Quasilinear problems with critical Sobolev exponent for the Grushin p-Laplace operator

We study the following class of quasilinear degenerate ellipticequations with critical nonlinearity{-Delta(gamma,p)u=lambda|u|(q-2)u+|u|(p gamma*-2)u in Omega subset of R-N,u=0 on partial derivative Omega,where Delta(gamma,p)u:=Sigma(N)(i=1) X-i(|del(gamma)u|(p-2)X(i)u) is the Grushinp-Laplace opera-tor, z := (x, y) is an element of R-N, N=m+n, m, n >= 1, where del gamma=(X-1, ..., X-N) is the Grushin gradient, defined as the system of vector fields X-i=partial derivative/partial derivative(xi), i=1, ..., m, Xm+j=|x|(gamma) partial derivative/partial derivative yj, j=1, ..., n, where gamma>0. Here, Omega subset of R-N is a smooth bounded domain such that Omega boolean AND{x=0}not equal, lambda>0, q is an element of[p, p(gamma)*), where p(gamma)*=pN(gamma)/N-gamma-p and N-gamma=m+(1+gamma)n denotes the homogeneous dimension attached to the Grushin gradient. The results extend to thep-case the Brezis-Nirenberg type results in Alves-Gandal-Loiudice-Tyagi [J. Geom. Anal. 2024, 34(2),52]. The main crucial step is to preliminarily establish the existence of the extremals for the involved Sobolev-type inequalityintegral(RN)|del(gamma)u|(p)dz >= S-gamma,S-p(integral(RN)|u|(p gamma*)dz)(p/p gamma)* and their qualitative behavior as positive entire solutions to the limit problem-Delta(u)(gamma,p)=u(p gamma*-1) on R-N,whose study has independent interest.