Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations

In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity:

$$\{\begin{array}{l}M({\int }_{{\mathbb{R}}^N}|(-\mathrm{△}{)}^{\frac{\alpha }{2}}u{|}^{2}dx)(-\mathrm{△}{)}^{\alpha }u+\lambda V(x)u=f(x,u)\text{in}{\mathbb{R}}^N,\\ u\in H^{\alpha }({\mathbb{R}}^N),N\ge 1,(1.1)\end{array}$$

where $\lambda >0$ is a parameter, $a$ and $b$ are positive numbers satisfying $M(t)=am(t)+b$, $m:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is continuous. $V:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is continuous. $f$ satisfies $\underset{|t|\to \mathrm{\infty }}{lim}f(x,t)/|t{|}^{k-1}=Q(x)$ uniformly in $x\in {\mathbb{R}}^N$ for each $2<k<2_{\alpha }^{\ast },(2_{\alpha }^{\ast }=\frac{2N}{N-2\alpha })$. We investigated the effects of functions $m$ and $Q$ on the solution. By applying the variational method, we obtain the existence of multiple solutions. Furthermore, it is worth mentioning that the ground state solution has also been obtained.