A family of nonlocal degenerate operators: maximum principles and related properties

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases.