Existence and multiplicity of solutions for critical Choquard-Kirchhoff type equations with variable growth

1.   Introduction

We study the critical nonlocal Choquard equations with variable exponents of the form:

where

$K$: $\mathbb{R}^{+}_0\rightarrow \mathbb{R}^{+}_0$ is the Kirchhoff function, $V \in C\left(\mathbb{R}^{N}, \mathbb{R}^{+}\right)$, $\alpha: \mathbb{R}^{N} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^+$, $f$ is a continuous function, $\lambda$ is a real parameter, $p: \mathbb{R}^{N} \rightarrow \mathbb{R}$ is a function and $p^{*}(x) = N p(x)/(N-p(x))$ is the critical Sobolev exponent.

In the sequel, if $h_{1}, h_{2} \in C\left(\mathbb{R}^{N}\right)$, we say that $h_{1} \ll h_{2}$ if $\inf \left\{h_{2}(x)-h_{1}(x):\right. \left.x \in \mathbb{R}^{N}\right\} > 0$. And $C > 0$ may represent different constants.

Throughout this paper, we consider the following hypotheses:

$(\mathcal{P})$ $p: \mathbb{R}^N \rightarrow \mathbb{R}$ is continuous and $p$ satisfies

$(\mathcal{V})$ $V\in C(\mathbb{R}^N, \mathbb{R})$ satisfies $\underset{x\in\mathbb{R}^N}{\inf}V(x)\ge V_0 > 0$, with $V_0$ being a positive constant. Moreover, for any $\mathcal {D} > 0$, meas$\{x \in \mathbb{R}^N: V(x) \leq \mathcal {D}\} < \infty$, where meas$(\cdot)$ denotes the Lebesgue measure in $\mathbb{R}^N$.

$(\mathcal{K})$ $(K_1)$ $K: \mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ is continuous and there exists $k_0 > 0$ satisfying $\inf_{t\geq0} K(t) = k_0$.

$(K_2)$ For all $t\geq0$, there exists $\sigma \in [1, p^{\ast}(x)/2p^{+})$ such that $\sigma\mathcal{K}(t)\geq K(t)t$, where$\mathcal{K}(t) = \int_0^tK(s)ds$.

$(K_3)$ For all $t \in \mathbb{R}^+$, there exists $k_1 > 0$ such that $K(t) \geq k_1 t^{\sigma-1}$ and $K(0) = 0$.

$(\mathcal {G})$ $(g_1)$ $g: \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function and $g$ is odd for the second variable.

$(g_2)$ $r \in C(\mathbb{R}^N)$ and there exist $r(x)\geq 0$ satisfying $p(x)\ll r(x)q^-\leq r(x)q^+ \ll p^{\ast}(x)$. There exist $a\geq0$ such that

and

For all $x, y \in\mathbb R^N$,

where

$(g_3)$ For all $t \in \mathbb{R}^+, g(x, t)$ and $G(x, t) = \int_0^t g(x, s)ds$, there exists $\theta$ satisfying $0 < \theta G(x, t) \leq 2g(x, t)t$ where $p^{+}/\sigma < \theta < p^{\ast}(x)$.

In 1931, variable exponents Lebesgue spaces appeared in ^[34]. It is known that the $p(\cdot)$-Laplacian is derived from the $p$-Laplacian, especially to the Laplacian $(p = 2)$. From a practical point of view, variable exponents problem has many applications in the life, such as in image processing ^[11] and electrorheological fluids ^[40]. For these reasons, many authors have begun to study the existence of solutions to variable exponents problem, such as the books of R$\breve{\mbox{a}}$dulescu-Repov$\breve{\mbox{s}}$ ^[39] and Diening et al. ^[13]. When it comes to critical problem, we know Brézis and Nirenberg studied in ^[8] at first in 1983 and then it is a nature extensions of ^[8]. However, many critical problems are confronted with the lack of compactness. In 1984, Lions in ^[26,27] initially introduce the concentration-compactness principles. And in ^[5,6,10], authors show that there exists a minimizing or a (PS) sequence at infinity. In recent decades, it is nature for many scholars to consider more results for critical exponents $p(x)$-Laplacian equations. In ^[7,15], they study the variable exponents second concentration-compactness principles in $\Omega$. Moreover, there are much more results regarding $p(x)$-Laplacian and fractional $p(x)$-Laplacian equations, such as ^{[1,16,18,20,21,22,29]}.

