Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity

1.   Introduction

In this paper, we investigate the existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian and local nonlinearity. More precisely, we consider the following Kirchhoff-type equation

where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator which, up to a normalization constant, may be defined as

along any $\phi\in\mathcal{C}_{0}^{\infty}({\mathbb R}^N)$, where $B_{\varepsilon}(x): = \{y\in {\mathbb R}^N: |x-y| < \varepsilon\}$. In fact, many scholars have paid more attention to the fractional and nonlocal operators in recent years. This type of operator occurs naturally in many field of science, such as finance, continuum mechanics, free boundary obstacle problems, population dynamics, plasma physics and anomalous diffusion. For more details on this type of operator, we refer to ^[2,5,6] and the references therein. For the fractional Sobolev spaces and the study of the fractional Laplacian by using variational methods, we refer the readers to ^[9,20,27]. In order to simplify our statements, we first suppose on the potential function $V$ that

$(V_{1}) \; V\in\mathcal{C}({\mathbb R}^{N})$ and there exists a constant $V _{0} > 0$ such that $\inf\limits_{x\in {\mathbb R}^{N}}V(x)\geq V_{0}$;

$(V_{2})$ there exists $R > 0$ such that

Condition $(V_{2})$ was originally from ^[3], it can be used to solve the problem of the lack of compactness in the whole space ${\mathbb R}^N$. Moreover, the condition $(V_{2})$ is weaker than the coercivity condition $\lim\limits_{|x|\rightarrow\infty}V(x) = \infty$.

As we all know, Kirchhoff equations was first proposed by Kirchhoff in 1883 (see ^[15]), which was related to the celebrated D'Alembert wave equation

for free vibrations of elastic strings. Here, $\rho$ is the mass density, $P_{0}$ is the initial tension, $h$ is the area of the cross section, $E$ is the Young modulus of the material and $L$ is the length of the string. Bernstein ^[4] and Pohozaev ^[24] were the early scholars devoted to study Kirchhoff equations. After Lions ^[18] proposed an abstract framework for Kirchhoff problems, many scholars have studied Kirchhoff equations by using variational methods. For example, we refer to ^[12,19] for Kirchhoff equations involving subcritical nonlinearities; we also collect some articles, see ^[17,21] for Kirchhoff equations involving critical and supercritical nonlinearities.

Recently, Fiscella and Valdinoci ^[11] proposed a stationary Kirchhoff type variational model, which considered the nonlocal aspect of the tension arising from nonlocal measurements of fractional length of the string. More precisely, they studied the following fractional Kirchhoff type problem involving critical growth

By combining a truncated technique with the mountain pass theorem, they obtained the existence of nontrivial solutions for the above equation when $\lambda$ is large enough. Afterwards, many scholars have studied the existence and multiplicity of nontrivial solutions, ground state solutions, sign-changing solutions for fractional Kirchhoff type equations. For the subcritical case, Pucci et al. ^[25] obtained the existence of multiple solutions for fractional p-Laplacian Schrödinger-Kirchhoff type equations via the Ekeland variational principle and the mountain pass theorem. For the critical case, Fiscella ^[10] provided the existence of two solutions for fractional Kirchhoff equation with singular term and critical nonlinearity via variational methods. By using the concentration compactness principle in fractional Sobolev spaces, Xiang et al. ^[30] established the existence and multiplicity of solutions for a class of fractional $p$-Laplacian Kirchhoff type problems involving critical exponent. Moreover, there are few results about the existence of solutions for fractional Kirchhoff problems involving the supercritical term. Fortunately, Ambrosio and Servadei ^[1] first studied the existence of nontrivial solutions for factional Kirchhoff problems with supercritical growth by using a truncation argument, the mountain pass theorem and Moser iterative method. For more related results on fractional Kirchhoff type equations, we refer the interested reader to ^{[23,26,29,31]} and the references therein.

