  We obtain the existence of ground state solution for the nonlocal problem $$
m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-\Delta u
+ b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a
Kirchhoff-type function, $b$ may be negative and noncoercive, $A$ is locally
bounded and the function $f$ has critical exponential growth. We also obtain
new results for the classical Schr\"odinger equation, namely the local case
$m\equiv 1$. In the proofs we apply Variational Methods beside a new
Trudinger-Moser type inequality.