On the other hand, the study of the Choquard equation began with Fröhlich ^[17] and Pekar ^[35] who dealt with the following quantum polaron model:

Then in the following Choquard equation:

In particular, when $N = 3, p = 2$ and $\lambda = 1$, Lieb in ^[25] used problem (1.3) to get some significant results about plasma. As is known to all, Penrose ^[31,36] applied Eq (1.3) as the model to solve gravity problem. Recently, more and more works pay attention to the problem (1.3) of existence and multiplicity of solutions. When it comes to the whole domain $\mathbb{R}^N$ of Choquard equations, we can cite ^[33,43] to get more details. For the critical case in bounded domains $\Omega$, Gao and Yang in ^[45] considered about the following critical Choquard problem:

Then in ^[46], we got the existence of solutions for a series of equations by using variational methods. When it comes to the Choquard problems with variable exponents, we found there is lack of relevant results. Therefore, we call attention to ^[28], it is the first time to consider the nonhomogeneous Choquard equation with $p(x)$-Laplacian operator by using variational methods. Secondly, in combination with the truncation function and Krasnoselskii's genus, they found the multiplicity of solutions for the Choquard-type $p(x)$-Laplacian equations with non-degenerate Kirchhoff term. In ^[41], the authors proved the existence of at least two nontrivial solutions for nonhomogeneous Choquard equations by using of Nehari manifold and minimax methods.

Recently, Alves and Tavares ^[2] considered the following quasilinear variable exponent Choquard equations:

The existence of solutions for Eq (1.4) was deduced from using the Hardy-Littlewood-Sobolev inequality together with variational methods. Zhang et al. in ^[47] considered the following equation:

where $p : \mathbb{R}^N \rightarrow \mathbb{R}$ is radially symmetric and $\mu > 0$. The existence of infinitely many solutions for problem (1.5) was obtained by variational methods, Hardy-Littlewood-Sobolev inequality and the concentration-compactness principle. The results of critical Choquard-Kirchhoff equations with variable exponents Eq (1.1) does not obtain, especially for the degenerate cases.

The research complete and improve results for the critical Choquard-Kirchhoff type equations involving variable exponents. Especially we discuss the results in non-degenerate and degenerate cases which are treated in many papers, for example, see a well-known paper ^[12]. In the recent decades, more and more attention were paied to degenerate Kirchhoff problem. For example, in 2015, Autuori et al. in ^[4] used the mountain pass theorem to demonstrate the asymptotic behavior of non-negative solutions for Kirchhoff equations. Then Pucci et al. in ^[37] considered entire solutions for the stationary Kirchhoff equations. Not long after, in 2016, Caponi and Pucci in ^[9] also investigate existence of entire solutions for a class of Kirchhoff fractional equations. And we can refer to ^{[23,24,30,42,44]} to get related content and details.

And $u \in W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)$ is a weak solution of Eq (1.1) if

for all $v \in W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)$. The space $W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)$ will be introduced in Section 2.

Now we are in a position to give the main theorems of this paper.

Theorem 1.1. Assume $p, V, K$ and $g$ respectively satisfy $(\mathcal{P})$, $(\mathcal{V})$, $(K_1)$, $(K_2)$ and $(g_1)$–$(g_3)$, respectively. In $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, there exists $\lambda_1 > 0$ and $\lambda \geq \lambda_1$, Eq (1.1) admits a nontrivial solution.

Theorem 1.2. Assume $p, V, K$ and $g$ satisfy $(\mathcal{P})$, $(\mathcal{V})$, $(K_1)$, $(K_2)$ and $(g_1)$–$(g_3)$, respectively. In $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, there exists constant $\lambda_2 > 0$ and $\lambda > \lambda_2$, Eq (1.1) admits at least $s$ pairs of nontrivial solutions.

Therefore, we obtain similar results in the degenerate case.

Theorem 1.3. Assume $p, V, K$ and $g$ satisfy $(\mathcal{P})$, $(\mathcal{V})$, $(K_2)$, $(K_3)$ and $(g_1)$–$(g_3)$, respectively. In $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, there exists $\lambda_3 > 0$ and $\lambda \geq \lambda_3$, Eq (1.1) admits a nontrivial solution in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$.

