@Article{Aikyn2026,
author = "Aikyn, Yergen and Kassymov, Aidyn and Ruzhansky, Michael",
title = "Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups",
journal = "Opuscula Math.",
year = "2026",
volume = "46",
number = "5",
pages = "599--622",
doi = "https://doi.org/10.7494/OpMath.202610081",
url = "https://doi.org/10.7494/OpMath.202610081",
abstract = "In this paper, we study the existence and multiplicity of solutions to the following subelliptic problem with Hardy potential \[\begin{cases}-\Delta_{\mathbb{G}} u(\xi)-\mu \frac{u(\xi)}{|x^{\prime}|^{2}}=u^{q}(\xi)+\lambda u^{r}(\xi)&\text{in}\; \Omega , \\ u(\xi)\gt 0 &\text{in}\; \Omega, \\ u(\xi)=0 &\text{on}\; \partial \Omega,\end{cases}\] where \(\Omega \subset \mathbb{G}\) is a bounded domain containing the origin with smooth boundary \(\partial \Omega\), \(0\lt\mu\lt\bar{\mu}:=\left(\frac{N-2}{2}\right)^{2}\) with \(N\gt 2\), and \(\lambda\gt 0\). Here, the Hardy potential is defined on the first stratum of a stratified Lie group \(\mathbb{G}\). Specifically, we have \(|x^{\prime}|=\sqrt{x_{1}^{2}+\ldots+x_{N}^{2}},\) \(x^{\prime}=(x_{1},\ldots, x_{N}) \in \mathbb{R}^{N}\), where \(N\) is the dimension of the first stratum of the stratified Lie group \(\mathbb{G}\). Under the assumptions \(1\lt r+1\lt 2\lt q+1\lt 2^*:=\frac{2Q}{Q-2}\), we prove the existence of at least two positive solutions provided that \(\lambda\) is sufficiently small.",
keywords = "subelliptic problem, Hardy potential, Palais-Smale condition, concentration-compactness principle",
issn = "1232-9274"
}