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

TL;DR

This paper investigates bifurcation and multiple solutions for a nonlocal subelliptic Brezis-Nirenberg problem on stratified Lie groups, extending classical results to fractional operators in this geometric setting.

Contribution

It introduces the analysis of the spectrum and bifurcation phenomena for fractional subelliptic operators on stratified Lie groups, including the Heisenberg group, using variational methods.

Findings

Established existence of multiple solutions for the nonlocal problem.

Proved the spectrum of the fractional p-sub-Laplacian is closed.

Characterized the second eigenvalue variationally.

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} (-\Delta_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+\lambda u \quad &\text{in}\quad \Omega, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash \Omega, \end{cases} \end{equation*} where $(-\Delta_{\mathbb{G}})^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $\Omega$ 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, $\lambda>0$ 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 $(-\Delta_{p,{\mathbb{G}}})^s$ \begin{align*} (-\Delta_{p,{\mathbb{G}}})^s u&=\lambda |u|^{p-2}u,~\text{in}~\Omega,\nonumber u&=0~\text{ in }~{\mathbb{G}}\setminus\Omega, \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 $(-\Delta_{p,{\mathbb{G}}})^s$ is closed and the second eigenvalue $\lambda_2(\Omega)$ with $\lambda_2(\Omega)>\lambda_1(\Omega)$ is well-defined and provides a variational characterization of $\lambda_2(\Omega)$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group.