Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

TL;DR

This paper develops pointwise a priori estimates for solutions to p-Laplacian equations, extending classical results and analyzing singularity, decay, and higher-order derivative estimates for various related quasilinear equations.

Contribution

It introduces new pointwise estimates for p-Laplacian solutions, including gradient bounds and decay rates, generalizing existing results to broader classes of equations.

Findings

Interior gradient estimates for p-harmonic functions

Singularity and decay estimates for sign-changing solutions

Pointwise estimates for higher derivatives when p=2

Abstract

In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $\Delta_{p}u=0,\ x\in\Omega$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-\Delta_{p}u=|u|^{\lambda-1}u, \ x\in\Omega$, which are then generalized for the equation with general right hand term $f(x,u)$, under some asymptotic conditions of $f$. Lastly, we get pointwise estimates for higher order derivatives of the solution of $-\Delta u=u^{\lambda},x\in\Omega$, the case of $p=2$ for p-Laplace equation.