关键词: Carnot group, fractional sub-Laplacian, perturbation methods, fractional Dirichlet eigenvalue, Morse index

Abstract: In this paper, we investigate a Dirichlet boundary value problem for a class of fractional degenerate elliptic equations on homogeneous Carnot groups $\mathbb{G}=(\mathbb{R}^n ,\circ)$, namely $$\begin{equation*} \left\{ \begin{array}{cc} (-\triangle_{\mathbb{G}})^s u=f(x,u)+g(x,u) & \mbox{in} \Omega; \\[2mm] u\in {\cal{H}}_0^s(\Omega), \end{array} \right. \end{equation*}$$ where $s\in(0,1)$, $\Omega\subset\mathbb{G}$ is a bounded open domain, $(-\Delta_{\mathbb{G}} )^s$ is the fractional sub-Laplacian, ${\cal{H}}_0^s (\Omega)$ denotes the fractional Sobolev space, $f(x,u)\in C(\overline{\Omega}\times\mathbb{R}), g(x,u)$ is a Carathéodory function on $\Omega\times\mathbb{R}$. Using perturbation methods and Morse index estimates in conjunction with fractional Dirichlet eigenvalue estimates, we establish the existence of multiple solutions to the problem.

Key words: Carnot group, fractional sub-Laplacian, perturbation methods, fractional Dirichlet eigenvalue, Morse index