A nonlocal concave-convex problem with nonlocal mixed boundary data
  The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data. 
Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci
math.AP