Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity

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 $(-{\mathrm{\Delta }}_{\mathbb{G}}{)}^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $\mathrm{\Omega }$ is a open bounded subset of $\mathbb{G},$ $s\in (0,1)$, $Q>2s,$ $2_s^{\ast }:=\frac{2Q}{Q-2s}$ is subelliptic fractional Sobolev critical exponent, $\lambda >0$ 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 $(-{\mathrm{\Delta }}_{p,\mathbb{G}}{)}^s$ \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 $0<s<1<p<\mathrm{\infty }$ 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 $(-{\mathrm{\Delta }}_{p,\mathbb{G}}{)}^s$ is closed and the second eigenvalue ${\lambda }_2(\mathrm{\Omega })$ with ${\lambda }_2(\mathrm{\Omega })>{\lambda }_1(\mathrm{\Omega })$ is well-defined and provides a variational characterization of ${\lambda }_2(\mathrm{\Omega })$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group.