Embedding theorems for variable exponent fractional Sobolev spaces and an application

$ (-\varDelta)_{p(\cdot)}^{s(\cdot)}u+V(x)|u|^{p(x)-2}u = f(x,u)+g(x) $ where $ x\in\Omega\subset \mathbb{R}^n $, $ (-\varDelta)_{p(\cdot)}^{s(\cdot)} $ is $ s(x) $-$ p(x) $-Laplacian operator with $ 0 < s(x) < 1 < p(x) < \infty $ and $ p(x)s(x) < n $, the nonlinear term $ f: \Omega \times \mathbb{R} \to \mathbb{R} $ is a Carathéodory function, $ V:\mathbb{R}^n\to \mathbb{R} $ is a potential function and $ g:\mathbb{R}^n\to \mathbb{R} $ is a perturbation term