Liouville type theorem for nonlinear elliptic system involving Grushin operator

Abstract

We study the degenerate elliptic system of the form{−ΔGu=vp−ΔGv=uq on RN=RN1×RN2, where ΔG:=Δx+|x|2αΔy is the Grushin operator, α≥0 and p≥q>1. We establish some Liouville type results for stable solutions of the system. In particular, we prove the comparison principle – a crucial step to establish such results. As consequences, we obtain a Liouville type theorem for the scalar equation and provide a counterpart of the previous result in C. Cowan (2013) [7].