This article deals with the study of the following singular quasilinear equation:

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with \(C^2\) boundary \(\partial \Omega \), \(1< q< p<\infty \), \(\delta >0\) and \(f\in L^\infty _{loc}(\Omega )\) is a non-negative function which behaves like \(\text {dist}(x,\partial \Omega )^{-\beta },\) \(\beta \ge 0\) near the boundary of \(\Omega \). We prove the existence of a weak solution in \(W^{1,p}_{loc}(\Omega )\) and its behaviour near the boundary for \(\beta <p\). Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case \(\beta <2-\frac{1}{p}\). Subsequently, for the case \(\beta \ge p\), we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see (1.6)). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of (P) for the case \(\beta +\delta \ge 1\) that has not been fully answered in former contributions even for p-Laplace operators.

本文涉及对以下奇异拟线性方程的研究：

其中\(\Omega \)是\({\mathbb {R}}^N\) 中的有界域，具有\(C^2\)边界\(\partial \Omega \)，\(1< q< p <\infty \) , \(\delta >0\)和\(f\in L^\infty _{loc}(\Omega )\)是一个非负函数，其行为类似于\(\text {dist} (x,\partial \Omega )^{-\beta },\) \(\beta \ge 0\)靠近\(\Omega \)的边界。我们证明了\(W^{1,p}_{loc}(\Omega )\) 中弱解的存在及其在\(\beta <p\)边界附近的行为. 因此，我们获得了弱解的最优 Sobolev 正则性。通过建立比较原理，证明了\(\beta <2-\frac{1}{p}\)情况下弱解的唯一性。随后，对于\(\beta \ge p\) 的情况，我们证明了不存在的结果。此外，我们证明了一类更一般的涉及奇异非线性和低阶项的拟线性方程的弱解梯度的 Hölder 正则性（见（1.6））。这个结果是全新的并且具有独立意义。除此之外，我们证明了 ( P )的最小弱解的 Hölder 正则性，在之前的贡献中甚至对于p也没有完全回答的情况\(\beta +\delta \ge 1\)-拉普拉斯运算符。