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

Abstract We study the following Choquard type equation in 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 = I 2 ∗ F , f ∈ R where $$I_2$$ is Newton logarithmic kernel, V a bounded Schrödinger potential and nonlinearity f ( x , u ), whose primitive vanishing at zero F exhibits highest possible growth which of exponential type. The competition between kernel demands for new tools. A proper function space setting provided by weighted version Pohozaev–Trudinger inequality enables us to prove existence variational, particular finite energy solutions C ).