Existence results for critical problems with Hardy term involving the Grushin operator

We provide existence results for the subelliptic critical problem involving a Hardy-type perturbation -Delta(gamma) u - mu psi(2) /d(2) u = |u| (2 & lowast;) (gamma) -2u + lambda u in Omega, u = 0 on partial derivative Omega, where Delta(gamma) = Delta(x) + |x| (2 gamma) Delta(y), gamma > 0 is the so-called Grushin operator, Omega is a smooth bounded domain of R-N = R-m (x) x R-n (y) , 0 is an element of Omega, d is the gauge norm naturally associated with gamma , psi = |del(gamma) d|, where del(gamma) is the Grushin gradient, 2(& lowast;) (gamma) is the critical Sobolev exponent in this context, and 0 < (-)mu, where (-)mu is the best Hardy constant for gamma . In particular, we extend to the Grushin context well-known results obtained by Jannelli in [J Differ Equ. 1999;156] for the classical Laplacian case, and generalize to the case with Hardy perturbation some of the existence results proved for the Grushin unperturbed operator by Alves-Gandal-Loiudice-Tyagi in [J Geom Anal. 2024;34(2):52].