Nonlocal fractional $p(\cdot)$-Kirchhoff systems with variable-order: Two and three solutions

1.   Introduction and the main results

The study of the existence and multiplicity solutions for nonlocal elliptic systems boundary value problems and variational problems has attracted intense research interests for several decades. In this paper, we investigate the existence of nontrivial solutions for a class of $p(\cdot)$-Kirchhoff systems

where $M_{1}(t), M_{2}(t):[0, +\infty)\rightarrow [0, +\infty)$ are the models of Kirchhoff coefficient, $\Omega$ is a bounded smooth domain in $\mathbb R^{N}$, $\lambda, \mu$ are two real parameters. $F, G$ are differentiable and measurable functions in $\mathbb{R}^{2}$ for all $x\in \Omega$, $F_{\eta}, F_{\xi}$ are the partial derivatives of $F$, $G_{\eta}, G_{\xi}$ are the partial derivatives of $G$, whose assumptions will be introduced later.

The fractional $p(\cdot)$-Laplace operator $(-\Delta)_{p(\cdot)}^{s(\cdot)}$ with variable $s(\cdot)$-order is defined as

where $v\in C_{0}^{\infty}(\mathbb{R}^{N})$ and $P.V.$ stands for the Cauchy principal value. $(-\Delta)_{p(\cdot)}^{s(\cdot)}$ is a nonlocal operator of elliptic type, which is connected with the Sobolev space of variable exponent. Concerning this kind of operator problems, here we just list a few pieces of literatures, see ^[1,2,3,4]. Especially, Biswas et al. ^[5] firstly proved a continuous embedding result and Hardy-Littlewood-Sobolev-type result, and then the existence and multiplicity of solutions were obtained by variational approaches. When $s(\cdot) \equiv$ constant and $p(\cdot)\equiv$ constant, $(-\Delta)_{p(\cdot)}^{s(\cdot)}$ in $(1.2)$ reduce to the usual fractional Laplace operator $(-\Delta)_{p}^{s}$, see ^[6,7,8] for the essential knowledge.

Throughout this paper, $s(\cdot), p(\cdot) \in C_{+}(\mathfrak{D})$ are two continuous functions that the following assumptions are satisfied.

$\text{(S)}:$ $s(\cdot):\overline{\Omega}\times\overline{\Omega}$ $\rightarrow (0, 1) \;\text{ is symmetric, namely, }\; s(x, y)$=$s(y, x) \text{ for any }$ $(x, y)\in \overline{\Omega}\times\overline{\Omega}$ $\text{ with }\; \overline{s}(x)$=$s(x, x);$

$\text{(P)}:$ $p(\cdot):\overline{\Omega}\times\overline{\Omega} \rightarrow$ $(1, +\infty) \;\text{ is symmetric, namely, }\; p(x, y)$=$p(y, x) \text{ for any }\; (x, y)\in \overline{\Omega}$ $\times\overline{\Omega} \;\text{ with }\; \overline{p}(x)$=$p(x, x).$

Kirchhoff in ^[9] introduced the following Kirchhoff equation

where $\rho, p_{0}, h, E, L$ with physical meaning are constants. A characteristic of Eq (1.3) is the fact that it contains a nonlocal item $\frac{p_{0}}{h}+\frac{E}{2L}\int^{L}_{0}\left|\frac{\partial \xi(x)}{\partial t}\right|^{2}dx,$ and then this type of equation is called nonlocal problem. From then on, the existence, multiplicity, uniqueness, and regularity of solutions for various Kirchhoff-type equations have been studied extensively, such as, see ^{[10,11,12,13,14]} for further details.

The continuous Kirchhoff terms $M_{i}(t): R_{0}^{+}\rightarrow R^{+}, (i = 1, 2)$ are strictly increasing functions, which the following conditions are satisfied.

$\text{(M)}:$ There exist $m_{i} = m_{i}(\iota) > 0$ and $M_{i} = M_{i}(\iota) > 0, (i = 1, 2)$ for any $\iota > 0$ such that

and put

In recent years, a multitude of scholars has devoted themselves to the study of Kirchhoff-type systems. When $M_{1}(t) = 1$ and $M_{2}(t) = 1$, Chen et al. in ^[15] consider the nontrivial solutions for the following elliptic systems.

by utilizing Nehari manifold method and Fibering maps, they studied the existence of weak solutions for this kind of problem (1.4). Moreover, it has been applied in the local case $s = 1$ in ^[16].

