Bifurcation and multiplicity results for critical Grushin-Choquard problems

We consider the following nonlocal Br\'ezis-Nirenberg type critical Choquard problem involving the Grushin operator \begin{equation*} \left\{ \begin{aligned} -\Delta_\gamma & u =\lambda u + \left(\displaystyle\int_\Omega \frac{|u(w)|^{2^*_{\gamma,\mu}}}{d(z-w)^\mu}dw\right) |u|^{2^*_{\gamma,\mu}-2}u \quad &&\text{in} \ \Omega, u &= 0 \quad &&\text{on} \, \partial \Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, with $\Omega \cap \{ x=0\} \neq \emptyset$, and $\lambda >0$ is a parameter. Here, $\Delta_\gamma$ represents the Grushin operator, defined as $$\Delta_\gamma u(z) = \Delta_x u(z) +(1+\gamma)^2 |x|^{2\gamma} \Delta_y u(z), \quad \gamma \geq 0,$$ where $z=(x,y)\in \Omega \subset \mathbb{R}^m\times \mathbb{R}^n$, $m+n=N \geq 3$ and $2^*_{\gamma,\mu}= \frac{2N_\gamma-\mu}{N_\gamma-2}$ is the Sobolev critical exponent in the Hardy-Littlewood context with $N_\gamma= m+(1+\gamma)n$ is the homogeneous dimension associated to the Grushin operator and $0<\mu<N_\gamma$. The homogeneous norm related to the Grushin operator is denoted by $d(\cdot)$. In this article, we prove the existence of bifurcation from any eigenvalue $\lambda^*$ of $-\Delta_\gamma$ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of $\lambda^*$, the number of nontrivial solutions to the problem is at least twice the multiplicity of $\lambda^*$.