Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality

We study the following Choquard type equation in the whole plane (C)-Δu+V(x)u=(I2∗F(x,u))f(x,u),x∈R2where I2 is the Newton logarithmic kernel, V is a bounded Schrödinger 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).