Theorem 1.4. Assume $p, V, K$ and $g$ satisfy $(\mathcal{P})$, $(\mathcal{V})$, $(K_2)$, $(K_3)$ and $(g_1)$–$(g_3)$, respectively. In $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, there exists constant $\lambda_4 > 0$ and $\lambda \geq \lambda_4$, Eq (1.1) admits at least $s$ pairs of nontrivial solutions in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$.

The paper is organized as follows. Section 2 contains fundamental knowledge of spaces with variable exponents. In Section 3, we verify the $(PS)_c$ condition. Section 4 and Section 5 respectively prove Theorems 1.1–1.4.

2.   Preliminaries

In this section, we give fundamental knowledge on the Lebesgue spaces and the Sobolev spaces with variable exponents. We refer to ^[13,14] for more details.

Assume $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and

We define

And we define the variable exponent Lebesgue space as

endowed with the norm

The Lebesgue-Sobolev space with variable exponents $W^{1, p(x)}\left(\mathbb{R}^{N}\right)$ is defined by:

with the norm

For problem (1.1), we study in $W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)$ which is more suitable, with the norm

where

Proposition 2.1 (^[14]). (1) Denote by $L^{p^{\prime}(x)}(\Omega)$ the conjugate space of $L^{p(x)}(\Omega)$ with $\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)} = 1$, there holds

(2) $\rho: L^{p(x)}(\Omega) \rightarrow \mathbb{R}$ and $\rho(u) = \int_{\Omega}|u|^{ p(x)} d x$,

Proposition 2.2 (^[2]). Assume $p, q \in C^{+}\left(\mathbb{R}^{N}\right), w\in L^{p^{+}}\left(\mathbb{R}^{N}\right) \cap L^{p^{-}}\left(\mathbb{R}^{N}\right), z \in L^{q^{+}}\left(\mathbb{R}^{N}\right) \cap L^{q^{-}}\left(\mathbb{R}^{N}\right)$, and $\alpha: \mathbb{R}^{N} \times \mathbb{R}^{N} \rightarrow \mathbb{R}$ be a continuous function satisfying $0 < \alpha^{-} : = \inf _{x \in \mathbb{R}^{N}} \alpha(x)\leq \alpha^{+}: = \sup _{x \in \mathbb{R}^{N}} \alpha(x) < N$ and for $\forall x, y \in \mathbb{R}^{N}$, there is

Then, we have

where $C > 0$ is irrelevant $w$ and $z$.

Corollary 2.1. For $w(x) = z(x) = |v(x)|^{\tau(x)}\in L^{r^{+}}(\mathbb{R}^{N})\cap L^{r^{-}}(\mathbb{R}^{N})$, there exists $C > 0$ which is irrelevant $r$ such that

$\tau, \, r\in C_{+}(\overline{\mathbb{R}^{N}})$ satisfying $1 < \tau^{-}r^{-}\leq \tau(x)r^{-}\leq \tau(x)r^{+} < p^{*}(x)$.

Remark 2.1. If $(\mathcal{P})$ and $(\mathcal{V})$ hold, then for all $s \in C^{+}\left(\mathbb{R}^{N}\right)$ and $p(x) \leq s(x) \leq p^{*}(x), \forall x \in \mathbb{R}^{N}$,

is compact embedding. Hence,

where S is the best Sobolev constant.

Remark 2.2. We can find that there is $b > 0$ satisfying

3.   $(PS)_c$ condition

Let's first recall the definition of the $(P S)_{c}$ condition. The functional $J_{\lambda}$ satisfies the $(PS)_c$ condition if any sequence $J_{\lambda}(u_n) \rightarrow c$ and $J_{\lambda}^{\prime}(u_n) \rightarrow 0$ has a convergent subsequence. In this section, we will prove the functional $J_{\lambda}$ satisfies the $(P S)_{c}$ condition.

The energy functional $J_\lambda:W_V^{1, p(x)}(\mathbb{R}^{N}) \rightarrow \mathbb{R}$ is

where

$G(x, t) = \int_0^t g(x, s)ds$. Obviously, $J_\lambda \in C^1(W^{1, p(x)}_{V}(\mathbb{R}^{N}))$. Moreover, for all $u, v \in W^{1, p(x)}_{V}(\mathbb{R}^{N})$, we deduce that

Hence, the solutions of problem (1.1) are the critical points of $J_\lambda$.

