We study the following class of quasilinear degenerate elliptic equations with critical nonlinearity \begin{ align*} \begin{ cases}-\Delta_{ \gamma,p} u= \lambda |u|^{ q-2}u+|u|^{ p_{ \gamma}^{ *}-2}u & \text{ in } \Omega\subset \mathbb{ R}^N, \\ u=0 & \text{ on } \partial \Omega, \end{ cases} \end{ align*} where $\Delta_{ \gamma, p}v:=\sum_{ i=1}^N X_i(|\nabla_\gamma u|^{ p-2}X_i u)$ is the Grushin $p$-Laplace operator, $z:=(x, y) \in \mathbb{ R}^N$, $N=m+n,$ $m,n \geq 1,$, where $\nabla_\gamma=(X_1, \ldots, X_N)$ is the Grushin gradient, defined as the system of vector fields $X_i=\frac{ \partial}{ \partial x_i}, i=1, \ldots, m$, $X_{ m+j}=|x|^\gamma \frac{ \partial}{ \partial y_j}, j=1, \ldots, n$, where $\gamma>0$. Here, $\Omega \subset \mathbb{ R}^{ N}$ is a smooth bounded domain such that $\Omega\cap \{ x=0\}\neq \emptyset$, $\lambda>0$, $q \in [p,p_\gamma^*)$, where $p_{ \gamma}^{ *}=\frac{ pN_\gamma}{ N_\gamma-p}$ 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_{ \gamma} u|^p dz \geq S_{ \gamma,p} \left ( \int_{ \mathbb{ R}^N} |u|^{ p_\gamma^*} dz \right )^{ p/p_\gamma^*} \end{ equation*} and their qualitative behavior as positive entire solutions to the limit problem \begin{ equation*} -\Delta_{ \gamma,p} u= u^{ p_{ \gamma}^{ *}-1}\quad \mbox{ on}\, \mathbb{ R}^N, \end{ equation*} whose study has independent interest.
Categories: math.AP