A new fractional function space Xk(⋅),α(⋅)(Ω) with variable exponents k, α and its relaxed properties are established in this paper. Under this configuration, the comparison principle of the fractional α(⋅)-Laplace operator and a priori estimate of weak solutions for a variable-order fractional α(⋅)-Laplacian equation are obtained. Then, the application of a fixed point theorem ensures the existence of weak solutions for a nonlinear elliptic equation with the variable-order fractional α(⋅)-Laplacian.

中文翻译：

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变阶分数 Sobolev 空间和变指数非线性椭圆方程

本文建立了一个新的具有变指数k、α及其松弛性质的分数函数空间Xk(⋅),α(⋅)(Ω)。在此配置下，得到分数阶α(⋅)-拉普拉斯算子的比较原理与变阶分数阶α(⋅)-拉普拉斯方程弱解的先验估计。然后，不动点定理的应用确保了具有变阶分数 α(⋅)-Laplacian 的非线性椭圆方程的弱解的存在性。