The combined effects of two singular nonlinearities in some degenerate elliptic equations

In this article, we deal with a class of nonlinear degenerate elliptic equations involving singular nonlinearities of the form \(\begin{aligned}\begin{aligned} \left\{ \begin{array}{ll} -\textrm{div}\left( \dfrac{\vert \nabla u\vert ^{p-2}\nabla u}{(1+u)^{\lambda }}\right) +\dfrac{\vert \nabla u\vert ^{p}}{u^{\theta }}=\dfrac{f}{u^{\gamma }}& \text{ in }\ \Omega ,\\ u>0& \text{ in }\ \Omega , \\ u=0& \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}-\text{div}\left({\displaystyle \frac{{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u}{{(1+u)}^{\lambda }}}\right)+{\displaystyle \frac{{|\mathrm{\nabla }u|}^p}{u^{\theta }}}={\displaystyle \frac{f}{u^{\gamma }}}\hfill & \text{in}\mathrm{\Omega },\hfill \\ u>0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset of \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ , \(N\ge 2\) $N\ge 2$ , \(1<p<N\) $1<p<N$ , \(\lambda \ge 0\) $\lambda \ge 0$ , \(0<\theta <1\) $0<\theta <1$ and \(0<\gamma \le 1\) $0<\gamma \le 1$ . We study the interaction between two regularizing singular terms and we show that their combined effect improves the regularity of the solutions under various hypotheses on the nonnegative datum \(f\in L^{m}(\Omega )\) $f\in L^m(\mathrm{\Omega })$ , with \(m\ge 1\) $m\ge 1$ .