Lemma 3.1. Assume $(\mathcal{P})$, $(\mathcal{V})$, $(\mathcal{G})$, $(K_1)$ and $(K_2)$ hold. Let $(u_n)_n\subset W^{1, p(x)}_{V}(\mathbb{R}^{N})$ be a $(PS)$ sequence of $J_\lambda$, then

as $n \rightarrow \infty$, where $(W^{1, p(x)}_{V}(\mathbb{R}^{N}))'$ is the dual of $W^{1, p(x)}_{V}(\mathbb{R}^{N})$. If there is $\tau(x) = \frac{p^\ast(x)}{p^\ast(x)-p^+}$ such that

where $S$ is Sobolev constant. In $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, $(u_n)_n\rightarrow u$ strongly.

Proof. We prove $(u_n)_n$ is bounded in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$.

Assume $(u_n)n$ and $c_\lambda$ satisfy (3.3) and (3.4), respectively. Then, from $(f_3)$, we can deduce that

So $(u_n)_n$ is bounded in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$.

Then we need to demonstrate

In fact, since $u_{n} \rightarrow u$ weakly in $W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)$ as $n \rightarrow \infty$, when $\Lambda^{\prime}(u) \in\left(W_{V}^{1, p(x)}\left(\mathbb{R}^{N}\right)\right)^{'}$, we yield that

So we only need to prove that

We deduce from Proposition 2 that

Combining $(g_2)$ and $(u_{n})_n$,

and

According to $(g_2)$ and Remark 2.1, we can yield that

and

Combining (3.6)–(3.10), we can obtain $\langle\Lambda^{\prime}\left(u_{n}\right), u_{n}-u\rangle \rightarrow 0 \ {\rm { as }}\ n \rightarrow \infty.$ So we deduce that

Then, in view of the concentration-compactness principle for variable exponents in ^[19], we get

where

and

Furthermore, we yield that

where

Now we demonstrate

We assume that $I \neq \emptyset$. For any $i \in I$ and any $\varepsilon > 0$ small, we define a function $\phi_{\varepsilon, i}$ centered at $z_i$ satisfying

Combining with $\langle J_\lambda'(u_n), u_n\phi_{\varepsilon, i}\rangle \rightarrow 0$, we deduce that

We deduce from $u_{n} \rightarrow u$ in $L^{p(x)}\left(B_{2\varepsilon}(z_i)\right)$ that

So,

and in $\mathbb{R}^{N}$, we can choose $w_{N}$ to be the unit sphere,

Next, as $n \rightarrow \infty$, we claim

According to Proposition 2.2 and the Lebesgue dominated convergence theorem,

And

Combining (3.13)–(3.17), we deduce that

Since $\phi_{\varepsilon, i}$ has compact support and $(K_1)$, choosing $n\to\infty$ and $\varepsilon\rightarrow 0$ in (3.18), we get

In view of (3.11), we yield that

where $\tau(x) = \frac {p^{\ast}(x)}{p^{\ast}(x)-p^{+}(x)}$. On account of (3.3), (3.19) and (3.5),

We get a contradiction, so $I = \emptyset$.

Then we show $\nu_\infty = 0$. In the first, we assume $\nu_\infty > 0$. Similarly, we define $\phi_R \in C_0^\infty(\mathbb{R}^N)$ satisfying $\phi_R(x) = 0$ in $B_R$ and $\phi_R(x) = 1$ in $B_{R+1}^c$. According to $\langle J_\lambda'(u_n), u_n\phi_R\rangle \rightarrow 0$ as $n \rightarrow \infty$, we deduce

We have

and

So we get

Letting $R \rightarrow \infty$ in (3.22), we deduce

According to (3.12) and (3.24), we can also infer

Then we get a contradiction, so $\nu_\infty = 0$.

