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

Abstract

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 $0 ps$, 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.