Liouville type theorem for nonlinear elliptic system involving Grushin operator

We study the degenerate elliptic system of the form {-Delta(G)u = v(p) on R-N = R-N1 x R-N2, {-Delta(G)u = u(q) where Delta(G) := Delta(x) + vertical bar x vertical bar(2 alpha)Delta(y), is the Grushin operator, alpha >= 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]. (C) 2017 Elsevier Inc. All rights reserved.