Inserted: 26 apr 2016
Last Updated: 16 nov 2016
Journal: Math. Ann.
We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron's Method in nonlinear Potential Theory.
Keywords: obstacle problem, fractional Sobolev spaces, quasilinear nonlocal operators, Caccioppoli estimates, nonlocal tail, comparison estimates, Fractional Superharmonic functions, Perron Method