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 \(\left\{\begin{array}{cc}(-\triangle_{\mathbb{G}})^s u=f(x,u)+g(x,u) & \mbox{in}~\Omega; \\[2mm]u\in {\cal{H}}_0^s(\Omega),\end{array}\right.\) where s ∈ (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 Ω × 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.