In the famous literature ^[17], the three critical points theorem was established by Ricceri. Starting from this paper, Marano and Motreanu in ^[18] extended the result of Ricceri to non-differentiable functionals. Subsequently, Fan and Deng in ^[19] studied the version of Ricceri's result including variables exponents. Ricceri's result in ^[20] has been successfully applied to Sobolev spaces $W_{0}^{1, p}(\Omega)$, and then at least three solutions are obtained. Furthermore, Bonanno in ^[21] established the existence of two intervals of positive real parameters $\lambda$ for which the functional $\Phi-\lambda J$ has three critical points, and applied the result to obtain two critical points.

By using three critical points theorem, Azroul et al. ^[22] discussed the fractional $p$-Laplace systems with bounded domain

thus, the existence and multiplicity of solutions were obtained by Azroul et al. In addition, there are many scholars who have used different methods to study the existence of elliptic systems on bounded and unbounded regions, for instance, see ^[23,24,25] for details.

With respect to the fractional $p(\cdot)$-Laplace operators, Azroul et al. ^[26] dealt with the class of Kirchhoff type elliptic systems in nonlocal fractional Sobolev spaces with variable exponents and constant order

where

Based on the three critical points theorem introduced by Ricceri and on the theory of fractional Sobolev spaces with variable exponents, the existence of weak solutions for a nonlocal fractional elliptic system of $(p(x, \cdot), q(x, \cdot))$-Kirchhoff type with homogeneous Dirichlet boundary conditions was obtained. By using Ekeland's variational principle and dual fountain theorem, Bu et al. in ^[27] obtained some new existence and multiplicity of negative energy solutions for the fractional $p(\cdot)$-Laplace operators with constant order without the Ambrosetti-Rabinowitz condition.

Previous studies have shown that the fractional $p(\cdot)$-Laplace operators with variable-order are much more complex and difficult than $p$-Laplace operators. The investigation of these problems has captured the attention of a host of scholars. For example, Wu et al. in ^[28] considered the fractional Kirchhoff systems with a bounded set $\Omega$ in $\mathbb R^{N}$, as follows:

by applying Ekeland variational principle, they obtained the existence of a solution for this class of problem.

When $\mu = 0$, problem (1.1) reduces to the following fractional Kirchhoff-type elliptic systems

Motivated by the above cited works, we take into account the nonlocal fractional Kirchhoff-type elliptic systems with variable-order. Our aims are to establish the existence of at least three solutions for problem (1.1) by utilizing Ricceri's result in ^[29] and obtain the existence of at least two solutions for problem (1.8) with the help of the multiple critical points theorem in ^[37]. The primary consideration of the paper is an extension of the results found in the literatures and our results are new to the Kirchhoff-type systems in some ways.

For simplicity, $C_{j}\; (j = 1, 2, ..., N)$ are used in various places to denote distinct constants, $i = 1, 2$, and we denote

where $\mathcal{H}(\cdot)$ is a real-valued function and

$F:\Omega\times \mathbb R^{2}\rightarrow \mathbb R$ is a $C^{1}$-function, whose partial derivatives are $F_{\eta}, F_{\xi},$ which satisfy the following conditions.

$\text{(F1)}:$ For some positive constant $C$, there exist $\alpha(x), \beta(x)\in C_{+}(\mathfrak{D})$ and $2+\alpha^{+}+\beta^{+} < p^{-}$ such that

$\text{(F2)}:$ $F(x, s, t) > 0\; \;\text{ for any}\; (x, s, t)\in \overline{\Omega}\times[1, +\infty)\times[1, +\infty)$,$\; \text{and}\; F(x, s, t) < 0\; \;\text{ for any}\;$($x, s, t)\in \overline{\Omega}\times(0, 1)\times(0, 1)$, $F(x, 0, 0)$=$0 \; \;\text{ for a.e.}\; x\in \overline{\Omega}$.

$G:\Omega\times \mathbb R^{2}\rightarrow \mathbb R$ is a $C^{1}$-function, whose partial derivatives are $G_{\eta}, G_{\xi}$, which satisfy assumptions, as follows:

$\text{(G)}:$ $\sup_{|s|\leq \sigma, |t|\leq \sigma}(|G_{\eta}(x, s, t)|$+$|G_{\xi}(x, s, t)|)\in L^{1}(\Omega)$ $\text{for all }\; \sigma > 0.$

Definition 2. We say that $(\eta, \xi)\in X_{0}$ is a (weak) solution of nonlocal Kirchhoff systems $(1.1)$, if

for any $(\varphi, \psi) \in X_{0}$, and we will introduce $X_{0}$ in Section 2.

