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 {align*} {cases}-Δγ,p u= λ|u|q-2u+|u|^{p_γ*-2}u & in Ω⊂ R^N, \\ u=0 & on ∂ Ω, {cases} {align*} where Δγ, pv:=∑ᵢ₌₁N Xᵢ(|∇_γu|ᵖ⁻²Xᵢ u) is the Grushin p-Laplace operator, z:=(x, y) ∈ RN, $N=m+n,$ m,n ≥ 1,, where ∇_γ=(X₁, …, XN) is the Grushin gradient, defined as the system of vector fields Xᵢ=∂/∂ xᵢ, i=1, …, m, Xₘ₊ⱼ=|x|^γ∂/∂ yⱼ, j=1, …, n, where $γ>0$. Here, Ω⊂ RN is a smooth bounded domain such that Ω∩ =0\≠ ∅, $λ>0$, q ∈ [p,p_γ^*), where p_γ*=pN_γ/N_γ-p and N_γ=m+(1+γ)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 {equation*} ∫R^N |∇_γ u|^p dz ≥ Sγ,p ( ∫R^N |u|p_γ^* dz )p/p_γ^* {equation*} and their qualitative behavior as positive entire solutions to the limit problem {equation*} -Δγ,p u= u^{p_γ*-1} on\, R^N, {equation*} whose study has independent interest.