In this paper, we mainly discuss the embedding theory of variable exponent fractional Sobolev space $ W^{s(\cdot), p(\cdot)} (\Omega) $, and apply this theory to study the $ s(x) $-$ p(x) $-Laplacian equation:
$ (-\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.
Citation: Haikun Liu, Yongqiang Fu. Embedding theorems for variable exponent fractional Sobolev spaces and an application[J]. AIMS Mathematics, 2021, 6(9): 9835-9858. doi: 10.3934/math.2021571
In this paper, we mainly discuss the embedding theory of variable exponent fractional Sobolev space $ W^{s(\cdot), p(\cdot)} (\Omega) $, and apply this theory to study the $ s(x) $-$ p(x) $-Laplacian equation:
$ (-\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.
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Haikun Liu, Yongqiang Fu. Embedding theorems for variable exponent fractional Sobolev spaces and an application[J]. AIMS Mathematics, 2021, 6(9): 9835-9858. doi: 10.3934/math.2021571
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