Define the corresponding functional $I: X_{0}\rightarrow \mathbb R$ associated with Kirchhoff systems $(1.1)$, by

for all $(\eta, \xi)\in X_{0}$, where

The functions $\Phi, \Psi, J:$ $X_{0}\rightarrow \mathbb{R}$ are well defined, and we define their Gâteaux derivatives at $(\eta, \xi)\in X_{0}$, by

for all $(\varphi, \psi)\in X_{0}$, where

for all $(\upsilon, \phi)\in X_{0}$.

Hence, $(\eta, \xi)\in X_{0}$ is a (weak) solution of Kirchhoff systems $(1.1)$ if and only if $(\eta, \xi)$ is a critical point of the functional $I$, that is

Definition 2. For $s^{+}p^{+} < N$, and denote by $\mathcal{A}$: there exists a kind of functions $\mathcal{F}: \Omega\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ such that two Carathéodory functions $\mathcal{F}_{\eta} = \frac{\partial \mathcal{F}}{\partial \eta}$ and $\mathcal{F}_{\xi} = \frac{\partial \mathcal{F}}{\partial \xi}$, satisfying

for any $\vartheta(x)\in [1, p^*_s(x)).$

Now, let us show our results in this article.

Theorem 1.1. For $s(\cdot), p(\cdot) \in C_{+}(\mathfrak{D})$ with $s^{+}p^{+} < N$ and $F\in\mathcal{A}$, assume that (S), (P), (M), (F1) and (F2) are satisfied. There exist three constants $a, c_{1}, c_{2} > 0$ with $0 < \gamma\leq1 < c_{1} < c_{2}$ such that

Then, for any

there exists a positive real number $\rho$ such that the system (1.8) has at least two weak solutions $w_{j} = (\eta_{j}, \xi_{j})\in X_{0}(j = 1, 2)$ whose norms $\|w_{j}\|$ in $X_{0}$ are less than some positive constant $\rho$.

Theorem 1.2. For $s(\cdot), p(\cdot) \in C_{+}(\mathfrak{D})$ with $s^{+}p^{+} < N$ and $F\in\mathcal{A}$, assume that (S), (P), (M), (F1) and (F2) are satisfied. Then there exists an open interval $\Lambda\subseteq (0, +\infty)$ and a positive real number $\rho$ with the following property: For each $\lambda\in \Lambda$ and for two Carathéodory functions $G_{\eta}$, $G_{\xi}$ :$\Omega\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ satisfying (G), there exists $\delta > 0$ such that for each $\mu \in [0, \delta]$, problem $(1.1)$ has at least three weak solutions $w_{j} = (\eta_{j}, \xi_{j})\in X_{0}(j = 1, 2, 3)$ whose norms $\|w_{j}\|$ in $X_{0}$ are less than some positive constant $\rho$.

Remark 1.1. Existence results for the Kirchhoff-type elliptic systems with both boundary value problems and variational problems were obtained according to using critical points theorem by Ricceri and Bonanno, respectively, where the condition of Palais-Smale is not satisfied.

Remark 1.2. The nonlocal Kirchhoff coefficient $M_{1}(t), \; M_{2}(t)$ stir up some of the fundamental difficulties. To deal with these difficulties, we suppose that $M_{1}(t), \; M_{2}(t)$ are strictly increasing functions, and then prove that the function $\Phi'$ is a homeomorphism.

The remaining of this article is organized as follows: Some fundamental results about the fractional Lebesgue spaces and Sobolve spaces are given in Section 2. In Section 3, in order to use critical point theory, we prove some technical lemmas. Theorem 1.1 and Theorem 1.2 are proved in Section 4. Finally, we make a conclusion in Section 5.

2.   Preliminary

2.1.   Variable exponents Lebesgue spaces

To study Laplacian problems with variable exponents, we need to recall a slice of preliminary theories on generalized Lebesgue spaces $L^{\vartheta(x)}(\Omega)$ and give some necessary lemmas and propositions.

For any $\vartheta(x)\in C_{+}(\mathfrak{D})$, the generalized Lebesgue spaces with variable exponents is defined by

with respect to the norm

then, the spaces $(L^{\vartheta(x)}(\Omega), \|\cdot\|_{\vartheta(x)})$ is a separable and reflexive Banach space, see ^[30,31].