However, many scholars usually supposed that the nonlinearity $f(u)$ satisfies the growth condition $|f(u)|\leq C(|u|+|u|^{q-1})$, $q\in(p, p_{s}^{*})$ and the following conditions

$(S1) \; \frac{f(u)}{|u|^{3}}$ is an increasing function of $u\in {\mathbb R}\setminus\{0\}$;

$(S2)$ $\lim\limits_{|u|\rightarrow+\infty}\frac{F(u)}{u^{4}} = +\infty$;

or the Ambrosetti-Rabinowitz condition

$(AR)$ there exists $\mu\in(p, p_{s}^{*})$ such that $0 < \mu F(u)\leq f(u)u$ for all $u\in {\mathbb R}$.

Indeed, under the above conditions, it is easy to obtain the existence and multiplicity of solutions for fractional Kirchhoff type equations by using variational methods. For example, Cheng and Gao ^[7] established the existence of least energy sign-changing solutions when $f(u)$ satisfies $(S1)$ and $(S2)$. Under the nonlinearity $f(u)$ satisfies $(AR)$ condition, Nyamoradi and Zaidan ^[22], they proved the existence of nontrivial solutions. In addition, some scholars studied the existence of solutions for fractional Kirchhoff type equations when $f(u)$ satisfies the Berestycki-Lions type conditions, see for instance ^[14,33].

As far as we know, there are no papers dealing with the fractional Kirchhoff type equation with local nonlinearity, which the nonlinearity $f(u)$ is superlinear just in a neighborhood of $u = 0$. There is no doubt that serious difficulties will be encountered, because there is no assumption about the function $f(u)$ at infinity. This difficulty makes the study of fractional Kirchhoff type equations become more interesting and challenging. In order to overcome this difficulty, Li and Su ^[16] used a truncation argument due to Costa and Wang (see ^[8]) to get the existence and multiplicity of solutions for Kirchhoff type equations. In ^[13], Huang and Jia also obtained the existence of positive solutions for quasilinear Schrödinger equations via the truncation argument in ^[8].

Motivated by the above works, the purpose of this paper is to give existence results for Eq $\rm(1.1)$. To the best of our knowledge, the existence of solutions for fractional Kirchhoff equations with local nonlinearity has not been studied yet. Before stating our result, we consider $M: {\mathbb R}_{0}^{+}\rightarrow {\mathbb R}_{0}^{+}$, ${\mathbb R}_{0}^{+} = [0, +\infty)$, is supposed to satisfy the following conditions:

$(M_{1}) \; M\in\mathcal{C}({\mathbb R}_{0}^{+})$ and there exists $m_{0} > 0$ such that $M(t)\geq m_{0}$ for any $t\in {\mathbb R}_{0}^{+}$;

$(M_{2})$ the function $t\mapsto M(t)$ is increasing;

$(M_{3})$ for each $t_{1}\geq t_{2} > 0$, it holds

Moreover, we assume the following assumptions on $f$:

$(f_{1})$ $f\in\mathcal C({\mathbb R}^{N}\times {\mathbb R}, {\mathbb R})$, $f(x, t) = 0$ for all $t\leq0$ and there exists $\alpha\in(2p, p_{s}^{*})$ such that

$(f_{2})$ there exists $\sigma\in(2p, p_{s}^{*})$ and $\sigma > \alpha$ such that

$(f_{3})$ there exists $\mu\in(2p, p_{s}^{*})$ such that

where $p_{s}^{*} = \frac{Np}{N-sp}$ and $F(x, t) = \int_{0}^{t}f(x, s)\text{d}s$.

In the following, let's state our result.

Theorem 1.1. Suppose that $(V_{1})$, $(V_{2})$, $(M_{1})$–$(M_{3})$ and $(f_{1})$–$(f_{3})$ are satisfied. Then there exists $\lambda_{0} > 0$ such that Eq $\rm(1.1)$ admits a nontrivial solution for all $\lambda > \lambda_{0}$.

