  In this paper, we deal with the following elliptic type problem $$
\begin{cases}
  (-\Delta)_{q(.)}^{s(.)}u + \lambda Vu = \alpha \left\vert
u\right\vert^{p(.)-2}u+\beta \left\vert u\right\vert^{k(.)-2}u & \text{ in
}\Omega,
  \\[7pt]
  u =0 & \text{ in }\mathbb{R}^{n}\backslash \Omega ,
  \end{cases} $$ where $q(.):\overline{\Omega}\times
\overline{\Omega}\rightarrow \mathbb{R}$ is a measurable function and
$s(.):\mathbb{R}^n\times \mathbb{R}^n\rightarrow (0,1)$ is a continuous
function, $n>q(x,y)s(x,y)$ for all $(x,y)\in \Omega \times \Omega $,
$(-\Delta)_{q(.)}^{s(.)}$ is the variable-order fractional Laplace operator,
and $V$ is a positive continuous potential. Using the mountain pass category
theorem and Ekeland's variational principle, we obtain the existence of a least
two different solutions for all $\lambda>0$. Besides, we prove that these
solutions converge to two of the infinitely many solutions of a limit problem
as $\lambda \rightarrow +\infty $.