Let $\vartheta(x)$ be the conjugate exponent of $\widetilde{\vartheta}(x)$, namely

Lemma 2.1. (see ^[31]) Assume that $\xi\in L^{\vartheta(x)}(\Omega)$ and $\eta\in L^{\widetilde{\vartheta}(x)}(\Omega)$, then

Proposition 2.1. (see ^[30,32]) If we define

then for all $\xi_{n}, \xi\in L^{\vartheta(x)}(\Omega)$, the following properties are possessed.

Remark 2.1. Note that for any function $\vartheta_{1}(x), \; \vartheta_{2}(x) \in C_{+}(\overline{\Omega})$ and $\vartheta_{1}(x) < \vartheta_{2}(x)$, there exists an embedding $L^{\vartheta_{2}(x)}(\Omega)\hookrightarrow L^{\vartheta_{1}(x)}(\Omega)$ for any $x\in \overline{\Omega}$. Especially, when $\vartheta(x)\equiv$ constant, the results of Proposition $2.1$ still hold.

2.2.   Fractional Sobolev spaces with variable-order

From now on, we briefly review a slice of essential lemmas and propositions about the Sobolev spaces, which will be used later. The readers are invited to consult ^[33,34,35] and the references therein.

The fractional Sobolev spaces $W^{s(\cdot), p(\cdot)}(\Omega)$ is defined as

and it can be endowed with the norm

where

then, the spaces $(W, \|\cdot\|_{W})$ is a separable and reflexive Banach space, see ^[2,5] for a more detailed.

We define the new fractional Sobolev spaces $W'$ concerning variable exponent and variable-order for some $\chi > 0$.

where $\mathcal{Q} = \mathbb{R}^{2N}\backslash (\Omega^{c}\times\Omega^{c})$ and it can also be endowed with the norm

where

Remark 2.2. Note that the norm $\|\cdot\|_{W}$ is different from $\|\cdot\|_{W'}$ for the reason that $\Omega\times\Omega\subset\mathcal{Q}$ and $\Omega\times\Omega \neq \mathcal{Q}$.

Let

with the norm

$W^{*}_{0}$ denotes the dual spaces of $W_{0}$.

In what follows, $X_{0}$ will denote the Cartesian product of two fractional Sobolev spaces $W_{0}$ and $W_{0}$, i.e. $X_{0} = W_{0}\times W_{0}$. Defined the norm

where $\|\eta\|_{W_{0}}, \; \|\xi\|_{W_{0}}$ is the norm of $W_{0}$.

Theorem 2.1. (see ^[2]) Let $s(\cdot)$, $p(\cdot)\in C_{+}(\mathfrak{D})$ satisfy (S) and (P), with $N > p(x, y)s(x, y)$ for all $(x, y)\in\overline{\Omega}\times\overline{\Omega}$. Let $\phi(x)\in C_{+}(\mathfrak{D})$ satisfy

where $\overline{p}(x) = p(x, x)$ and $\overline{s}(x) = s(x, x)$. Then, there exists a constant $C_\phi = C_\phi(N, s, p, \phi, \Omega) > 0$ such that

for any $\xi\in W_{0}$. Moreover, the embedding $W_{0}\hookrightarrow L^{\phi(\cdot)}(\Omega)$ is compact.

Proposition 2.2. (see ^[36]) If we define

then for all $\xi_n, \xi\in W_{0}$, the following properties hold.

In order to prove our main results, we present two multiple critical theorems: The first ensures the existence of two critical points, while the second establishes the existence of three critical points due to Ricceri.

Theorem 2.2. (see ^[37]) Let X be a reflexive real Banach space, and let $\Phi, \Psi: X \rightarrow R$ be two sequentially weakly lower semicontinuous functions. Assume that $\Phi$ is (strongly) continuous and satisfies $\lim_{\|\eta\|\rightarrow \infty}\Phi(\eta) = +\infty$. Assume also that there exist two constants $r_{1}$ and $r_{2}$ such that

where

for $i = 1, 2$ and $\overline{\Phi^{-1}(-\infty, r_{i}})^{\omega}$ is the closure of $\Phi^{-1}(-\infty, r_{i})$ in the weak topology. Then, for each

the functional $\Phi+\lambda\Psi$ has two local minima which lie in $\Phi^{-1}(-\infty, r_{1})$ and $\Phi^{-1}[r_{1}, r_{2})$, respectively.

