This paper deals with a class of singular problems involving the fractional \(p(x,\cdot )\)-Laplace operator of the form

where \(\Omega \) is a smooth bounded domain in \({\mathbb {R}}^N\) (\(N\ge 3\)), \(0<s<1\), \(\lambda \) is a positive parameter and \(\gamma : {\mathbb {R}}^N \longrightarrow (0,1)\) is a continuous function, \(p:\; {\mathbb {R}}^{2N} \longrightarrow \;(1,\infty )\) is a bounded, continuous and symmetric function, \(q: {\mathbb {R}}^N \longrightarrow (1,\infty )\) is a continuous function. Using the direct method of minimization combined with the theory of fractional Sobolev spaces with variable exponents, we prove that the problem has one positive solution for \(\lambda >0\) small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional \(p(x,\cdot )\)-Laplace operators.

本文处理一类涉及分数\(p(x,\cdot)\)-拉普拉斯算子形式的奇异问题

其中\(\Omega \)是\({\mathbb {R}}^N\) ( \(N\ge 3\) ), \(0<s<1\) , \(\ lambda \)是一个正参数，\(\gamma : {\mathbb {R}}^N \longrightarrow (0,1)\)是一个连续函数，\(p:\; {\mathbb {R}}^ {2N} \longrightarrow \;(1,\infty )\)是一个有界连续对称函数，\(q: {\mathbb {R}}^N \longrightarrow (1,\infty )\)是一个连续的功能。使用直接最小化法结合变指数分数索博列夫空间理论，我们证明该问题对于\(\lambda >0\)有一个正解足够小。据我们所知，本文是研究涉及分数\(p(x,\cdot )\) -Laplace 算子的奇异问题的首批尝试之一。