On semilinear Grushin--Schr\"odinger equation in $\mathbb{R}^N$

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -\Delta_\gamma u + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $\Delta_\gamma$ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^\gamma$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.