Comparison results for a nonlocal singular elliptic problem

Abstract

We prove comparison results for the solution $u$ to the following nonlocal singular problem $$\begin{array}{rlrl}{(-\mathrm{\Delta })}^{s}u& ={\displaystyle \frac{f}{u^{\gamma }}}& & \text{in }\mathrm{\Omega }\\ u& >0& & \text{in }\mathrm{\Omega }\\ u& =0& & \text{in }{\mathbb{R}}^n\setminus \mathrm{\Omega }.\end{array}$$

Here $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$, $0<s<1$, $\gamma >0$ and $f\in L^1(\mathrm{\Omega })$, $f\ge 0$. Some interesting consequences are $L^p$ regularity results and energy estimates for $u$ depending on the value of $\gamma$.

This talk is based on joint work with I. De Bonis, V. Ferone and B. Volzone.