda }{G}_i(t,u,v)]\mathrm{d}t}{a_n^2+b_n^2}}=A.$$

(3)

Let $r_n={\displaystyle \frac{a_n^2+b_n^2}{2{\Lambda }^2}}$ for all $n\in \mathbb{N}$. For all $(\omega ,\varpi )\in X$ with ${\parallel (\omega ,\varpi )\parallel }_X^2\le 2r_n$, one has

$$r_n\ge \mathsf{\Psi }(\omega ,\varpi )={\displaystyle \frac{1}{2}}{\parallel (\omega ,\varpi )\parallel }_X^2\ge {\displaystyle \frac{1}{2{\Lambda }^2}}{\parallel (\omega ,\varpi )\parallel }_{\infty }^2,$$

which means that ${\parallel (\omega ,\varpi )\parallel }_{\infty }<\Lambda {(2r_n)}^{{\displaystyle \frac{1}{2}}}={(a_n^2+b_n^2)}^{{\displaystyle \frac{1}{2}}}$. Following from $(H_3)$ and $F_i(t,0,0)=G_i(t,0,0)=0(i=0,1,\cdots ,m)$, we get

$$\begin{array}{cc}\hfill \phi (r_n)=& \underset{\mathsf{\Psi }(u,v)<r_n}{inf}{\displaystyle \frac{({sup}_{\mathsf{\Psi }(\omega ,\varpi )<r_n}\mathsf{{\rm Y}}(\omega ,\varpi ))-\mathsf{{\rm Y}}(u,v)}{r_n-\mathsf{\Psi }(u,v)}}\le {\displaystyle \frac{({sup}_{\mathsf{\Psi }(\omega ,\varpi )<r_n}\mathsf{{\rm Y}}(\omega ,\varpi ))-\mathsf{{\rm Y}}(0,0)}{r_n}}\hfill \\ \hfill \le & {\displaystyle \frac{{sup}_{{\parallel (\omega ,\varpi )\parallel }_X^2\le 2r_n}\mathsf{{\rm Y}}(\omega ,\varpi )}{r_n}}\hfill \\ \hfill \le & {\displaystyle \frac{1}{r_n}}\underset{{\parallel (\omega ,\varpi )\parallel }_{\infty }<{(a_n^2+b_n^2)}^{{\displaystyle \frac{1}{2}}}}{max}[-{\displaystyle \frac{b}{\lambda a}}(H(\omega (0))+H(\varpi (0)))-{\displaystyle \frac{d}{\lambda c}}(H(\omega (T))+H(\varpi (T)))\hfill \\ & -{\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{i=1}^m}{\int }_0^{\omega (t_i)}{I}_i(s)\mathrm{d}s-{\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{i=1}^m}{\int }_0^{\varpi (t_i)}{J}_i(s)\mathrm{d}s+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,\omega (t),\varpi (t))\mathrm{d}t\hfill \\ & +{\displaystyle \frac{\mu }{\lambda }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,\omega (t),\varpi (t)))\mathrm{d}t]\hfill \\ \hfill \le & {\displaystyle \frac{2{\Lambda }^2}{a_n^2+b_n^2}}[({\displaystyle \frac{b}{\lambda a}}+{\displaystyle \frac{d}{\lambda c}})h_0{(a_n^2+b_n^2)}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{1}{\lambda }}{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k}{\iota +1}}{(a_n^2+b_n^2)}^{{\displaystyle \frac{\iota +1}{2}}}+{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}\underset{|\omega (t)|\le a_n,|\varpi (t)|\le b_n}{max}\hfill \\ & \left(F_i(t,\omega (t),\varpi (t))+{\displaystyle \frac{\mu }{\lambda }}{G}_i(t,\omega (t),\varpi (t))\right)\mathrm{d}t].\hfill \end{array}$$

