In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type (P) (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in \(W_{loc}^{s,p}(\Omega )\) via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for \(0 \le \delta < 1+s- \frac{1}{p}\) and furthermore the nonexistence of weak solution for \(\delta \ge sp.\) Moreover, by virtue of barrier arguments we study the behavior of weak solutions in terms of distance function. Consequently, we prove Hölder regularity up to the boundary and optimal Sobolev regularity for weak solutions.

在本文中，我们研究了（P）型非线性分数椭圆问题（见下文）的弱解的存在性，唯一性，不存在性和正则性，该问题涉及光滑有界域中的奇异非线性和奇异权重。我们通过逼近方法证明了\（W_ {loc} ^ {s，p}（\ Omega）\）中弱解的存在。建立独立的利益的一个新的比较原理，我们证明弱溶液的用于唯一\（0 \文件\增量<1个+ S- \压裂{1} {P} \）并且此外弱溶液为不存在\（\三角洲\ ge sp。\）此外，借助障碍论证，我们根据距离函数研究了弱解的行为。因此，我们证明了直至边界的Hölder正则性和弱解的最佳Sobolev正则性。