Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity
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2025
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| author | Ghosh, Sekhar Kumar, Vishvesh |
| author_facet | Ghosh, Sekhar Kumar, Vishvesh |
| contents | In this paper, we explore the bifurcation phenomena and establish the existence of multiple solutions for the nonlocal subelliptic Brezis-Nirenberg problem:
\begin{equation*}
\begin{cases} (-Δ_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+λu \quad &\text{in}\quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases}
\end{equation*} where $(-Δ_{\mathbb{G}})^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is a open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $Q> 2s,$ $2_s^*:=\frac{2Q}{Q-2s}$ is subelliptic fractional Sobolev critical exponent, $λ>0$ is a real parameter. This work extends the seminal contributions of Cerami, Fortunato, and Struwe to nonlocal subelliptic operators on stratified Lie groups. A key component of our study involves analyzing the subelliptic $(s, p)$-eigenvalue problem for the (nonlinear) fractional $p$-sub-Laplacian $(-Δ_{p,{\mathbb{G}}})^s$ \begin{align*}
(-Δ_{p,{\mathbb{G}}})^s u&=λ|u|^{p-2}u,~\text{in}~Ω,\nonumber
u&=0~\text{ in }~{\mathbb{G}}\setminusΩ, \end{align*} with $0<s<1<p<\infty$ and $Q>ps$, over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups applying variational methods. Particularly, we prove that the $(s, p)$-spectrum of $(-Δ_{p,{\mathbb{G}}})^s$ is closed and the second eigenvalue $λ_2(Ω)$ with $λ_2(Ω)>λ_1(Ω)$ is well-defined and provides a variational characterization of $λ_2(Ω)$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity Ghosh, Sekhar Kumar, Vishvesh Analysis of PDEs 35R03, 35H20, 22E30, 35J20, 35R11 In this paper, we explore the bifurcation phenomena and establish the existence of multiple solutions for the nonlocal subelliptic Brezis-Nirenberg problem: \begin{equation*} \begin{cases} (-Δ_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+λu \quad &\text{in}\quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases} \end{equation*} where $(-Δ_{\mathbb{G}})^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is a open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $Q> 2s,$ $2_s^*:=\frac{2Q}{Q-2s}$ is subelliptic fractional Sobolev critical exponent, $λ>0$ is a real parameter. This work extends the seminal contributions of Cerami, Fortunato, and Struwe to nonlocal subelliptic operators on stratified Lie groups. A key component of our study involves analyzing the subelliptic $(s, p)$-eigenvalue problem for the (nonlinear) fractional $p$-sub-Laplacian $(-Δ_{p,{\mathbb{G}}})^s$ \begin{align*} (-Δ_{p,{\mathbb{G}}})^s u&=λ|u|^{p-2}u,~\text{in}~Ω,\nonumber u&=0~\text{ in }~{\mathbb{G}}\setminusΩ, \end{align*} with $0<s<1<p<\infty$ and $Q>ps$, over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups applying variational methods. Particularly, we prove that the $(s, p)$-spectrum of $(-Δ_{p,{\mathbb{G}}})^s$ is closed and the second eigenvalue $λ_2(Ω)$ with $λ_2(Ω)>λ_1(Ω)$ is well-defined and provides a variational characterization of $λ_2(Ω)$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group. |
| title | Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity |
| topic | Analysis of PDEs 35R03, 35H20, 22E30, 35J20, 35R11 |
| url | https://arxiv.org/abs/2501.12791 |