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

In this paper, we consider weighted p -Laplacian equations with a gradient term and a nonlinear term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfenced> <m:mi>W</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>∇</m:mi> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>F</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:math> Δp,fu+(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">R</m:mi> <m:mi mathvariant="normal">i</m:mi> <m:mi mathvariant="normal">c</m:mi> </m:mrow> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msubsup> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:math> Ricfᵐ≥ 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.