In this paper, we give some properties of the new fractional Sobolev spaces with variable exponents and apply them to study a class of eigenvalue problems involving the fractional p(·)-Laplace operator. We obtain sequences of eigenvalues going asymptotically to infinity and we also establish sufficient conditions to get zero value for the principal eigenvalue, which is a striking difference between the variable exponent case and the constant exponent case. As an application, we obtain several existence and nonexistence results for the eigenvalue problem according to the asymptotic growth of the nonlinearity and the range of the spectral parameter.

中文翻译：

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关于分数p（·）-Laplacian特征值问题的评论

在本文中，我们给出了具有可变指数的新分数Sobolev空间的一些性质，并将其用于研究一类涉及分数p（·）-Laplace算子的特征值问题。我们获得了渐近到无穷大的特征值序列，并且还建立了足够的条件来使主特征值获得零值，这是可变指数情况与恒定指数情况之间的显着差异。作为应用，我们根据非线性的渐近增长和谱参数的范围，获得了特征值问题的若干存在和不存在结果。