By (3) and $1\le \iota +1<2$, we have

$$\begin{array}{cc}\hfill {\phi }^{+}\le \underset{n\to \infty }{lim\; inf}\phi (r_n)\le & {\displaystyle \frac{2{\Lambda }^2}{a_n^2+b_n^2}}\left[{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}\underset{|\omega (t)|\le a_n,|\varpi (t)|\le b_n}{max}\left(F_i(t,\omega (t),\varpi (t))+{\displaystyle \frac{\mu }{\lambda }}{G}_i(t,\omega (t),\varpi (t))\right)\mathrm{d}t\right]\hfill \\ \hfill \le & 2A{\Lambda }^2<\infty .\hfill \end{array}$$

Now, we claim that $\mathsf{\Phi }=\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$ is unbounded from below for $\lambda \in ({\lambda }_1,{\lambda }_2)=\left({\displaystyle \frac{K_{2}T}{2B}},{\displaystyle \frac{K_2}{2A{\Lambda }^2}}\right)\subset \left(0,{\displaystyle \frac{1}{{\phi }^{+}}}\right)$.

Let ${\left\{(c_n,d_n)\right\}}_{n\in \mathbb{N}}$ be a real sequence such that ${lim}_{n\to \infty }{c}_n=\infty$ and ${lim}_{n\to \infty }{d}_n=\infty$, then

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{F}_i(t,c_n,d_n)\mathrm{d}t+\frac{\mu }{\lambda }{\sum }_{i=0}^m{\int }_{s_i}^{t_{i+1}}{G}_i(t,c_n,d_n)\mathrm{d}t}{c_n^2+d_n^2}}=B.$$

(4)

For all $n\in \mathbb{N}$, define $({\omega }_n(t),{\varpi }_n(t))=(c_n,d_n)$ for $t\in [0,T]$. Therefore, $({\omega }_n,{\varpi }_n)\in X$ for $n\in \mathbb{N}$ and $\parallel ({\omega }_n,{\varpi }_n){\parallel }_X^2\le K_{2}T(c_n^2+d_n^2)$. Then,

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\omega }_n,{\varpi }_n)=& \mathsf{\Psi }({\omega }_n,{\varpi }_n)-\lambda \mathsf{{\rm Y}}({\omega }_n,{\varpi }_n)\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{\parallel ({\omega }_n,{\varpi }_n)\parallel }_X^2+{\displaystyle \frac{b}{a}}\left(H(\omega (0))+H(\varpi (0))\right)+{\displaystyle \frac{d}{c}}\left(H(\omega (T))+H(\varpi (T))\right)+{\displaystyle \sum _{i=1}^m}{\int }_0^{\omega (t_i)}{I}_i(s)\mathrm{d}s\hfill \\ & +{\displaystyle \sum _{i=1}^m}{\int }_0^{\varpi (t_i)}{J}_i(s)\mathrm{d}s-\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,\omega (t),\varpi (t))\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,\omega (t),\varpi (t))\mathrm{d}t\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{K}_{2}T(c_n^2+d_n^2)+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0(|c_n|+|d_n|)+{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k}{\iota +1}}\left(|c_n{|}^{\iota +1}+{|d_n|}^{\iota +1}\right)\hfill \\ & -\lambda {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,c_n,d_n)\mathrm{d}t-\mu {\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,c_n,d_n)\mathrm{d}t.\hfill \end{array}$$

Now, if $B<\infty$, from (4), for any $M\in (0,B-{\displaystyle \frac{T{K}_2}{2\lambda }})$, there exists $N_M\in \mathbb{N}$, for all $n>N_M$, such that

$${\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,c_n,d_n)\mathrm{d}t+{\displaystyle \frac{\mu }{\lambda }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,c_n,d_n)\mathrm{d}t>(B-M)(c_n^2+d_n^2).$$

