On the Liouville property for fully nonlinear equations with superlinear first-order terms

We consider in this note one-side Liouville properties for viscos- ity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is F (x, D2u) ≥ Hi(x, u, Du) in RN , where Hi has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases H1(u,Du) = uq + |Du|γ, H2(u,Du) = uq|Du|γ and H3(x,u,Du) = ±uq|Du|γ − b(x) · Du, where q ≥ 0, γ > 1 and b is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.