Regularizing effect of absorption terms in singular problems

Abstract

We prove existence of solutions to problems whose model is $$\{\begin{array}{ll}-{\Delta }_{p}u+u^q=\frac{f}{u^{\gamma }}& \text{in}\Omega ,\\ u\ge 0& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$), ${\Delta }_{p}u$ is the $p$-laplacian operator for $1\le p<N$, $q>0$, $\gamma \ge 0$ and $f$ is a nonnegative function in $L^m(\Omega )$ for some $m\ge 1$. 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 $\gamma >1$.