Liouville-type theorems for a class of semilinear elliptic equations with nonlinear gradient terms on smooth metric measure spaces

Abstract

In this paper, we consider the following f-Laplacian equation with a gradient term and a nonlinear term

$\Delta_{f} u+W(u)|\nabla u|^{2}+F(u)=0$

on smooth metric measure spaces with nonnegative m-Bakry-èmery Ricci curvature, where W(t) is a continuous function for t > 0 and F(t) is a differentiable function in (0, ∞). Under certain assumptions on W and F, by using the Saloff-Coste’s Sobolev inequality and the Nash-Moser iteration method, we derive some Liouville-type theorems for positive solutions to the above equation. Our results extend those of Lu (2025) and McCoy (2007).

Keywords

nonlinear elliptic equation / Nash-Moser iteration / Liouville theorem

References

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