We prove comparison results for the solution \(u\) to the following nonlocal singular problem \begin {align*} \left (-\Delta \right )^s u &=\dfrac {f}{u^\gamma } &&\text {in }\Omega \\ u&>0 && \text {in } \Omega \\ u&=0 && \text {in }\mathbb R^n\setminus \Omega . \end {align*}
Here \(\Omega \) is a bounded domain in \(\mathbb R^n\), \(0<s<1\), \(\gamma >0\) and \(f \in L^1(\Omega )\), \(f \ge 0\). 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.