Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity

Abstract In this paper, when studying the connection between the fractional convexity and the fractional p-Laplace operator, we deduce a nonlocal and nonlinear equation. Firstly, we will prove the existence and uniqueness of the viscosity solution of this equation. Then we will show that u ⁢ ( x ) {u(x)} is the viscosity sub-solution of the equation if and only if u ⁢ ( x ) {u(x)} is so-called ( α , p ) {(\alpha,p)} -convex. Finally, we will characterize the viscosity solution of this equation as the envelope of an ( α , p ) {(\alpha,p)} -convex sub-solution. The technique involves attainability of the exterior datum and a comparison principle for the nonlocal and nonlinear equation.