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.
On the Liouville property for fully nonlinear equations with superlinear first-order terms / Cirant, Marco; Goffi, Alessandro. - STAMPA. - 781:(2023), pp. 7-39. [10.1090/conm/781/15707]
On the Liouville property for fully nonlinear equations with superlinear first-order terms
Goffi, Alessandro
2023
Abstract
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.File | Dimensione | Formato | |
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