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

We study the following Choquard type equation in the whole plane (C) - Delta u + V(x)u=(I-2*F(x, u))f(x, u), x is an element of R-2 where is the Newton logarithmic kernel, V is a bounded Schrodinger 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).

Automated summary

In this study, we investigate a Choquard type equation in the whole plane, where the competition between the logarithmic kernel and exponential nonlinearity requires new tools. By applying a new weighted version of the Pohozaev-Trudinger inequality, a proper function space setting is provided to prove the existence of variational, particularly finite energy solutions.