In this paper, we consider a class of fractional Laplacian problems involving fractional \(q_{i}(x)\)-laplacian operators \( (i=1:2)\), and a singular nonlinearity. By using variational methods and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces, we prove the existence of solutions for such problems. An illustrative example is presented to validate the main results of this paper.

中文翻译：

---

涉及分数q(x,.)-拉普拉斯算子的奇异双相基尔霍夫型问题解的存在性

在本文中，我们考虑一类分数拉普拉斯问题，涉及分数\(q_{i}(x)\) -拉普拉斯算子\( (i=1:2)\)和奇异非线性。通过使用变分方法和单调性论证并结合广义Lebesgue Sobolev空间理论，我们证明了此类问题解的存在性。给出了一个说明性例子来验证本文的主要结果。