Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -\Delta_p u+ \frac{a(x)}{u^{\gamma}}=\mu f(x) \ &\text{ in }\Omega, \newline u>0&\text{ in }\Omega, \newline u = 0 \ &\text{ on } \partial\Omega, \end{cases} \end{equation} where $\Omega$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $\Delta_p u$ is the $p$-Laplacian with $p>1$, $0<\gamma\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $f\in L^{1}(\Omega)$ we show that problem \eqref{eintro} is solvable for any $\mu >\mu_0>0$, for some $\mu_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<\mu<\mu_{0*}$, for some small $\mu_{0*}$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $\gamma=1$).