Critical Ambrosetti-Prodi type problems on Carnot groups

TL;DR

This paper studies critical Ambrosetti-Prodi problems involving the sub-Laplacian on Carnot groups, establishing existence, multiplicity, and bifurcation results depending on the parameter relative to eigenvalues.

Contribution

It extends classical Ambrosetti-Prodi results to the setting of Carnot groups, analyzing critical problems with the sub-Laplacian and eigenvalue-based bifurcation phenomena.

Findings

Existence of solutions for mbda<mbda_1 and mbda>mbda_1.

Multiplicity of solutions when mbda<mbda_1.

Bifurcation from eigenvalues mbda_k for k>1.

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 \[ \left\{ \begin{aligned} -\Delta_{\mathbb{G}} u &= \lambda u + u_{+}^{2_{Q}^{*}-1} + f(\xi) \quad &&\text{in } \Omega,\\[2mm] u &= 0 \quad &&\text{on } \partial\Omega, \end{aligned} \right. \] where $\Delta_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $\Omega \subset \mathbb{G}$ is an open bounded domain with smooth boundary, $\lambda>0$ is a real parameter, $f\in L^{\infty}(\Omega)$, $u_{+}$ denotes the positive part of $u$, and $2_{Q}^{*}$ 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 $\lambda<\lambda_{1}$ and $\lambda>\lambda_{1}$, where $\lambda_{k}$ denotes the $k$-th Dirichlet eigenvalue of $-\Delta_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $\lambda=\lambda_{1}$ and show that bifurcation occurs from each eigenvalue $\lambda_{k}, k >1$.