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

Abstract

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 ${\Delta }_{\gamma ,p}v:={\sum }_{i=1}^N{X}_i(|{\nabla }_{\gamma }u{|}^{p-2}{X}_{i}u)$ is the Grushin $p$-Laplace operator, $z:=(x,y)\in {\mathbb{R}}^N$, $N=m+n,$ $m,n\ge 1,$, where ${\nabla }_{\gamma }=(X_1,\dots ,X_N)$ is the Grushin gradient, defined as the system of vector fields $X_i=\frac{\partial }{\partial x_i},i=1,\dots ,m$, $X_{m+j}=|x{|}^{\gamma }\frac{\partial }{\partial y_j},j=1,\dots ,n$, where $\gamma >0$. Here, $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain such that $\Omega \cap \{x=0\}\ne \varnothing$, $\lambda >0$, $q\in [p,p_{\gamma }^{\ast })$, 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.

28 pages