Embedding theorems for variable exponent fractional Sobolev spaces and an application

• Published in: AIMS mathematics Vol. 6; no. 9; pp. 9835 - 9858
• Main Authors: Liu, Haikun, Fu, Yongqiang
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: embedding; fractional sobolev spaces; s(x)-p(x)-laplacian equation; variable exponent
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021571

$ (-\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.