Multiplicity of solutions for fractional $q(\mathrm{.})$-Laplacian equations

In this paper, we deal with the following elliptic type problem $$\{\begin{array}{ll}(-\mathrm{\Delta }{)}_{q(\mathrm{.})}^{s(\mathrm{.})}u+\lambda Vu=\alpha {\mid u\mid }^{p(\mathrm{.})-2}u+\beta {\mid u\mid }^{k(\mathrm{.})-2}u& \text{ in }\mathrm{\Omega },\\ u=0& \text{ in }{\mathbb{R}}^n\mathrm{\backslash }\mathrm{\Omega },\end{array}$$ where $q(.):\overline{\Omega}\times \overline{\Omega}\rightarrow $\mathbb{R}$$isameasurablefunctionand$s(.):$\mathbb{R}$^n\times $\mathbb{R}$^n\rightarrow (0,1)$ is a continuous function, $n>q(x,y)s(x,y)$ for all $(x,y)\in \mathrm{\Omega }\times \mathrm{\Omega }$, $(-\mathrm{\Delta }{)}_{q(\mathrm{.})}^{s(\mathrm{.})}$ 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 $\lambda \to +\mathrm{\infty }$.