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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 thatu⁢(x){u(x)}is the viscosity sub-solution of the equation if and only ifu⁢(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.

advances in calculus of variations

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