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

Abstract

In this paper, we deal with the following elliptic type problem $$\{\begin{array}{ll}(-\Delta {)}_{q(.)}^{s(.)}u+\lambda Vu=\alpha {\left|u\right|}^{p(.)-2}u+\beta {\left|u\right|}^{k(.)-2}u& \text{ in }\Omega ,\\ u=0& \text{ in }{\mathbb{R}}^n\backslash \Omega ,\end{array}$$ where $q(.):\overline{\Omega }\times \overline{\Omega }\to \mathbb{R}$ is a measurable function and $s(.):{\mathbb{R}}^n\times {\mathbb{R}}^n\to (0,1)$ 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 $\lambda \to +\infty$.