The energy functional associated with Eq $\rm(1.1)$ is given by

Due to we assume that the nonlinearity $f(x, u)$ is superlinear only in a neighborhood of $u = 0$, the energy functional may be not well defined. Therefore, we can not directly use the variational method to prove the existence of solutions. In order to prove Theorem 1.1, we will use a truncation argument, which came from ^[8,13,16]. More precisely, we first show that the existence of nontrivial solutions for the revised equation via the mountain pass theorem. By Moser iteration method and $L^{\infty}$-estimate, we can obtain solution of revised equation, which is the solution of the original Eq $\rm(1.1)$ when $\lambda$ is sufficiently large. In addition, our assumptions on the nonlinearity $f$ just in a neighborhood of the origin, which are greatly relax. The results of this paper are new and can enrich the previous ones in the literature.

Remark 1.2. It is worth mentioning that this paper is the first time to assume the nonlinearity $f(x, u)$ just in a neighborhood at $u = 0$ and discuss existence of solutions for fractional Kirchhoff-type equations. Therefore, our results are new, and enrich the previous ones in the literature.

This paper is organized as follows. In Section 2, we recall some basic properties of the fractional Sobolev spaces and introduce a truncation argument. In Section 3, we give the proof of Theorem 1.1.

2.   Preliminaries

In this section, we first recall some basic results of the fractional Sobolev spaces. Let $0 < s < 1 < p < \infty$ be real numbers and $N > sp$. The fractional Sobolev space $W^{s, p}({\mathbb R}^N)$ is given by

where $[u]_{s, p}$ is the Gagliardo semi-norm, namely,

and $W^{s, p}({\mathbb R}^N)$ is equipped with the following norm

It is well known that $W^{s, p}({\mathbb R}^N) = \left(W^{s, p}({\mathbb R}^N), \|\cdot\|_{W^{s, p}({\mathbb R}^N)}\right)$ is a uniformly convex Banach space.

Moreover, let $p\in[1, \infty)$ be a real number, $L^{p}({\mathbb R}^N, V)$ denotes the Lebesgue space of real-valued functions and equipped with the norm

Under the condition $(V_{1})$, we can know that $L^{p}({\mathbb R}^N, V) = (L^{p}({\mathbb R}^N, V), \|\cdot\|_{p, V})$ is a uniformly convex Banach space.

Now, let $X$ denotes the completion of $C_{0}^{\infty}({\mathbb R}^N)$ in $W^{s, p}({\mathbb R}^N)$ and endowed with the norm

According to Lemma 10 in the Appendix of ^[25], we also know that $X = (X, \|\cdot\|_{X})$ is a uniformly convex Banach space. Furthermore, $X$ is a reflexive Banach space. The dual space of $(X, \|\cdot\|_{X})$ is denoted by $(X^{*}, \|\cdot\|_{X^{*}})$.

Denote the best fractional Sobolev constant:

Lemma 2.1. (^[25]) Assume that $V$ satisfies $(V_{1})$. If $r\in[p, p_{s}^{*}]$, then the embeddings

are continuous. Thus, there exists a constant $C_{r} > 0$ such that

Moreover, if $r\in[p, p_{s}^{*})$, then the embedding $X\hookrightarrow\hookrightarrow L^{r}(B_{R})$ is compact for any $R > 0$.

Lemma 2.2. (^[25]) Suppose that $(V_{1})$ and $(V_{2})$ are satisfied. Let $r\in[p, p_{s}^{*})$ be a fixed exponent. If $\{u_{n}\}$ is a bounded sequence in $X$, then there exists $u_{0}\in X\cap L^{r}({\mathbb R}^N)$ such that up to a subsequence,

as $n\rightarrow+\infty$, for all $r\in[p, p_{s}^{*})$.

Next, we show that the definitions of $(PS)_{c}$ condition and the mountain pass theorem.

Definition 2.3. Let $\mathcal{E}\in\mathcal{C}^{1}(X, {\mathbb R})$. We say the $\mathcal{E}$ satisfies the $(PS)_{c}$ condition at level $c\in {\mathbb R}$ in $X$, if any $(PS)_{c}$ sequence $\{u_{n}\}\subset X$, that is, $\mathcal{E}(u_{n})\rightarrow c$, $\mathcal{E}'(u_{n})\rightarrow0$ in $X^{*}$, as $n\rightarrow\infty$, admits a convergent subsequence in $X$.

