Bifurcation and multiplicity results for critical Grushin-Choquard problems
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| Format: | Preprint |
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2025
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| _version_ | 1866910222942994432 |
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| author | Kanungo, Suman Mishra, Pawan Kumar Bisci, Giovanni Molica |
| author_facet | Kanungo, Suman Mishra, Pawan Kumar Bisci, Giovanni Molica |
| contents | 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 $\mathbb{R}^N$, $N \geq 3$, with $Ω\cap \{ x=0\} \neq \emptyset$, 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 $λ^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcation and multiplicity results for critical Grushin-Choquard problems Kanungo, Suman Mishra, Pawan Kumar Bisci, Giovanni Molica Analysis of PDEs 2020 Mathematics Subject Classification. 35J70, 35H20, 35A15 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 $\mathbb{R}^N$, $N \geq 3$, with $Ω\cap \{ x=0\} \neq \emptyset$, 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 $λ^*$. |
| title | Bifurcation and multiplicity results for critical Grushin-Choquard problems |
| topic | Analysis of PDEs 2020 Mathematics Subject Classification. 35J70, 35H20, 35A15 |
| url | https://arxiv.org/abs/2510.13299 |