On criticality coupled sub-Laplacian systems with Hardy type potentials on Stratified Lie groups

In this work, our main concern is to study the existence and multiplicity of solutions for the following sub-elliptic system with Hardy type potentials and multiple critical exponents on Carnot group

$$\{\begin{array}{rlr}& -{\mathrm{\Delta }}_{\mathbb{G}}u=\frac{{\psi }^{\alpha }|u{|}^{2^{\ast }(\alpha )-2}u}{d(z{)}^{\alpha }}+\frac{p_1}{2^{\ast }(\gamma )}\frac{{\psi }^{\gamma }|u{|}^{p_1-2}u|v{|}^{p_2}}{d(z,z_0{)}^{\gamma }}+\lambda h(z)\frac{{\psi }^{\sigma }|u{|}^{q-2}u}{d(z{)}^{\sigma }}& \text{in}\mathrm{\Omega },\\ & -{\mathrm{\Delta }}_{\mathbb{G}}v=\frac{{\psi }^{\beta }|v{|}^{2^{\ast }(\beta )-2}v}{d(z{)}^{\beta }}+\frac{p_2}{2^{\ast }(\gamma )}\frac{{\psi }^{\gamma }|u{|}^{p_1}|v{|}^{p_2-2}v}{d(z,z_0{)}^{\gamma }}+\lambda h(z)\frac{{\psi }^{\sigma }|v{|}^{q-2}v}{d(z{)}^{\sigma }}& \text{in}\mathrm{\Omega },\\ & u=v=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $-{\mathrm{\Delta }}_{\mathbb{G}}$ is a sub-Laplacian on Carnot group $\mathbb{G}$, $\alpha ,\beta ,\gamma ,\sigma \in [0,2)$, $d$ is the ${\mathrm{\Delta }}_{\mathbb{G}}$-natural gauge, $\psi =|{\mathrm{\nabla }}_{\mathbb{G}}d|$ and ${\mathrm{\nabla }}_{\mathbb{G}}$ is the horizontal gradient associated to ${\mathrm{\Delta }}_{\mathbb{G}}$. The positive parameters $\lambda$, $q$ satisfy $0<\lambda <\mathrm{\infty }$, $1<q<2$, and $p_1$, $p_2>1$ with $p_1+p_2=2^{\ast }(\gamma )$, here $2^{\ast }(\alpha ):=\frac{2(Q-\alpha )}{Q-2}$, $2^{\ast }(\beta ):=\frac{2(Q-\beta )}{Q-2}$ and $2^{\ast }(\gamma )=\frac{2(Q-\gamma )}{Q-2}$ are the critical Hardy-Sobolev exponents, $Q$ is the homogeneous dimension of the space $\mathbb{G}$. By means of variational methods and the mountain-pass theorem of Ambrosetti and Rabonowitz, we study the existence of multiple solutions to the sub-elliptic system.