Theorem 2.3. (see ^[29]) Let $X$ be a reflexive real Banach space $\Phi: X \rightarrow R$ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on $X^{*}$ and $\Phi$ is bounded on each bounded subset of $X$; $\Psi: X \rightarrow R$ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, $I\subseteq R$ an interval. Assume that

for $\lambda \in I$, and that there exists $r \in \mathbb{R}$ and $\eta_{0}, \eta_{1}\in X$ such that

Then there exists an open interval $\Lambda \subseteq I$ and a positive real number $\rho$ with the following property: For every $\lambda\in \Lambda$ and every $C^{1}$-functional $J: X \rightarrow \mathbb{R}$ with compact derivative, there exists $\delta > 0$ such that for each $\mu\in [0, \delta]$ the equation

has at least three solutions in $X$ whose norms are less than $\delta$.

3.   Some technical lemmas

In this section, in order to use critical point theory for Kirchhoff systems $(1.1)$, we need the following crucial lemmas, which will play an important role in the proof of our results.

Lemma 3.1. Assume that the functions $G, F \in \mathcal{A}$, then $J, \Psi \in C^{1}(X_{0}, \mathbb{R })$ and their derivatives are defined as (1.14) and (1.15) for all $(\varphi, \psi)\in X_{0}$. Moreover $J', \Psi': X_{0}\rightarrow \mathbb{R}$ is compact.

Proof. Here we follow the approach in ^[22], for completeness, we give the proof process. Suppose that $s^{+}p^{+} < N$, we prove that $J'(\eta, \xi)$ is continuous operator. Set $\{(\eta_{n}, \xi_{n})\}\subset X_{0}$ with $(\eta_{n}, \xi_{n})\rightarrow (\eta, \xi)$ strongly in $X_{0}$. According to Theorem 2.1, we obtain

So, for a subsequence denoted by $\{(\eta_{n}, \xi_{n})\}$, there exist functions $\widetilde{\eta}, \widetilde{\xi} \in L^{\vartheta(x)}(\Omega)$ such that

for all $n \in \mathbb{N}$.

Fix $(\widetilde{\eta}, \widetilde{\xi})\in X_{0}$ with $\|(\widetilde{\eta}, \widetilde{\xi})\|_{X_{0}}\leq1$. Since $G\in \mathcal{A }$, combining Lemma 2.1 with Theorem 2.1, we obtain

Consequently, for $\|(\widetilde{\eta}, \widetilde{\xi})\|_{X_{0}}\leq1$, we get

According to Definition 2, we deduce

and

using the dominate convergence theorem, we have

So, this prove that $J'(\eta, \xi)$ is continuous operator.

We show the operator $J'(\eta, \xi)$ is compact. Let $\{(\eta_{n}, \xi_{n})\}$ be a bounded subsequence in $X_{0}$. Arguing in the same way as above, we obtain that the sequence $\{J'(\eta_{n}, \xi_{n})\}$ converges strongly, therefor the operator $J'(\eta, \xi)$ is compact.

Similarly, we also deduce that $\Psi'(\eta, \xi)$ is continuous and compact.

Lemma 3.2. Assume that (M) is satisfied. Then

(i) & $\Phi$ is sequentially weakly lower semicontinuous and bounded on each bounded subset,

(ii) & $\Phi': X_{0}\rightarrow X_{0}^{*}$ is a strictly monotone and continuous operator,

(iii) & $\Phi': X_{0}\rightarrow X_{0}^{*}$ is a homeomorphism.

Proof. (i) Since $\widetilde{M}_{i}'(t) > m_{i} > 0,$ $\widetilde{M}_{i}(t)$ are increasing function on $[0, +\infty)$. Argue in a similar way from [38,Lemma 2.4], the operators $\eta\mapsto \delta'_{p(\cdot)}(\eta)$ and $\xi\mapsto \delta'_{p(\cdot)}(\xi)$ are strictly monotone. From [39,Proposition 25.10], $\delta_{p(\cdot)}(\eta)$ and $\delta_{p(\cdot)}(\xi)$ are strictly convex. Set $\{(\eta_{n}, \xi_{n})\}\subset X_{0}$ be a subsequence such that

Based on the convexity of $\delta_{p(\cdot)}(\eta)$ and $\delta_{p(\cdot)}(\xi)$, we obtain

and

Thus, we have

and

namely, the operators $\eta\mapsto \delta_{p(\cdot)}(\eta)$ and $\xi\mapsto \delta_{p(\cdot)}(\xi)$ are sequentially weakly lower semicontinuous.