Therefore, combination of $I = \emptyset$ and $\nu_\infty = 0$,

According to the Brézis-Lieb type lemma, we get

thus $\left\|u_{n}-u\right\|_{L^{p^{*}(x)}\left(\mathbb{R}^{N}\right)} \rightarrow 0$. Consequently, we have

Then we get

where $w \in W^{1, p(x)}_{V}(\mathbb{R}^{N})$, $L(v)$ in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$ was

Combining the weak convergence of $(u_n)_n$ in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$ with the boundedness of $(K(T_{p(x)}(u_n))-K(T_{p(x)}(u)))_n$ in $\mathbb{R}^{N}$, we obtain that

In view of $\langle J_\lambda(u_n), u_n -u\rangle\to 0\ (n\rightarrow \infty)$,

Hence, we obtain that

Therefore, in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$, we get $(u_n)_n\rightarrow u$ strongly.

4.   Non-degenerate case for Eq (1.1)

We respectively demonstrate Theorems 1.1 and 1.2 by using the mountain pass theorem ^[3] and the Krasnoselskii genus ^[38].

4.1.   Proof of Theorem 1.1

First of all, we demonstrate there exists mountain pass structure for $J_\lambda$.

Lemma 4.1. Let $E$ is a real Banach space, and $J_\lambda\in C^1(E)$, with $J_\lambda(0) = 0$. If the following conditions hold true:

(1) For any $u\in E$ and $\|u\|_{E} = \rho$, there is $\rho, \chi > 0$ satisfying $J_\lambda(u)\geq \chi$;

(2) For any $\|\omega\|_{E} > \rho$, there is $\omega\in E$ satisfying $J_\lambda(\omega) < 0$.

We can define $\Gamma = \{\gamma\in C([0, 1], E)\, :\, \gamma(0) = 1, \gamma(1) = \omega\}$. Hence,

and $(u_n)_n\subset E$.

Proof. First, we verify condition (1) of Lemma 4.1. In view of $(K_1)$, we have $\inf_{t\geq0} K(t) = k_0$. Then according to Remark 2.1, we get $\|u\|_{W_{V}^{1, p(x)} \ \ \ \left(\mathbb{R}^{N}\right)}\leq S |u|_{L^{s(x)} \ \ \ \left(\mathbb{R}^{N}\right)}.$ So,

where $\|u\|\leq 1$. Hence, let $\rho$, $\chi > 0$, $\|u\| = \rho$ and the fact $p^- \leq q^{-}r(x)\leq q^{+}r(x)\ll p^*$ satisfying $J_\lambda(u)\geq\chi$. So we prove $(1)$ of Lemma 4.1.

In order to prove the conclusion $(2)$ of Lemma 4.1, we choose $\psi \in C_0^\infty(\mathbb{R}^N)$ and $\psi > 0$, combination with $(K_2)$, for all $t\geq 1$, we get

According to the conditions of $f$, we obtain that

In view of $\sigma p^+ < p^{*}(x)$ and for $t_0$ large enough, we obtain $J_\lambda(t_0\psi) < 0$ and $t_0\|\psi\| > \rho$. Set $\omega = t_0\psi$, so $e$ satisfies the conditions and the conclusion (2) is true.

Proof of Theorem 1.1. Next, if $\lambda$ large enough, we prove that

Combining (4.3), Lemmas 3.1 and 4.1, it is obvious that we can deduce $J_\lambda$ exists nontrivial critical points. So we need to demonstrate (4.3). Let $v_0 \in W^{1, p(x)}_{V}(\mathbb{R}^{N})$ such that

Hence, for some $t_\lambda > 0$, we get $\sup_{t\geq0}J_\lambda(tv_0) = J_\lambda(t_\lambda v_0)$. And

Next, we need to prove the boundedness of $\{t_\lambda\}_{\lambda > 0}$. First, for any $\lambda > 0$, let $t_\lambda \geq 1$. According to (4.4), we obtain that

Since $\sigma \in [1, p^{\ast}(x)/2p^{+})$, so $2p^+\sigma < p^{*}(x)$ and (4.5), hence we get the boundedness of $\{t_\lambda\}_\lambda$.

