Existence of infinitely many solutions for critical sub-elliptic systems via genus theory

Abstract

We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities $ \begin{equation*} \left\{\begin{aligned} -\Delta_{\mathbb{G}}u-\mu_{1}\frac{\psi^{2}u}{\text{d}(z)^{2}} = \lambda_{1}\frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{\text{d}(z)^{\alpha}}+\beta p_{1}f(z)\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G},\\ -\Delta_{\mathbb{G}}v-\mu_{2}\frac{\psi^{2}v}{\text{d}(z)^{2}} = \lambda_{2}\frac{\psi^{\alpha}|v|^{2^*(\alpha)-2}v}{\text{d}(z)^{\alpha}}+\beta p_{2}f(z)\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G}, \end{aligned}\right. \end{equation*} $ where $ -\Delta_{\mathbb{G}} $ is the sub-Laplacian on Carnot group $ \mathbb{G} $, $ \mu_{1} $, $ \mu_{2}\in [0, \mu_{\mathbb{G}}) $, $ \alpha, \, \gamma\in (0, 2) $, $ \lambda_{1} $, $ \lambda_{2} $, $ \beta $, $ p_{1} $, $ p_{2} > 0 $ with $ 1 < p_{1}+p_{2} < 2 $, $ \text{d}(z) $ is the $ \Delta_{\mathbb{G}} $-gauge, $ \psi = |\nabla_{\mathbb{G}}\text{d}(z)| $, $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $ is the critical Sobolev-Hardy exponents, and $ \mu_{\mathbb{G}} = (\frac{Q-2}{2})^{2} $ is the best Hardy constant on $ \mathbb{G} $. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when $ \beta $ or $ \lambda_{1} $, $ \lambda_{2} $ belong to a suitable range.

Keywords

• sub-laplacian problem
• critical exponents
• genus theory
• carnot groups