Inserted: 2 sep 2015
Last Updated: 2 sep 2015
We prove a weighted fractional inequality involving the solution $u$ of a nonlocal semilinear problem in $\mathbb R^n$. Such inequality bounds a weighted $L^2$-norm of a compactly supported function $\phi$ by a weighted $H^s$-norm of $\phi$. In this inequality a geometric quantity related to the level sets of $u$ will appear. As a consequence we derive some relations between the stability of $u$ and the validity of fractional Hardy inequalities.