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 $${\mathrm{\Delta }}_{p}v+(v-a_1)(v-a_2)(v-a_3)=0$$ on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1<a_2<a_3$ are real constants and ${\mathrm{\Delta }}_{p}v=\mathrm{div}(\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^{p-2}\mathrm{\nabla }v)$ 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.