So

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\omega }_n,{\varpi }_n)=& \mathsf{\Psi }({\omega }_n,{\varpi }_n)-\lambda \mathsf{{\rm Y}}({\omega }_n,{\varpi }_n)\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{K}_{2}T(c_n^2+d_n^2)+({\displaystyle \frac{b}{a}}+{\displaystyle \frac{d}{c}})h_0(|c_n|+|d_n|)+{\displaystyle \sum _{i=1}^m}{\displaystyle \frac{k}{\iota +1}}\left(|c_n{|}^{\iota +1}+{|d_n|}^{\iota +1}\right)\hfill \\ & -\lambda (B-M)(c_n^2+d_n^2)\to -\infty asn\to \infty .\hfill \end{array}$$

If $B=\infty$, fix $\overline{M}>{\displaystyle \frac{T{K}_2}{2\lambda }}$. From (4) there exists $N_{\overline{M}}\in \mathbb{N}$, for all $n>N_{\overline{M}}$, such that

$${\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{F}_i(t,c_n,d_n)\mathrm{d}t+{\displaystyle \frac{\mu }{\lambda }}{\displaystyle \sum _{i=0}^m}{\int }_{s_i}^{t_{i+1}}{G}_i(t,c_n,d_n)\mathrm{d}t>\overline{M}(c_n^2+d_n^2).$$

Then we also have $\mathsf{\Phi }({\omega }_n,{\varpi }_n)\to -\infty$ as $n\to \infty$.

It is not difficult to say that $\mathsf{\Psi }$ is a continuous, coercive, sequentially weakly lower semi-continuous and Gâteaux differentiable functional, and $\mathsf{{\rm Y}}$ is a sequentially weakly upper semi-continuous and Gâteaux differentiable functional.

Based on conclusion $(1)$ of Lemma 5, it can be concluded that the functional $\mathsf{\Phi }=\mathsf{\Psi }-\lambda \mathsf{{\rm Y}}$ admits a sequence ${\left\{(u_n,v_n)\right\}}_{n\in \mathbb{N}}$ of critical points; that is, the BVP (1) admits an unbounded sequence of solutions in X. □

4. Example

Example 1.

Let $\theta ={\displaystyle \frac{1}{2}}$, $\eta ={\displaystyle \frac{1}{3}}$, $T=1$, $m=1$, $\lambda ={\displaystyle \frac{1}{2}}$, $\mu =a=2$, $b=c=1$ and $d=3$. Consider the following fractional boundary value problem

$$\left\{\begin{array}{c}-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t)))+K(t)u(t)={\displaystyle \frac{1}{2}}{F}_{iu}(t,u(t),v(t))\hfill \\ +2G_{iu}(t,u(t),v(t)),t\in (s_i,t_{i+1}],i=0,1,\hfill \\ -{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t)))+K(t)v(t)={\displaystyle \frac{1}{2}}{F}_{iv}(t,u(t),v(t))\hfill \\ +2G_{iv}(t,u(t),v(t)),t\in (s_i,t_{i+1}],i=0,1,\hfill \\ \Delta \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t_1))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t_1))\right)=I_1(u(t_1)),\hfill \\ \Delta \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t_1))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t_1))\right)=J_1(v(t_1)),\hfill \\ {}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t))+{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t))={}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t_1^{+}))+{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(t_1^{+})),t\in (t_1,s_1],\hfill \\ {}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(s_1^{-}))+{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(s_1^{-}))={}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(s_1^{+}))+{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(s_1^{+})),\hfill \\ {}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t))+{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t))={}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t_1^{+}))+{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(t_1^{+})),t\in (t_1,s_1],\hfill \\ {}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(s_1^{-}))+{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(s_1^{-}))={}_{0}D_t^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(s_1^{+}))+{}_{t}D_1^{-{\displaystyle \frac{1}{3}}}(v^{\prime }(s_1^{+})),\hfill \\ 2\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(0))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(0))\right)-h(u(0))=0,\hfill \\ \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(1))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(u^{\prime }(1))\right)+3h(u(1))=0,\hfill \\ 2\left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(v^{\prime }(0))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(v^{\prime }(0))\right)-h(v(0))=0,\hfill \\ \left({\displaystyle \frac{1}{2}}{}_{0}D_t^{-{\displaystyle \frac{1}{2}}}(v^{\prime }(1))+{\displaystyle \frac{1}{2}}{}_{t}D_1^{-{\displaystyle \frac{1}{2}}}(v^{\prime }(1))\right)+3h(v(1))=0,\hfill \end{array}\right.$$

