Gradient estimates for $p$-Laplacian equation with cubic polynomial nonlinearity on Riemannian manifolds

This paper studies a class of $p$-Laplace equations with cubic polynomial nonlinearity $$\Delta_p v + (v-a_1)(v-a_2)(v-a_3) = 0$$ on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1 < a_2 < a_3$ are real constants and $\Delta_p v = \operatorname{div}(|\nabla v|^{p-2}\nabla v)$ denotes the $p$-Laplace operator. Depending on whether the solution lies in the intervals $(a_1,a_2), (a_2,a_3)$ or $(a_1,a_3)$, we employ, respectively, a logarithmic transformation or a hyperbolic tangent transformation to convert the original equation to another one for further analysis. Through a detailed analysis of the lower-bound estimate for the linearized operator of the new equation, and by combining Saloff-Coste's Sobolev inequality with a Moser iteration, we establish Cheng-Yau type gradient estimates under an additional assumption on $p$. As applications, the Liouville theorem and a Harnack inequality are further proved.