  We study the existence of large solutions for nonlocal Dirichlet problems
posed on a bounded, smooth domain, associated to fully nonlinear elliptic
equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with
subcritical power $0<p<2s$.
  Due to the nonlocal nature of the diffusion, new blow-up phenomena arise
within the range $0<p<2s$, involving a continuum family of solutions and/or
solutions blowing-up to $-\infty$ on the boundary. This is in striking
difference with the local case studied by Lasry-Lions for the case subquadratic
case $1<p<2$.