Theorem 2.4. (^[32]) Let $X$ be a real Banach space, suppose $\mathcal{E}\in\mathcal{C}^{1}(X, {\mathbb R})$ satisfies the $(PS)_{c}$ condition with $\mathcal{E}(0) = 0$. Moreover,

(i) there exist $\varrho, \eta > 0$ such that $\mathcal{E}(u)\geq\eta$ for all $u\in X$, with $\|u\|_{X} = \varrho$,

(ii) there exists $e\in X$ satisfying $\varrho < \|e\|_{X}$ such that $\mathcal{E}(e) < 0$.

Define

Then

is a critical value of $\mathcal{E}(u)$.

In order to prove Theorem 1.1, we need to modify and extend $f$ to a suitable $\widetilde{f}$. The argument was developed in ^[8,13,16]. Based on this, we make the following truncation argument.

According to $(f_{1})$ and $(f_{2})$, we know that there exist two positive constants $A, B > 0$ such that

for $0 < t\leq2\delta$, where $\delta$ is a positive constant that satisfies $0 < \delta < \frac{1}{2}$ and $x\in {\mathbb R}^N$.

For fixed $\delta > 0$, let $d(t)\in\mathcal C^{1}({\mathbb R}, [0, 1])$ be an even cut-off function satisfying $td'(t)\leq0$,

and $|td'(t)|\leq\frac{2}{\delta}$ for $t\in {\mathbb R}$. Set

where

It follows from the definition of $d(t)$ and $(f_{1})$–$(f_{3})$ that $\widetilde{f}(x, t)$ has the following properties.

Lemma 2.5.(^[13,16]) If $(f_{1})$–$(f_{3})$ are satisfied, then

(i) there exists a constant $C > 0$ such that

(ii) it hold that

Finally, we introduced the well-known Simon inequality as follows, which will be used later.

Lemma 2.6.(^[28]) There exist constants $c_{p}, C_{p} > 0$ such that for any $x, y\in {\mathbb R}^N$, it holds

3.   Proof the Theorem 1.1

In this section, we will complete the proof of Theorem 1.1. We first prove that the existence of nontrivial solutions for the modified equation. More precisely, we consider the following fractional Kirchhoff type equation

where $\widetilde{f}$ is given by $\rm(2.5)$. The energy functional $\widetilde{I}_{\lambda}: X\rightarrow {\mathbb R}$ associated with Eq $\rm(3.1)$

where $\mathcal{M}(t) = \int_{0}^{t}M(\tau)\text{d}\tau$. Under the assumptions of Theorem 1.1, by using the similar proof method in ^[25], we can know that $\widetilde{I}_{\lambda}\in\mathcal{C}^{1}(X, {\mathbb R})$ and

for any $u, v\in X$. Obviously, the critical points of the energy functional $\widetilde{I}_{\lambda}$ are exactly the weak solutions of Eq $\rm(3.1)$.

In the following, let us first verify that $\widetilde{I}_{\lambda}$ has mountain pass geometry.

Lemma 3.1. Suppose that $(V_{1})$, $(V_{2})$, $(M_{1})$ and $(f_{1})$–$(f_{3})$ are satisfied. Then there exist $\zeta_{\lambda}, \rho_{\lambda} > 0$ such that $\widetilde{I}_{\lambda}(u)\geq \zeta_{\lambda}$ for any $u\in X$, with $\|u\|_{X} = \rho_{\lambda}$.

Proof. By means of $\alpha > 2p$ and Lemma 2.1, there exists $C_{\alpha} > 0$ such that $\|u\|_{L^{\alpha}({\mathbb R}^N)}\leq C_{\alpha}\|u\|_{X}$. From $(M_{1})$, $\rm(2.3)$, $\rm(3.2)$ and Lemma 2.5 $(i)$, we get

Since $\alpha > 2p$, we can find $\zeta_{\lambda}, \rho_{\lambda} > 0$ such that $\widetilde{I}_{\lambda}(u)\geq\zeta_{\lambda}$ for any $u\in X$, with $\|u\|_{X} = \rho_{\lambda}$. This completes the proof.

