Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

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