The next step is to demonstrate $t_\lambda \rightarrow 0$ as $\lambda \rightarrow \infty$. Similarly, we get $\lambda_n \rightarrow \infty$ and $t_{\lambda_n} \rightarrow t_0$. we yield

as $n \rightarrow \infty$. And

So it results $K\left(T_{p(x)}(t_0 v_0)\right) = \infty$ thanks to (4.4). Hence, $t_\lambda \rightarrow 0$ as $\lambda \rightarrow \infty$. Then we have

and

Moreover, an easy computation gives that

Then we can find $\lambda_1 > 0$ satisfying for any $\lambda \geq \lambda_1$, we have

Let $\omega = \tau v_0$ satisfying $J_\lambda(\omega) < 0$, so choosing$\gamma(t) = t \tau v_0$, we get

Finally, if $\lambda$ large enough, we obtain

Hence, Eq (1.1) admits a nontrivial solution.

4.2.   Proof of Theorem 1.2

In this part, we demonstrate Theorem 1.2 by using similar method in ^[38]. Assume $\Pi$ is a set which includes closed subsets A. A are symmetric and A are subsets of $X\setminus\{0\}$ which is an infinite dimensional Banach space.

Lemma 4.2 (^[38]). X is shown above. There exists $U$ satisfying $X = U \oplus V$. Assume $J_\lambda\in C^{1}(X)$ be an even functional and $J_\lambda(0) = 0$. $J_\lambda$ meets

$(I_1)$ for any $u\in \partial B_{\rho}\bigcap Z$, there is constant $\rho$, $\chi > 0$ satisfying $J_\lambda(u)\ge \chi$;

$(I_2)$ for any $c$ and $c\in(0, \xi)$, there is constant $\xi > 0$, $J_\lambda$ satisfying the $(PS)_c$ condition;

$(I_3)$ there exists $R = R(\widetilde X) > 0$ satisfying $J_\lambda(u)\leq 0$ on $\widetilde X\setminus B_R$ where $\widetilde X\subset X$ ia any finite dimensional subspace.

Let $U$ is $s$ dimensional Banach space and we give the definition of $U = span\{u_1, \cdots, u_s\}$. When $n\geq s$, we have $u_{n+1} \not\in Q_n = span\{u_1, \cdots, u_n\}$. So we assume $R_n = R(Q_n)$ and $\Omega_n = B_{R_n}\bigcap Q_n$. Then we give the definition of $Q_n$, that is

We have

where $\gamma(D)$ is genus of $D$. Let

Therefore, when $t > s$, we find $0\leq c_t\leq c_{t+1}$, with $c_t < \xi$. For any $t > s$ such that $c_t = c_{t+1} = \cdots = c_{t+\alpha} = c < \xi$, $T_c$ denote the set of critical points in $X$ and we have $\gamma(T_c)\ge \alpha+1$.

Proof of Theorem 1.2. According to Lemma 4.2, we can prove Theorem 1.2.

First, we verify conditions. Since $J_\lambda \in C^1\big(W^{1, p(x)}_{V}(\mathbb{R}^{N})\big)$. According to (3.2), we have $J_\lambda(0) = 0$. Therefore, the demonstration is analogous to (1) and (2) in Lemma 4.1. Since $J_\lambda$ meets conditions $(I_1)$ and $(I_3)$ of Lemma 4.2.

Therefore, we can demonstrate there exists $(\Upsilon_n)_n\subset\mathbb R^+$, and $\Upsilon_n\leq \Upsilon_{n+1}$, satisfying

According to the definition of $c_n^{\lambda}$, we deduce that

And

so that $\Upsilon_n < \infty$ and $\Upsilon_n\leq\Upsilon_{n+1}$. We show that Eq (1.1) has at least $s$ pairs of solutions. And we have two possibilities:

(Ⅰ) In the first case, we assume $\lambda > 0$. There exists $k_0$ such that

(Ⅱ) In the second case, similarly, in (4.3) and $\lambda > \lambda_2$, we deduce from $\lambda_2 > 0$ that

Hence, we yield

By means of Proposition 9.30 in ^[38], we get that $J_\lambda$ has many critical values. And there exists $t = 1, 2, \cdots, s-1$ such that$c_t^\lambda = c_{t+1}^\lambda$, $T_{c^{\lambda}_{t}}$ has a lot of critical points. So we demonstrate Theorem 1.2.

5.   Degenerate case for Eq (1.1)

In this part, we consider Eq (1.1) in the degenerate case. We give a crucial lemma at first.

Lemma 5.1. $J_\lambda$ has a $(PS)$ sequence $(u_n)_n$ in $W^{1, p(x)}_{V}(\mathbb{R}^{N})$. Let $\tau_\sigma(x): = \frac{p^{\ast}(x)}{p^{\ast}(x)-p^{+}\sigma}$,

then $(u_n)_n\rightarrow u$ strongly.