Lemma 3.2. Suppose that $(V_{1})$, $(V_{2})$, $(M_{1})$–$(M_{3})$ and $(f_{1})$–$(f_{3})$ are satisfied. Then there exists $\bar{u}\in X$ such that $\widetilde{I}_{\lambda}(\bar{u}) < 0$.

Proof. It follows from Lemma 2.5 $(ii)$ that

From $(M_{1})$–$(M_{3})$, there exists a constant $b > 0$ such that

For $u\in\mathcal{C}_{0}^{\infty}({\mathbb R}^N)$, with $\|u\|_{X} = 1$. According to $k > 2p$, $\rm(3.4)$ and $\rm(3.5)$, we get

Consequently, we can take $\bar{u} = tu\in X$ such that $\widetilde{I}_{\lambda}(\bar{u}) < 0$ for $t$ sufficiently large.

Now, similar to the proof of Lemma 6 in ^[25], we prove that $\widetilde{I}_{\lambda}$ satisfies $(PS)_{c}$ condition.

Lemma 3.3. Suppose that $(V_{1})$, $(V_{2})$, $(M_{1})$–$(M_{3})$ and $(f_{1})$–$(f_{3})$ are satisfied. Then $\widetilde{I}_{\lambda}$ satisfies the $(PS)_{c}$ condition.

Proof. Let $\{u_{n}\}_{n\in\mathbb{N}}\subset X$ be a $(PS)_{c}$ sequence for the functional $\widetilde{I}_{\lambda}$ at level $c\in {\mathbb R}$, that is,

Let us first prove that $\{u_{n}\}$ is bounded in $X$. For this purpose, we assume that $\|u_{n}\|_{X}\rightarrow+\infty$, as $n\rightarrow\infty$. By $(M_{1})$–$(M_{3})$, one has

Hence, it follows from $(M_{1})$, $\rm(3.2)$, $\rm(3.3)$, $\rm(3.7)$ and Lemma 2.5 $(ii)$ that

as $n\rightarrow\infty$. By $k > 2p$, we get a contradiction. Therefore, $\{u_{n}\}_{n\in\mathbb{N}}$ is bounded in $X$.

Next, we prove that there exists $u\in X$ such that $u_{n}\rightarrow u$ in $X$, as $n\rightarrow\infty$. Combining Lemma 2.1-Lemma 2.2 and Theorem A.1 in ^[32], there exist a subsequence, still denoted by $\{u_{n}\}$ and a function $u$ in $X$ such that

as $n\rightarrow\infty$. In view of Lemma 2.5 $(i)$ and Hölder's inequality, we have

Then $\rm(3.8)$ implies that

Now, let $u\in X$ be fixed and denote by $\mathcal{B}_{u}$ the linear functional on $X$ defined by

By Hölder's inequality and the definition of $\mathcal{B}_{u}(\varphi)$, it is easy to verify that $\mathcal{B}_{u}$ is continuous and

Obviously, $\mathcal{B}_{u}$ is bounded. Furthermore, according to $\left\{M([u_{n}]_{s, p}^{p})-M([u]_{s, p}^{p})\right\}$ is bounded in ${\mathbb R}$ and $\rm(3.8)$, we have

From $u_{n}\rightharpoonup u$ in $X$ and $\widetilde{I}_{\lambda}'(u_{n})\rightarrow0$ in $X^{*}$, we obtain $\langle\widetilde{I}_{\lambda}'(u_{n})-\widetilde{I}_{\lambda}'(u), u_{n}-u\rangle\rightarrow0$ as $n\rightarrow\infty$. Therefore, it follows from $\rm(3.3)$ and $\rm(3.8)$–$\rm(3.11)$ that

as $n\rightarrow\infty$. By the fact that $M([u_{n}]_{s, p}^{p})[\mathcal{B}_{u_{n}}(u_{n}-u)-\mathcal{B}_{u}(u_{n}-u)]\geq0$ and $V(x)(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u)(u_{n}-u)\geq0$ for all $n\in\mathbb{N}$ by convexity. We get

Finally, we divided into two cases to prove $\|u_{n}-u\|_{X}\rightarrow0$, as $n\rightarrow\infty$. To this aim, we first assume that $p\geq2$. In view of Lemma 2.6, $\rm(3.10)$ and $\rm(3.14)$, we have

as $n\rightarrow\infty$. By $(V_{1})$, $\rm(3.13)$ and Lemma 2.6, we can show that

as $n\rightarrow\infty$. Hence, combining $\rm(3.15)$ with $\rm(3.16)$, we have $\|u_{n}-u\|_{X}\rightarrow0$ as $n\rightarrow\infty$.

