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

Zhen Qiu, Youde Wang, Jun Yang

Abstract

This paper studies a class of $p$-Laplace equations with cubic polynomial nonlinearity \[ Δ_p v + (v-a_1)(v-a_2)(v-a_3) = 0 \] on complete Riemannian manifolds $M$ with lower Ricci curvature bounds, where $a_1 < a_2 < a_3$ are real constants and $Δ_p v = \operatorname{div}(|\nabla v|^{p-2}\nabla v)$ 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.

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

Abstract

This paper studies a class of -Laplace equations with cubic polynomial nonlinearity on complete Riemannian manifolds with lower Ricci curvature bounds, where are real constants and denotes the -Laplace operator. Depending on whether the solution lies in the intervals or , 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 . As applications, the Liouville theorem and a Harnack inequality are further proved.
Paper Structure (11 sections, 11 theorems, 209 equations)

This paper contains 11 sections, 11 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Background and Motivations
  3. Main results
  4. Lower-bound estimate of $\mathcal{L}(f^{\alpha})$ for bounded solutions
  5. The low-bound under the logarithmic transformation
  6. The low-bound under the hyperbolic tangent transformation
  7. Proofs of main theorems
  8. The integral inequality to $f=|\nabla w|^2$
  9. Local $L^\gamma$-upper bound of the gradient
  10. The gradient estimate by Moser iteration: proof of Theorem \ref{['theorem1']}
  11. Proofs of Liouville theorem and Harnack inequality

Key Result

Theorem 1.2

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1) \kappa g$, where $n\geq 2$ and $\kappa$ is a non-negative constant. Suppose that $v$ is a solution in the sense of Definition definition to the $p$-Laplacian equation with cubic polynomial nonlinearity GA

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • ...and 12 more