On the other side, since $t\mapsto\widetilde{M}_{i}(t)$ are continuous and monotonous functions, we obtain

Consequently, the operator $\Phi$ is sequentially weakly lower semicontinuous.

Right now, we claim that $\Phi$ is bounded on each bounded subset of $X_{0}.$ Set $\{(\eta_{n}, \xi_{n})\}\subset X_{0}$ be a bounded subsequence. By Proposition 2.2, there exist constants $C_{6}, C_{7} > 0$ so that

Since $\widetilde{M}_{1}$ and $\widetilde{M}_{2}$ is monotone, we have

Hence, the operator $\Phi$ is bounded.

(ii) Let (M) is satisfied, then $\Phi\in C^{1}(X_{0}, \mathbb{R})$, and its derivatives are defined by (1.16), we have

where

for all $(\eta, \xi), (\varphi, \psi)\in X_{0}$. Therefore, $\Phi'(\eta, \xi)\in X^{*}_{0}$, where $X^{*}_{0}$ denotes the dual spaces of $X_{0}$.

In the first place, we show that $\Phi': X_{0}\rightarrow X_{0}^{*}$ is a strictly monotone operator. Since $\Phi_{\eta}(\eta)$ and $\Phi_{\xi}(\xi)$ are strictly monotone operators (see ^[40], Theorem 2.1), $\Phi'$ is a strictly monotone operator.

Next, we claim that $\Phi': X_{0}\rightarrow X_{0}^{*}$ is a continuous operator. Set $\{(\eta_{n}, \xi_{n})\}\subset X_{0}$ be a sequence, which converges strongly to $(\eta, \xi)$ in $X_{0}$. So, for a subsequence denoted by $\{(\eta_{n}, \xi_{n})\}$, we suppose

Then, the sequence

and

are bounded in $L^{p'}(\mathbb{R}^{N}\times\mathbb{R}^{N})$, and we have

and

According to the Brezis-Lieb lemma ^[41], combining (3.11) with (3.12) implies

and

Based on the fact that $\eta_{n}\rightarrow \eta$, $\xi_{n}\rightarrow \xi$ strongly in $X_{0}$ yields

and

Furthermore, $M_{1}(t)$ and $M_{2}(t)$ are continuous functions, which imply

and

With the help of Lemma 2.1, we get

as $n \rightarrow \infty,$ and then

Arguing in the similar way as above, we have

Thus, from (3.15)–(3.20), we obtain

(iii) Since the operator $\Phi'$ is a strictly monotone, it follows that $\Phi'$ is injective. Set $(\eta, \xi)\in X_{0}$ be such that $\|(\eta, \xi)\|_{X_{0}} > 1,$ from (M) and Proposition 2.2, we have

Thus, (3.22) implies that

Consequently, $\Phi'$ is coercive operator, thanks to the Minty-Browder Theorem (see ^[39], Theorem 26A), $\Phi'$ is a surjection. Due to its monotonicity, $\Phi'$ is an injection. So, $(\Phi')^{-1}$ exists.

Let us first prove that $\Phi'$ satisfies property:

Indeed, set $(\eta_{n}, \xi_{n})\rightharpoonup(\eta, \xi)$ in $X_{0}$, $\Phi'(\eta_{n}, \xi_{n})\rightarrow \Phi'(\eta, \xi)$ in $X^{*}_{0}$ and Theorem 2.1, which implies

Based on Fatou's Lemma yields

and

Applying Young's inequality, there exist $C_{8}, C_{9} > 0$ such that

The passage to the liminf implies in the above inequality, we get

This and combining (3.24) with (3.25), we obtain

and

Then, for a subsequence, we have

and

that is

Since $\eta_{n}\rightarrow \eta$, $\xi_{n}\rightarrow \xi$ in $X_{0}$, Proposition 2.2 imply that

that is

Consequently

Next, we prove that $(\Phi')^{-1}$ is continuous operator. Let $\{(f_{n}, g_{n})\}\subset X^{*}_{0}$ such that $f_{n}\rightarrow f_{0}$ in $X^{*}_{0}$ and $g_{n}\rightarrow g_{0}$ in $X^{*}_{0}$. Set $(\eta_{n}, \xi_{n}), (\eta, \xi)\in X_{0}$ such that

According to coercivity of $\Phi'$, we conclude that $(\eta_{n}, \xi_{n})$ is bounded. Then, up to subsequence $(\eta_{n}, \xi_{n})\rightharpoonup (\widetilde{\eta_{0}}, \widetilde{\xi_{0}})$, which implies

Combining the property of $(Q)$ and the continuity of $\Phi'$, we obtain

Since $\Phi'$ is an injection, we deduce that $(\eta, \xi) = (\widetilde{\eta_{0}}, \widetilde{\xi_{0}})$.

