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

Abstract

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

$$\begin{aligned}\begin{aligned} \left\{ \begin{array}{ll} -\textrm{div}\left( \frac{\vert \nabla u\vert ^{p-2}\nabla u}{(1+|u|)^{\gamma }}\right) +\frac{\vert \nabla u\vert ^{p}}{u^{\theta }}=\frac{u^{r}}{\vert x\vert ^{p}}+f &{}\text{ in }\ \Omega , \\ u>0&{} \text{ in }\ \Omega , \\ u=0&{} \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}\end{aligned}$$

where \(\Omega \) is a bounded open subset in \({\mathbb {R}}^{N}\) with \(0\in \Omega \), \(\gamma \ge 0\), \(1<p<N\), \(0<\theta <1\), and \(0<r<p-\theta \). We prove existence and regularity results for solutions under various hypotheses on the datum f.