Some evaluations of the fractional $ p $-Laplace operator on radial functions

Abstract

We face a rigidity problem for the fractional $ p $-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $ (-\Delta)^s(1-|x|^{2})^s_+ $ and $ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $ are constant functions in $ (-1, 1) $ for fixed $ p $ and $ s $. We evaluated $ (-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+ $ proving that it is not constant in $ (-1, 1) $ for some $ p\in (1, +\infty) $ and $ s\in (0, 1) $. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas.