Bifurcation and multiplicity results for critical Grushin-Choquard problems

Abstract

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 $\Omega$ is an open bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, with $\Omega \cap \{x=0\}\ne \varnothing$, and $\lambda >0$ is a parameter. Here, ${\Delta }_{\gamma }$ represents the Grushin operator, defined as \[ Δ_γu(z) = Δ_x u(z) +(1+γ)^2 |x|^{2γ} Δ_y u(z), \quad γ\geq 0, \] where $z=(x,y)\in \Omega \subset {\mathbb{R}}^m\times {\mathbb{R}}^n$, $m+n=N\ge 3$ 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 $-{\Delta }_{\gamma }$ 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 ${\lambda }^{\ast }$.

Accepted in Frac. Calc. Appl. Anal