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

We study the following class of quasilinear degenerate elliptic equations with critical nonlinearity \begin{align*} \begin{cases}-Δ_{γ,p} u= λ|u|^{q-2}u+|u|^{p_γ^{}-2}u & $\text{ in }$ Ω\subset $\mathbb{R}$^N, \ u=0 & $\text{ on }$ \partial Ω, \end{cases} \end{align} where ${\mathrm{\Delta }}_{\gamma ,p}v:={\sum }_{i=1}^N{X}_i(\mathrm{\mid }{\mathrm{\nabla }}_{\gamma }u{\mathrm{\mid }}^{p-2}{X}_{i}u)$ is the Grushin $p$-Laplace operator, $z:=(x,y)\in {\mathbb{R}}^N$, $N=m+n,$ $m,n\ge 1,$, where ${\mathrm{\nabla }}_{\gamma }=(X_1,\dots ,X_N)$ is the Grushin gradient, defined as the system of vector fields $X_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},i=1,\dots ,m$, $X_{m+j}=\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }\frac{\mathrm{\partial }}{\mathrm{\partial }{y}_j},j=1,\dots ,n$, where $\gamma >0$. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain such that $\mathrm{\Omega }\cap \{x=0\}\mathrm{\ne }\mathrm{\varnothing }$, $\lambda >0$, $q\in [p,p_{\gamma }^{\ast })$, where $p_{\gamma }^{\ast }=\frac{p{N}_{\gamma }}{N_{\gamma }-p}$ and $N_{\gamma }=m+(1+\gamma )n$ denotes the homogeneous dimension attached to the Grushin gradient. The results extends to the $p$-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 inequality \begin{equation*} \int_{$\mathbb{R}$^N} |\nabla_γ u|^p dz \geq S_{γ,p} \left ( \int_{$\mathbb{R}$^N} |u|^{p_γ^} dz \right )^{p/p_γ^} \end{equation*} and their qualitative behavior as positive entire solutions to the limit problem \begin{equation*} -Δ_{γ,p} u= u^{p_γ^{}-1}\quad \mbox{on}, $\mathbb{R}$^N, \end{equation} whose study has independent interest.