Bifurcation and multiplicity results for critical Grushin-Choquard problems

TL;DR

This work addresses a Brézis–Nirenberg-type Choquard problem driven by the Grushin operator on a bounded domain, featuring a nonlocal critical nonlinearity. The authors develop a Grushin-adapted Hardy–Littlewood–Sobolev framework and employ a variational approach together with an abstract critical point theorem to establish bifurcation from Grushin eigenvalues. They prove that, for eigenvalue $\lambda^*$ of multiplicity $m$, there exists a left neighborhood in which at least $m$ pairs of nontrivial solutions bifurcate and have norms vanishing as $\lambda \to \lambda^*$; this hinges on a sharp domain-dependent threshold involving the Grushin-HLS best constant $S_\Omega$, domain measure, and diameter. The results extend existing bifurcation and multiplicity theory to the nonlocal Grushin–Choquard setting (including the case $\gamma=0$) and provide new compactness tools for degenerate, anisotropic PDEs with critical nonlinearity.

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 $Ω$ is an open bounded domain in $\mathbb{R}^N$, with $N \geq 3$, and $λ>0$ is a parameter. Here, $Δ_γ$ 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 $z=(x,y)\in Ω\subset \mathbb{R}^m\times \mathbb{R}^n$, $m+n=N \geq 3$ and $2^*_{γ,μ}= \frac{2N_γ-μ}{N_γ-2}$ is the Sobolev critical exponent in the Hardy-Littlewood context with $N_γ= m+(1+γ)n$ is the homogeneous dimension associated to the Grushin operator and $0<μ<N_γ$. 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 $λ^*$ of $-Δ_γ$ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of $λ^*$, the number of nontrivial solutions to the problem is at least twice the multiplicity of $λ^*$.

Theorems & Definitions (16)

• Definition 1
• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Proposition 2.2: Grushin HLS inequality
• proof
• Remark 1
• Lemma 3.1
• proof