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 $Δ_{γ, p}v:=\sum_{i=1}^N X_i(|\nabla_γu|^{p-2}X_i u)$ is the Grushin $p$-Laplace operator, $z:=(x, y) \in \mathbb{R}^N$, $N=m+n,$ $m,n \geq 1,$, where $\nabla_γ=(X_1, \ldots, X_N)$ is the Grushin gradient, defined as the system of vector fields $X_i=\frac{\partial}{\partial x_i}, i=1, \ldots, m$, $X_{m+j}=|x|^γ\frac{\partial}{\partial y_j}, j=1, \ldots, n$, where $γ>0$. Here, $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain such that $Ω\cap \{x=0\}\neq \emptyset$, $λ>0$, $q \in [p,p_γ^*)$, where $p_γ^{*}=\frac{pN_γ}{N_γ-p}$ 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 \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.