On semilinear Grushin--Schrödinger equation in ${\mathbb{R}}^N$

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in $\mathbb{R}$^N, \end{equation*} where ${\mathrm{\Delta }}_{\gamma }$ 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 $(1+\mathrm{\mid }z\mathrm{\mid }{)}^a$, $a\in \mathbb{R}$. 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.