Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity
Sekhar Ghosh, Vishvesh Kumar
TL;DR
This work studies the nonlocal subelliptic Brezis-Nirenberg problem on stratified Lie groups with homogeneous dimension $Q>2s$, proving the existence of multiple solutions through variational methods. It develops and analyzes the $(s,p)$-spectrum of the fractional $p$-sub-Laplacian $(-\Delta_{p,\mathbb{G}})^s$, establishing that the spectrum is closed, characterizing the first and second eigenvalues variationally, and providing a minimax description for $\lambda_{n+1}$; these results are new in the subelliptic setting and novel for the Heisenberg group. The paper then uses an abstract even critical-point theorem to obtain bifurcation and multiplicity of solutions near a multiple eigenvalue, showing at least $m$ pairs of nontrivial solutions for appropriate $\lambda$ and demonstrating that $\|u_{\lambda,i}\|_{X_0^{s,2}(\Omega)}\to0$ as $\lambda$ approaches $\lambda^*$. Collectively, this extends classical Brezis-Nirenberg theory to nonlocal subelliptic operators on stratified Lie groups and lays groundwork for further nonlinear ($p$-Laplacian) extensions with implications for CR geometry and sub-Riemannian spectral theory.
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 $0<s<1<p<\infty$ 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.