Liouville-type theorems for weighted p -Laplacian equations with nonlinear gradient terms on smooth metric measure spaces

Abstract In this paper, we consider weighted p -Laplacian equations with a gradient term and a nonlinear term Δ p, f u + p − 1 W u ∇ u p + F u = 0 {}₏, ₅u+ (p-1) W (u) u ^p+F (u) =0 on smooth metric measure spaces (M n, g, e − f d V), where p > 1, W (t) is a continuous function for t > 0 and F (t) is a differentiable function in (0, ∞). By making some appropriate assumptions about W and F, we derive some Liouville-type theorems for positive solutions to the above equation, if (M n, g, e − f d V) satisfies R i c f m ≥ 0 Ric₅^m 0. As applications we derive Liouville-type theorems of positive solutions to some generalized static Fisher-KPP equation, Allen–Cahn equation, static Newell–Whitehead equation and Lichnerowicz equation.