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

Abstract

AbstractWe study the following Choquard type equation in the whole plane $$\begin{aligned} (C)\quad -\Delta u+V(x)u=(I_2*F(x,u))f(x,u),\quad x\in \mathbb {R}^2 \end{aligned}$$ ( C ) - Δ u + V ( x ) u = ( I 2 ∗ F ( x , u ) ) f ( x , u ) , x ∈ R 2 where $$I_2$$ I 2 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).