Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups

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.