Schr\"odinger-Newton equations in dimension two via a Pohozaev-Trudinger log-weighted inequality

We study the following Choquard type equation in the whole plane $© -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in$\mathbb{R}$^2$where$I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr"odinger potential and the nonlinearity $f(x,u)$, whose primitive in $u$ vanishing at zero is $F(x,u)$, exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to $(C)$.