Existence and multiplicity of solutions for fractional $p(x)$-Kirchhoff-type problems

In this paper, we deal with the existence and multiplicity of solutions for fractional $p(x)$-Kirchhoff-type problems as follows:

$$\{\begin{array}{l}M({\int }_Q\frac{1}{p(x,y)}\frac{|v(x)-v(y){|}^{p(x,y)}}{|x-y{|}^{d+sp(x,y)}}dxdy)(-{\mathrm{\Delta }}_{p(x)}{)}^{s}v(x)\\ =\lambda |v(x){|}^{r(x)-2}v(x),\text{in}\mathrm{\Omega },\\ v=0,\text{in}{\mathbb{R}}^d\mathrm{\setminus }\mathrm{\Omega },\end{array}$$

where $(-{\mathrm{△}}_{p(x)}{)}^s$ is the fractional $p(x)$-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of $M$ and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.