4.   Proof of the main results

In this subsection, we firstly prove Theorem 1.1 by applying Theorem 2.2.

Proof of Theorem 1.1. Let $X = X_{0},$ $\Psi$ and $\Phi$ are given as (1.11) and (1.12), respectively. Note that $\Psi'$ is a compact derivative from Lemma 3.1, Lemma 3.2 ensures that $\Phi$ is a weakly lower semicontinuous and bounded operator in $X_{0}$, and $\Phi'$ admits a continuous inverse operator $\Phi': X^{*}_{0} \rightarrow X_{0}$.

Let $(\eta_{*}, \xi_{*}) = (0, 0),$ then $\Phi(0, 0) = \Psi(0, 0) = 0.$ There exist a point $x_{0}\in \Omega$ and pick two positive constants $\mathfrak{R}_{2}, \mathfrak{R}_{1}(\mathfrak{R}_{2} > \mathfrak{R}_{1})$ such that $B(x_{0}, \mathfrak{R}_{2})\subset\Omega.$ Set $a, c$ be positive constants and define the function $\omega(x)$ by

where $B(x_{0}, \mathfrak{R})$ stands for the open ball in $\mathbb{R}^{N}$ of radius $\mathfrak{R}$ centered at $x_{0}$, then $(\omega(x), \omega(x))\in X_{0}$. Denote

Under condition (M) and by a simple computation, we obtain

When $\|\eta\|_{W_{0}}\rightarrow \infty, \|\xi\|_{W_{0}}$ bounded ($\|\xi\|_{W_{0}}\rightarrow \infty, \|\eta\|_{W_{0}}$ bounded) and $\|\eta\|_{W_{0}}\rightarrow \infty, \|\xi\|_{W_{0}}\rightarrow \infty$, we have

this implies that

Fix $\gamma$ such that $0 < \gamma\leq1 < c_{1} < c_{2}$ and set

By virtue of (4.1), set $\eta_{0} = \xi_{0} = \omega$ for $(\eta_{0}, \xi_{0})\in X_{0}$ and

Consequently, we have

Thus, (4.7) implies that

According to (F2), (1.20) and (4.1), we obtain

For each $(\eta, \xi)\in X_{0}$ with $\Phi(\eta, \xi)\leq r_{1}$, we conclude

Fix $\gamma_{0}$ such that $0 < \max\{\|\eta_{0}\|_{W_{0}}^{p^{+}}+\|\xi_{0}\|_{W_{0}}^{p^{+}}, \|\eta_{0}\|_{W_{0}}^{p^{-}}+\|\xi_{0}\|_{W_{0}}^{p^{-}}\} < \gamma_{0} < \gamma$. Thus, combining with (4.9) and (4.10), we have

From (4.8) and (4.11), we deduce

Similarly, for each $(\eta, \xi)\in X_{0}$ such that $\Phi(\eta, \xi)\leq r$, we get

According to $\Phi$ being sequentially weakly lower semicontinuous, then $\overline{\Phi^{-1}(-\infty, r)^{\omega}} = \Phi^{-1}(-\infty, r]$. Consequently, we have

this implies

Thus, by virtue of (4.12) and (4.15), we obtain

Therefore, the conditions of Theorem 2.2 are satisfied. Thus, the functional $\Phi+\lambda\Psi$ has two local minima $(\eta_{i}, \xi_{i})\in X_{0},$ which lie in $\Phi^{-1}(-\infty, r_{i})$, respectively. Since $I = \Phi+\lambda\Psi \in C^{1}(X_{0}, \mathbb{R }), (\eta_{i}, \xi_{i})\in X_{0}$ are the two solutions of the following equation

That is, $(\eta_{i}, \xi_{i})\in X_{0}$ are two weak solutions of the nonlocal fractional Kirchhoff-type elliptic systems (1.8). Since $\Phi(\eta_{i}, \xi_{i}) < r_{i}$, combining Theorem 2.1, we obtain

which implies that there is a real number $\rho > 0$ so that $w_{j} = (\eta_{j}, \xi_{j})\in X_{0}$ and $\|w_{j}\| < \rho.$

In what follows, we prove Theorem 1.2 Theorem by using Theorem 2.3.

