Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity

TL;DR

This paper proves the uniqueness of positive solutions for a class of anisotropic p-Laplacian equations with polynomial non-linearity, extending previous results using linearized methods.

Contribution

It introduces a new application of linearized techniques to establish solution uniqueness for anisotropic p-Laplacian equations with polynomial growth.

Findings

Uniqueness of positive solutions established for the specified PDE.

Extension of previous uniqueness results by Brasco and Lindgren.

Application of linearized method to anisotropic elliptic equations.

Abstract

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation} \nonumber \left\{ \begin{aligned} -\Delta^F_p u&=u^q \quad \text{in} \quad \Omega, u&=0 \quad \text{on} \quad \partial \Omega, \end{aligned} \right. \end{equation} where $p>\frac{3}{2}$ is a constant. We utilize the linearized method to derive the uniqueness results, which extends the conclusion obtained by L. Brasco and E. Lindgren.