Critical Ambrosetti-Prodi type problems on Carnot groups

In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -Δ_{\mathbb{G}} u &= λu + u_{+}^{2_{Q}^{*}-1} + f(ξ) \quad &&\text{in } Ω,\\[2mm] u &= 0 \quad &&\text{on } \