Critical Ambrosetti-Prodi type problems on Carnot groups

Abstract

In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -Δ_{\mathbb{G}} u &= λu + u_{+}^{2_{Q}^{*}-1} + f(ξ) \quad &&\text{in } Ω,\\[2mm] u &= 0 \quad &&\text{on } \partialΩ, \end{aligned} \right. \] where $\hat{I}{”}_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $\hat{I}\text{c◯}\subset \mathbb{G}$ is an open bounded domain with smooth boundary, $\hat{I}»>0$ is a real parameter, $f\in L^{\infty }(\hat{I}\text{c◯})$, $u_{+}$ denotes the positive part of $u$, and $2_Q^{\ast }$ is the critical Sobolev exponent associated with the homogeneous dimension $Q$. Motivated by the classical Ambrosetti-Prodi problem, we establish existence and multiplicity results for the cases $\hat{I}»<\hat{I}{»}_1$ and $\hat{I}»>\hat{I}{»}_1$, where $\hat{I}{»}_k$ denotes the $k$-th Dirichlet eigenvalue of $-\hat{I}{”}_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $\hat{I}»=\hat{I}{»}_1$ and show that bifurcation occurs from each eigenvalue $\hat{I}{»}_k,k>1$.

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