Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

Abstract

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 method in nonlinear Potential Theory.