TY  - JOUR
ID  - Aikyn2026
LB  - Aikyn2026
AU  - Aikyn, Yergen
AU  - Kassymov, Aidyn
AU  - Ruzhansky, Michael
TI  - Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups
ST  - Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 5
SP  - 599
EP  - 622
DO  - https://doi.org/10.7494/OpMath.202610081
UR  - https://doi.org/10.7494/OpMath.202610081
AB  - 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.
KW  - subelliptic problem
KW  - Hardy potential
KW  - Palais-Smale condition
KW  - concentration-compactness principle
SN  - 1232-9274
ER  -