Multiplicity of solutions for fractional $q(.)$-Laplacian equations

Abstract

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 $.