Regularizing effect of absorption terms in singular problems

We prove existence of solutions to problems whose model is $$\{\begin{array}{ll}{\displaystyle -{\mathrm{\Delta }}_{p}u+u^q=\frac{f}{u^{\gamma }}}& \text{in }\mathrm{\Omega },\\ u\ge 0& \text{in }\mathrm{\Omega },\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$), ${\mathrm{\Delta }}_{p}u$ is the $p$-laplacian operator for $1\le p<N$, $q>0$, $\gamma\ge 0$and$f$isanonnegativefunctionin$L^m(\Omega)$forsome$m\ge1$. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case $\gamma\le 1$. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local $W^{1,p}$-solutions in case $\gamma >1$.