Proof. If $\inf_{n\geq 1}\|u_n\| = 0$, then there is a subsequence of $(u_n)_n$ such that $u_n \rightarrow 0$ as $n \rightarrow \infty$. Hence, we suppose $d : = \inf_{n\geq 1}\|u_n\| > 0$ and $\|u_{n}\| > 1$. In view of the definition of the $(PS)$ sequence, we get

According to the conditions of $(K_2)$, $(K_3)$ and $(g_3)$, we can deduce

Since $p^+\sigma > 1$, we deduce $(u_n)_n$ is bounded.

Refer to Lemma 3.1, we can assume $I \neq \emptyset$. For any $i \in I$ and any $\epsilon > 0$ small, we define $\phi_{\epsilon, i}$ as Lemma 3.1. In view of $\langle J_\lambda'(u_n), u_n\phi_{\epsilon, i}\rangle \rightarrow 0$ as $n \rightarrow \infty$. Thus we have

Similarly,

and

By $(K_3)$ and (5.3), we have

According to (5.3), we get

Then we find that either $\nu_i = 0$ or

We deduce from $(K_2)$, $(g_3)$ and (5.5) that

So we get

Thus we get a contradiction with (5.1). Hence $\nu_i = 0$.

Next, we claim that $\nu_\infty = 0$. Similarly, we give a smooth cut-off function $\phi_{R}$ and because of $\langle J_\lambda'(u_n), u_n\phi_{R}\rangle\rightarrow 0$,

Similarly, we get

and

According to (5.5), we obtain that

and

Combining with (3.12), we find that either $\nu_\infty = 0$ or

Then we prove (5.9). Similarly, in view of (3.21), we deduce that

By (5.1), it is an obvious contradiction. Thus $\nu_\infty = 0$.

Hence, we have $I = \emptyset$ and

According to a Brézis-Lieb lemma with variable exponents (see ^[19], Lemma 3.9) and last equality, we get

and

In view of (3.27), (3.29) and $\langle J_\lambda(u_n), u_n -u\rangle\to0$, we deduce

So $(u_n)_n\rightarrow u$ strongly where $u \in W^{1, p(x)}_{V}(\mathbb{R}^{N})$.

Lemma 5.2. $J_\lambda$ satisfies the conditions (1) and (2) of Lemma 4.1.

Proof. By $(K_3)$, $(g_2)$ and Remark 2.1, for any $\lambda > 0$, $u \in W^{1, p(x)}_{V}(\mathbb{R}^{N})$ and $\|u\| < 1$, we deduce

Since $p^+\sigma < p^{\ast}(x)$ and $p(x)\ll rq^-\leq rq^+ \ll p^{\ast}(x)$, choosing $\rho, \chi > 0$ and $\|u\| = \rho$, we have $J_\lambda(u)\geq\chi$. Thus we prove (1) in Lemma 4.1. Similarly, we prove (2) of Lemma 4.1 is true.

Proof of Theorem 1.3. The proof is analogous to Lemma 4.1, we can obtain

The remaining steps are analogous to Theorem1.1.

Proof of Theorem 1.4. The proof of Theorem 1.4 is analogous to Theorem 1.2.

6.   Conclusions

In this paper, we consider the critical nonlocal Choquard-Kirchhoff type equations with variable exponents in the degenerate cases and non-degenerate cases. Because of the existence of critical reaction, we apply the concentration-compactness principle to overcome the lack of compactness. By using the variational methods, the Hardy-Littlewood-Sobolev inequality and Krasnoselskii genus, we get the results of existence and multiplicity of solutions for a class of critical Choquard-Kirchhoff type equations.

Acknowledgements

L. Tao was supported by the Graduate Scientific Research Project of Changchun Normal University (SGSRPCNU [2022], Grant No. 058). S. Liang was supported by the Research Foundation of Department of Education of Jilin Province (Grant no. JJKH20211161KJ) and Natural Science Foundation of Jilin Province (Grant no. YDZJ202201ZYTS582). R. Niu was supported by the National Natural Science Foundation of China (No. 11871199).

Conflict of interest

The authors declare that they have no competing interests.