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

Abstract

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \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 $p>1$, $0<\gamma \le 1$, and $a\ge 0$ 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 $0<\mu <{\mu }_{0\ast }$, for some small ${\mu }_{0\ast }$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\ne 2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $\gamma =1$).