Bifurcation and multiplicity results for critical Grushin-Choquard problems

We consider the following nonlocal Brézis-Nirenberg type critical Choquard problem involving the Grushin operator \begin{equation*} \left{ \begin{aligned} -Δ_γ& u =λu + \left(\displaystyle\int_Ω\frac{|u(w)|^{2^_{γ,μ}}}{d(z-w)^μ}dw\right) |u|^{2^_{γ,μ}-2}u \quad &&$\text{in}$ \ Ω, u &= 0 \quad &&$\text{on}$ , \partial Ω, \end{aligned} \right. \end{equation*} where $\mathrm{\Omega }$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, with $\mathrm{\Omega }\cap \{x=0\}\mathrm{\ne }\mathrm{\varnothing }$, and $\lambda >0$ is a parameter. Here, ${\mathrm{\Delta }}_{\gamma }$ represents the Grushin operator, defined as $${\mathrm{\Delta }}_{\gamma }u(z)={\mathrm{\Delta }}_{x}u(z)+(1+\gamma {)}^2\mathrm{\mid }x{\mathrm{\mid }}^{2\gamma }{\mathrm{\Delta }}_{y}u(z),\gamma \ge 0,$$ where $z=(x,y)\in \mathrm{\Omega }\subset {\mathbb{R}}^m\times {\mathbb{R}}^n$, $m+n=N\ge 3$ and $2_{\gamma ,\mu }^{\ast }=\frac{2N_{\gamma }-\mu }{N_{\gamma }-2}$ 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 ${\lambda }^{\ast }$ of $-{\mathrm{\Delta }}_{\gamma }$ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of ${\lambda }^{\ast }$, the number of nontrivial solutions to the problem is at least twice the multiplicity of ${\lambda }^{\ast }$.