The Dirichlet problem for nonlocal elliptic equations.

We study a class of nonlocal elliptic equations (− Δ) α / 2 ρ (x) − b (x) ⋅ ∇ ρ (x) = f (x) on a bounded domain Ω ⊂ R d . For f ∈ L q (Ω) ( q > d / α), by the Lax–Milgram theorem and the De Giorgi iteration, we prove the existence and uniqueness of L ∞ solution. Furthermore, we investigate the existence of densities for measure-valued solutions to nonhomogeneous measure-valued nonlocal elliptic equations. [ABSTRACT FROM AUTHOR]