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

Abstract

In this paper, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplace equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of ∆ p u = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplace equation −∆ p u = |u| λ−1 u, x ∈ Ω, which are then extended to the equation with general right hand term f (x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplace equation, in a case of p = 2, are also discussed.