  In this work we study the existence of nontrivial solution for the following
class of semilinear degenerate elliptic equations $$ -\Delta_{\gamma} u + a(z)u
= f(u) ~~ \mbox{in} ~~ \mathbb{R}^{N}, $$ where $\Delta_{\gamma}$ is known as
the {\it Grushin operator}, $z:=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{k}$ and
$m+k=N\geq 3$, $f$ and $a$ are continuous function satisfying some technical
conditions. In order to overcome some difficulties involving this type of
operator, we have proved some compactness results that are crucial in the proof
of our main results. For the case $a=1$, we have showed a Berestycki-Lions type
result.
