Solutions for fractional $p(x,\cdot )$-Kirchhoff-type equations in $\mathbb{R}^{N}$

In this paper, we discuss the fractional $p(x,\cdot )$ -Kirchhoff-type equations $$ M\left (\int _{\mathbb{R}^{N}\times \mathbb{R}^{N}} \frac{1}{p(x,y)} \frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}dxdy\right )(-\Delta _{p(x,.)})^{s} u+|u|^{\bar{p}(x)-2}u=f(x,u).$$ We weaken the conditions on the nonlinear term f and get the existence and multiplicity of solutions via variational methods, which improves some previous results.