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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 aKirchhoff-type function, $b$ may be negative and noncoercive, $A$ is locallybounded and the function $f$ has critical exponential growth. We also obtainnew results for the classical Schr\"odinger equation, namely the local case$m\equiv 1$. In the proofs we apply Variational Methods beside a newTrudinger-Moser type inequality.

Kirchhoff-Schr\"odinger equations in $\mathbb{R}^2$ with critical exponential growth and indefinite potential