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