In this paper, we are concerned with the multiplicity of nontrivial solutions for the following semilinear degenerate elliptic equation in $\mathbb{R}^N$ \begin{equation*} -\Delta_\lambda u + V(x) u = f(x,u) \;\text{ in } \mathbb{R}^N, N\ge 3, \end{equation*} where $V: \mathbb{R}^N\to \mathbb{R}$ is a potential function and allowed to be vanishing at infinitely, $f: \mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ is a given function and $\Delta_\lambda$ is the strongly degenerate elliptic operator. Some results on the multiplicity of solutions are proved under suitable assumptions on the potential $V$ and the nonlinearity $f.$ The proof is based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti-Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results \cite{AS13, Hamdani20, Luyen19, LT18, Tang13, TU20}.