Doubly critical problems involving Sub-Laplace operator on Carnot group

Abstract

<p>This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group $ \mathbb{G} $:</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{\mathbb{G}}u-\mu \frac{\psi^{2}(\xi) u }{\text{d}(\xi)^2} = \vert u\vert^{p-2}u +\psi^{\alpha}(\xi)\frac{\vert u\vert^{2^*(\alpha)-2}u}{\text{d}(\xi)^{\alpha}}, \quad u\in S^{1, 2}(\mathbb{G}). $\end{document} </tex-math></disp-formula></p><p>Here, $ p\in (1, 2^*] $, $ \alpha\in (0, 2) $, $ \mu\in [0, \mu_{\mathbb{G}}) $, $ 2^* = \frac{2Q}{Q-2} $, and $ 2^*(\alpha) = \frac{2(Q-\alpha)}{Q-2} $. By means of variational techniques, we extended the arguments developed in ^{[<xref ref-type="bibr" rid="b1">1</xref>]}. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms.</p>