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In this work we study the existence of nontrivial solution for the followingclass of semilinear degenerate elliptic equations $$ -\Delta_{\gamma} u + a(z)u= f(u) ~~ \mbox{in} ~~ \mathbb{R}^{N}, $$ where $\Delta_{\gamma}$ is known asthe {\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 technicalconditions. In order to overcome some difficulties involving this type ofoperator, we have proved some compactness results that are crucial in the proofof our main results. For the case $a=1$, we have showed a Berestycki-Lions typeresult.

A Berestycki-Lions type result for a class of degenerate elliptic problems involving the Grushin Operator