Abita Rahmoune, Umberto Biccari. **Multiplicity of solutions for fractional q(.)-Laplacian equations.** (2021)

**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 .

Arxiv: 2103.12600