(5)

where $0=s_0<t_1={\displaystyle \frac{1}{3}}<s_1={\displaystyle \frac{2}{3}}<t_2=1$. Choose $K(t)=ln(2+t^2)$, then, for $t\in [0,1]$, we have $0<K_1=ln2\le K(t)\le K_2=ln3$. Let $h(x)=arctanx+1$, $I_1(x)={|x|}^{{\displaystyle \frac{1}{3}}}$, $J_1(x)=-{|x|}^{{\displaystyle \frac{1}{2}}}$, there exist $l=2$, $\delta =1$, $k=3$ and $\sigma =\iota ={\displaystyle \frac{1}{2}}$, such that $(H_1)$–$(H_3)$ hold. Let $F_i(t,u,v)=2(u^4+v^4)$ and $G_i(t,u,v)=(1+t){(u^6+v^6)}^4$; then we can know that $\varsigma =3$, $\varphi =W=2$ and $Z=1$ make $(H_4)$ and $(H_5)$ true. Therefore, all conditions of Theorem 1 are satisfied; that is, the BVP (5) admits at least one nontrivial ground state solution.

5. Conclusions

In [6,27], fractional advection–dispersion equations under Dirichlet boundary conditions and Sturm–Liouville boundary conditions were investigated, respectively. Specifically, in [6], for the nonlinear term that was a $L^2$-Caratheódory function, the authors demonstrated the existence of the solution of the equation by the Bonanno’s critical point theorem. The system of [27] enhanced the influence of instantaneous impulses. In the case of non-homogeneous Sturm–Liouville conditions, the existence of the solution of the equation was proved by using the Mountain Pass Lemma and the minimization principle. When the Sturm–Liouville conditions were homogeneous, the multiplicity of solutions to the equation was given by the symmetric Mountain Pass Lemma. Additionally, ref. [23] took into account both instantaneous and non-instantaneous impulses in the fractional p-Laplacian differential equations under Dirichlet boundary value conditions. We provided the criterion for a multiplicity of solutions of the equation via Bonanno and Bisci’s critical point theorem.

This article investigates a class of fractional advection–dispersion coupled systems under nonlinear Sturm–Liouville conditions based on the variational method. The nonlinear Sturm–Liouville condition is more general than the ordinary Sturm–Liouville condition, and when the coefficients and constant terms in the Sturm–Liouville condition are correspondingly selected as 0 or 1, the Sturm–Liouville condition will degenerate into a Dirichlet boundary value condition. It should also be pointed out that the system has both instantaneous and non-instantaneous impulses. Therefore, the system in question not only considers the characteristics of sudden, rapid, and brief instantaneous impulses, but also takes into account the characteristics of certain durations, gentle amplitude, and the regularity of non-instantaneous impulse changes. This system is applied in fields such as communication systems, radar systems, signal processing, and pharmacy. In addition, the hypotheses set in this article are weaker than the Ambrosetti–Rabinowitz conditions.

Firstly, we establish the existence of the ground state solution for the system using the Nehari manifold method. As we all know, discussions on minimal energy solutions for fractional advection–dispersion coupled systems are scarce. Subsequently, Bonanno and Bisci’s critical point theorem demonstrates that there exist infinitely many solutions to the coupled system. The advantage of this theorem lies in its independence from requiring the symmetry or satisfaction of the Palais–Smale condition by the energy functional. Overall, the issues studied in this article have research value and significance, and the results presented in this article are also new.