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

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.