In the end, we assume that $1 < p < 2$. Let us introduce the following elementary inequality

From $\rm(3.8)$, there exists a constant $l > 0$ such that $[u_{n}]_{s, p}\leq l$ for all $n\in\mathbb{N}$. According to Hölder's inequality, $\rm(3.10)$, $\rm(3.14)$ and $\rm(3.17)$, we have

as $n\rightarrow\infty$, where $\overline{C} = 2C_{p}l^{\frac{p(2-p)}{2}}$. Similarly, it follows from $\rm(3.8)$ that there exists a constant $l_{*} > 0$ such that $\|u_{n}\|_{p, V}\leq l_{*}$ for all $n\in\mathbb{N}$. Therefore, we conclude from Hölder's inequality and $\rm(3.13)$ that

as $n\rightarrow\infty$, where $\widehat{C} = 2C_{p}l_{*}^{\frac{p(2-p)}{2}}$. Consequently, we have $\|u_{n}-u\|_{X}\rightarrow0$, as $n\rightarrow\infty$. This concludes the proof of Lemma 3.3.

Next, we prove that the existence of nontrivial solutions for Eq $\rm(3.1)$ via the mountain pass theorem.

Theorem 3.4. Suppose that $(V_{1})$, $(V_{2})$, $(M_{1})$–$(M_{3})$ and $(f_{1})$–$(f_{3})$ are satisfied. Then Eq $\rm(3.1)$ has a nontrivial solution.

Proof. By Lemma 3.1 and Lemma 3.2, we know that the functional $\widetilde{I}_{\lambda}$ satisfies the geometry of the mountain pass theorem. Moreover, in view of Lemma 3.3 and $\widetilde{I}_{\lambda}(0) = 0$, we know that $\widetilde{I}_{\lambda}$ satisfies all the condition of Theorem 2.4. Thus, we can see that $\widetilde{I}_{\lambda}$ has a critical value $c_{\lambda}$ and

where

and $\bar{u}$ is given by Lemma 3.2. Therefore, the Eq $\rm(3.1)$ has a nontrivial solution $u_{\lambda}\in X$ with $\widetilde{I}_{\lambda}(u_{\lambda}) = c_{\lambda}$.

From the truncation argument in Section 2, we know that if the solution $u_{\lambda}$ of Eq $\rm(3.1)$ satisfies $\|u_{\lambda}\|_{L^{\infty}({\mathbb R}^N)}\leq\delta$, then $u_{\lambda}$ is a nontrivial solution of the original Eq $\rm(1.1)$. Before we do that, we will use the following two lemmas to find an uniform boundedness of $\|u_{\lambda}\|_{X}^{p}$ and the critical level $c_{\lambda}$.

Lemma 3.5. Let $u_{\lambda}$ be a nontrivial solution of Eq $\rm(3.1)$. Then there exists a constant $\Pi > 0$ such that

Proof. By $u_{\lambda}$ is a critical point of $\widetilde{I}_{\lambda}$, $(M_{1})$, $k > 2p$, Lemma 2.5 $(ii)$ and $\rm(3.7)$, we get

Thus, there exists $\Pi > 0$ is a constant such

The proof is completed.

Lemma 3.6. There exists a constant $\Theta > 0$ independent of $\lambda$ such that

for all sufficiently large $\lambda$.

Proof. In view of $\rm(3.4)$, $\rm(3.5)$ and Theorem 3.4, we have

for all sufficiently large $\lambda$, where $\bar{u}$ is given by Lemma 3.2 and $\Theta > 0$ is a constant.

