Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere

TL;DR

This paper establishes a Liouville type theorem for a class of quasilinear p-Laplace equations on the sphere, providing insights into their asymptotic behavior and answering a question posed by L. Vèron.

Contribution

It introduces a Liouville theorem for p-Laplace type equations on the sphere, derived via integral by parts, and addresses an open question from Vèron's work.

Findings

Proves a Liouville type theorem for the class of equations.

Connects the theorem to asymptotic behavior near the origin.

Answers a longstanding question posed by Vèron.

Abstract

We use the integral by parts to get a Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere, this $p$-Laplace type equation arises from the study of asymptotic behavior near the origin for the semi-linear $p$-Laplace equation on the puncture ball $B_1(o) \subset R^n$. This gives a positive answer to L. V\'{e}ron's question in a paper \cite{Veron92} and his book \cite{Veron} at page 440.