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

Abstract

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{equation*} \left\{\begin{aligned} &-\Delta_{\mathbb{G}}u = \frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{d(z)^{\alpha}}+ \frac{p_{1}}{2^*(\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 } \, \, \Omega, \\ &-\Delta_{\mathbb{G}}v = \frac{\psi^{\beta}|v|^{2^*(\beta)-2}v}{d(z)^{\beta}}+ \frac{p_{2}}{2^*(\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 } \, \, \Omega, \\ &\quad u = v = 0\, \, &\text{on } \, \, \partial\Omega, \end{aligned}\right. \end{equation*} $ where $ -\Delta_{\mathbb{G}} $ is a sub-Laplacian on Carnot group $ \mathbb{G} $, $ \alpha, \beta, \gamma, \sigma\in [0, 2) $, $ d $ is the $ \Delta_{\mathbb{G}} $-natural gauge, $ \psi = |\nabla_{\mathbb{G}}d| $ and $ \nabla_{\mathbb{G}} $ is the horizontal gradient associated to $ \Delta_{\mathbb{G}} $. The positive parameters $ \lambda $, $ q $ satisfy $ 0 < \lambda < \infty $, $ 1 < q < 2 $, and $ p_{1} $, $ p_{2} > 1 $ with $ p_{1}+p_{2} = 2^*(\gamma) $, here $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $, $ 2^*(\beta): = \frac{2(Q-\beta)}{Q-2} $ and $ 2^*(\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.

Keywords

• sub-laplacian system
• critical exponents
• hardy-type potential
• carnot groups