The impact of a singular first-order term in some degenerate elliptic equations involving Hardy potential

In this paper, we study the regularizing effects of a singular first-order term in some degenerate elliptic equations with zero-order term involving Hardy potential. The model problem is \(\begin{aligned}\begin{aligned} \left\{ \begin{array}{ll} -\textrm{div}\left( \frac{\vert \nabla u\vert ^{p-2}\nabla u}{(1+|u|)^{\gamma }}\right) +\frac{\vert \nabla u\vert ^{p}}{u^{\theta }}=\frac{u^{r}}{\vert x\vert ^{p}}+f &{}\text{ in }\ \Omega , \\ u>0&{} \text{ in }\ \Omega , \\ u=0&{} \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}\end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}-\text{div}\left(\frac{{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u}{{(1+|u|)}^{\gamma }}\right)+\frac{{|\mathrm{\nabla }u|}^p}{u^{\theta }}=\frac{u^r}{{|x|}^p}+f\hfill & \text{in}\mathrm{\Omega },\hfill \\ u>0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ with \(0\in \Omega \) $0\in \mathrm{\Omega }$ , \(\gamma \ge 0\) $\gamma \ge 0$ , \(1<p<N\) $1<p<N$ , \(0<\theta <1\) $0<\theta <1$ , and \(0<r<p-\theta \) $0<r<p-\theta$ . We prove existence and regularity results for solutions under various hypotheses on the datum f.