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

TL;DR

This paper derives gradient estimates for solutions to a class of $p$-Laplacian equations with cubic polynomial nonlinearities on Riemannian manifolds with Ricci curvature bounds, using transformations, Sobolev inequalities, and Moser iteration.

Contribution

It introduces new transformation techniques and analytical methods to establish Cheng-Yau type gradient estimates for nonlinear $p$-Laplace equations on Riemannian manifolds.

Findings

Established Cheng-Yau type gradient estimates under certain conditions.

Proved Liouville theorem for solutions of the equation.

Derived a Harnack inequality for positive solutions.

Abstract

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.