A Liouville-type theorem for Lane-Emden system

Abstract

We provide a partial positive answer to a well-known conjecture about the nonexistence of positive solutions to Lane-Emden systems below the critical Sobolev hyperbola. Our proof is based on a monotonicity argument for suitable transformed functions. It relies on a special form of the Alexandrov-Serrin moving planes method, as well as some refined forms of the Maximum Principle for elliptic systems that we develop here. It is our hope that our method will shed some new light on this delicate problem.