A Brézis–Nirenberg Type Problem for a Class of Degenerate Elliptic Problems Involving the Grushin Operator

Abstract

Motivated by the seminal paper due to Brézis–Nirenberg (in: Commun Pure Appl Math 36:437–477, 1983), we will establish the existence of solutions for the following class of degenerate elliptic equations with critical nonlinearity:

$$\begin{aligned} {\left\{ \begin{array}{ll}-\Delta _{\gamma } v= \lambda |v|^{q-2}v+\left| v\right| ^{2_{\gamma }^{*}-2}v &{} \text { in } \Omega , \\ v=0 &{} \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Delta _{\gamma }:=\Delta _{x} + (1+\gamma )^2\left| x\right| ^{2\gamma } \Delta _{y}$ is the Grushin operator, $z:=(x, y) \in \mathbb {R}^N$, $N=m+n,$ $m,n \ge 1,$ $\Omega \subset \mathbb {R}^{N}$ is a smooth bounded domain, $\lambda >0$, $q \in [2,2_\gamma ^*)$ and $2_{\gamma }^{*}=\frac{2N_\gamma }{N_\gamma -2}$ is the critical Sobolev exponent in this context, where $N_\gamma =m+(1+\gamma )n$ is the so-called homogeneous dimension attached to the Grushin operator$\Delta _\gamma$. In order to prove our main results it was necessary to do a careful study involving the best constant $\mathcal {S}_\gamma (m, n)$ of the Sobolev embedding for the spaces associated with $\Delta _\gamma$. In order to do that, we prove a version of the Lions’ Concentration-Compactness Principle for the Grushin operator. We also provide existence results for a critical problem involving the Grushin operator on the whole space $\mathbb {R}^N$.