Study of the Existence of Supersolutions for Nonlocal Equations with Gradient Terms

We study the existence of positive supersolutions of nonlocal equations of type $$(-\Delta)^{s} u + |\Delta u|^{q} = \lambda f(u)$$ posed in exterior domains where the datum f can be comparable with $$u^p$$ near the origin. We prove that the existence of bounded supersolutions depends on the values of p, q and s.