Now, similar to the proof of Theorem 1.1 in ^[1], we use the Moser iteration technique to estimate $\|u_{\lambda}\|_{L^{\infty}({\mathbb R}^N)}$.

Lemma 3.7. If $u\in X$ is a nontrivial solution of Eq $\rm(3.1)$, then $u\in L^{\infty}({\mathbb R}^N)$. In addition, there exists a constant $\Xi > 0$ such that

for all $\lambda$ sufficiently large.

Proof. Let $L > 0$, we define

and

where $\beta > 1$ and $u_{L} = \min\{u, L\}$. Hence, we conclude from the functions introduced above that

By the definition of $\xi(u)$, we have $\left|uu_{L}^{p(\beta-1)}\right|\leq L^{p(\beta-1)}u$ in ${\mathbb R}^N$. Thus, $\xi(u)\in X$. Considering $\xi(u)$ as a test function in $\rm(3.1)$, we get

Now, we define

From $\rm(2.2)$ that

Hence, it follows from $(M_{1})$ and $\rm(3.18)$–$\rm(3.21)$ that

Assume first that $\frac{m_{0}}{\beta^{p}} < 1$. According to Lemma 2.5 $(i)$, $\rm(3.20)$–$\rm(3.22)$ and Hölder's inequality, we have

Since $\alpha\in(2p, p_{s}^{*})$ and $p > 1$, by simple calculation, we get

From Lemma 3.5 and Sobolev inequality, we obtain

It follows from the definition of $w_{L}$, we can show that $u_{L}\leq u$ in ${\mathbb R}^N$. By $\rm(3.23)$–$\rm(3.25)$, we have

where $C_{\lambda} = \frac{\lambda}{m_{0}}\left(S_{s, p}^{*}\right)^{-\frac{\alpha}{p}}\left(\Pi c_{\lambda}\right)^{\frac{\alpha-p}{p}}$. Letting $L\rightarrow+\infty$ in $\rm(3.26)$ and using the Fatou's Lemma, we have

Thus, we get

Taking $\vartheta = \frac{p_{s}^{*}}{\nu_{s}^{*}} > 1$ and $\beta = \vartheta$ in $\rm(3.27)$, we obtain

Taking $\beta = \vartheta^{2}$ in $\rm(3.27)$, we get

Combining $\rm(3.29)$ with $\rm(3.30)$, one has

We proceed the $i$ times iterations, by taking $\beta = \vartheta^{i}$, $i = 1, 2, \ldots,$. Then, by letting $i\rightarrow\infty$, we have

Next, we assume that $\frac{m_{0}}{\beta^{p}}\geq1$. Similarly, we get

Therefore, by $u\in L^{ p_{s}^{*}}({\mathbb R}^N)$, we have $u\in L^{\infty}({\mathbb R}^N)$.

Finally, by employing $\rm(3.32)$, $\rm(3.33)$, Lemma 3.6 and the Sobolev inequality, there exists a constant $\Xi > 0$ such that

for all sufficiently large $\lambda$. The lemma is now proved.

Proof of Theorem 1.1. Since $k, \alpha\in(2p, p_{s}^{*})$, we obtain

Hence, in view of $\rm(3.34)$ and Lemma 3.7, there exists $\lambda_{0} > 0$ such that

for all $\lambda > \lambda_{0}$, where $\delta$ is fixed in $\rm(2.4)$. Therefore, $u_{\lambda}$ is a nontrivial solution of Eq $\rm(1.1)$ for $\lambda > \lambda_{0}$.

4.   Conclusions

In this paper, we have considered a class of fractional Kirchhoff type equation. Under suitable assumptions on $V$ and $M$, using a truncation argument and the mountain pass theorem, we have established the existence of nontrivial solutions. It is expected that the results proved in this paper may be starting point further research in this field.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11661053, 11771198, 11901276 and 11961045) and supported by the Provincial Natural Science Foundation of Jiangxi, China (Nos. 20181BAB201003, 20202BAB201001 and 20202BAB211004). Authors are grateful to the referees for their very constructive comments and valuable suggestions.

Conflict of interest

The authors declare that they have no competing interests.