Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups

TL;DR

This paper studies subelliptic nonlocal Brezis-Nirenberg problems on stratified Lie groups, establishing existence results for solutions involving critical and subcritical nonlinearities using variational methods.

Contribution

It introduces new existence results for fractional subelliptic equations on stratified Lie groups, including the Heisenberg group, with critical and subcritical nonlinearities.

Findings

Existence of solutions for small positive parameter  under critical nonlinearity.

Existence of solutions for =0 with subcritical nonlinearity.

Results are novel even for the classical Heisenberg group when p=2.

Abstract

In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-\Delta_{\mathbb{G}, p})^s u&= \mu |u|^{p_s^*-2}u+\lambda h(x, u) \quad \text{in}\quad \Omega, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash \Omega, \end{align*} where $(-\Delta_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $\Omega$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $\mu, \lambda>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $\lambda$ is sufficiently small. Additionally, we examine the problem for $\mu = 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.