Regularity for non-linear integro-differential equations

The primary objective of this paper is to investigate regularity criteria for nonlinear equations governed by integro-differential operators. To achieve this goal, we utilize tools from potential analysis, including fractional relative Sobolev capacities, Wiener-type integrals, Wolff potentials, (s, p)-barriers, and (s, p)-balayages. Our approach begins with establishing the characterizations of fractional thinness and Perron boundary regularity. Subsequently, we present a generalized fractional Wiener criterion. Additionally, we demonstrate the continuity of fractional superharmonic functions, fractional resolutivity, the relationship between (s, p)-potentials and (s, p)-Perron solutions, and the existence of a capacitary function for any arbitrary condenser.