Abstract
We establish Liouville-type theorems for the elliptic inequality
u >= 0, G(alpha,beta,theta)(u(p(x,y)), u(q(x,y))) >= u(r(x,y)), (x,y) is an element of R-N1 x R-N2,
where G(alpha,beta,theta), 0 < alpha,beta < 2, theta >= 0, is the fractional Grushin operator of mixed orders alpha,beta, defined by
G(alpha,beta,theta)(u,v) = (-Delta(x))(alpha/2)u + vertical bar x vertical bar(2 theta)(-Delta(y))(beta/2)v,
where, (-Delta(x))(alpha/2) is the fractional Laplacian operator of order alpha with respect to the variable x is an element of R-N1, and (-Delta(y))(beta/2) is the fractional Laplacian operator of order beta with respect to the variable y is an element of R-N2. Here, p, q, r : R-N1 x R-N2 -> [1, infinity) are measurable functions satisfying certain conditions.