Proof of Theorem 1.2. Let $X = X_{0},$ $\Psi$ and $\Phi$ are given by (1.11) and (1.12), respectively. Note that $\Psi'$ is a compact derivative from Lemma 3.1, Lemma 3.2 ensures that $\Phi$ is a weakly lower semicontinuous and bounded operator in $X_{0}$, and $\Phi'$ admits a continuous inverse operator $\Phi': X^{*}_{0} \rightarrow X_{0}$. Moreover,

for any $\lambda\in(0, +\infty)$. Indeed,

By virtue of (F1), we obtain

consequently,

Since $W_{0}$ are continuously embedded in $C(\overline{\Omega})$, there exists a constant $C_{11}$ such that

When $\|\eta\|_{W_{0}}\rightarrow \infty, \|\xi\|_{W_{0}}$ bounded ($\|\xi\|_{W_{0}}\rightarrow \infty, \|\eta\|_{W_{0}}$ bounded) and $\|\eta\|_{W_{0}}\rightarrow \infty, \|\xi\|_{W_{0}}\rightarrow \infty$, we have

According to $2+\alpha^{+}+\beta^{+} < p^{-}$, there exist $p_{1} < p^{-}$ such that $\frac{1+\alpha^{+}}{p_{1}}+\frac{1+\beta^{+}}{p_{1}} = 1$. Therefore, from (4.23) and Young's inequality, we deduce

Hence, for $\lambda > 0$ and $p_{1} < p^{-}$, the combination of (4.20), (4.24) and (4.25) implies that

In view of (F2), we choose $\delta > 1$ such that

$\text{for all}\; s, t > \delta, \; x\in \Omega,$ and then

$\text{for all}\; \tau_{1}, \tau_{2}\in [0, 1).$ Set $a_{0}, b_{1}$ be two positive real numbers such that $a_{0} < \min(1, C_{\phi})$ with $C_{\phi}$ defined by Theorem 2.1 and $b_{1} > \delta$ with $b_{1}^{p^{-}}|\Omega| > 1$. From (4.28), we have

Let

Choosing $(\eta_{0}(x), \xi_{0}(x)) = (0, 0), (\eta_{1}(x), \xi_{1}(x)) = (b_{1}, b_{1})\, \text{ for all}\, x\in \Omega,$ such that we obtain

and

Thus, the combination of (4.31) and (4.32) implies that

On the other hand

Let $(\eta, \xi)\in X_{0}$ such that $\Phi(\eta, \xi)\leq r$, we get

which implies that

According to Proposition 2.2, we derive

Combining with (4.36), we deduce

Therefore

for all $x\in \Omega$. It follows from (4.34) that

Consequently, the conditions of Theorem 2.3 are satisfied. For every compact interval $\Lambda\subset(0, +\infty)$ and $G\in \mathcal{A}$, we fix $\lambda \in \Lambda$ and put

for all $(\eta, \xi)\in X_{0}$. Then, $J$ is a compact derivative. Therefore, there exists a positive number $\delta$ such that for every $\mu \in [0, \delta],$ $(\eta_{i}, \xi_{i})\in X_{0}$ are three solutions of the following equation

That is, $(\eta_{i}, \xi_{i})\in X_{0}$ are three weak solutions of the nonlocal fractional Kirchhoff-type elliptic systems (1.1).

5.   Conclusions

In this article, we consider a kind of $p(\cdot)$-fractional Kirchhoff systems. Under some reasonable assumptions, we obtain two and three solutions based on Bonanno's multiple critical points theorems and Ricceri's three critical points theorem, where the condition of Palais-Smale is not requested. Several recent results of literatures are extended and improved.

Acknowledgments

This work is supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX21$ _{-} 54), the Natural Science Foundation of Jiangsu Province (BK20180500), National Key Research and Development Program of China (2018YFC1508100), Special Soft Science Project of Technological Innovation in Hubei Province (2019ADC146), and Natural Science Foundation of China (11701595).

Conflict of interest

The authors declare that they have no competing interests.