Regularizing effect of lower order terms in nonlinear Dirichlet problems with singularity

We study the regularizing effect of two lower order terms to nonlinear Dirichlet problems with singularity. The simplest example model is \(\begin{aligned} {\left\{ \begin{array}{ll} -\operatorname {div}\left( \left( a(x)+|u|^{q}\right) | \nabla u|^{p-2} \nabla u \right) +u^{s}=\frac{f}{u^{\theta }} & \text{ in } \varOmega \\ u>0 & \text{ in } \varOmega \\ u=0 & \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-div\left(\left(a,(x),+,{|u|}^q\right),{|\mathrm{\nabla }u|}^{p-2},\mathrm{\nabla },u\right)+u^s=\frac{f}{u^{\theta }}\hfill & \mathrm{in}\Omega \hfill \\ u>0\hfill & \mathrm{in}\Omega \hfill \\ u=0\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right)\end{array}$$ where \(\varOmega \) $\Omega$ is a bounded open set of \(\mathbb {R}^{N}(N \ge 2),\) ${\mathbb{R}}^N(N\ge 2),$ \(1<p<N,\) $1<p<N,$ \(\theta >0,\) $\theta >0,$ \(q>0,\) $q>0,$ \(s\ge 1\) $s\ge 1$ and f is a nonnegative function belonging to a suitable Lebesgue space. In particular, we will prove how the presence of lower order terms may lead to an improvement of the summability of the solutions.