Positive solutions with prescribed mass for a planar Choquard equation with critical growth

Abstract

We study normalised solutions for a Choquard equation in the plane with polynomial Riesz kernel and exponential nonlinearities, which are critical in the sense of Trudinger-Moser. For all prescribed values of the mass, we prove existence of a positive radial solution by a variational argument, which exploits a delicate analysis on the mountain pass level. Under an additional monotonicity assumption on the nonlinearity, such a solution turns out to be also a ground state in $H^1(\mathbb R^2)$. Our work extends the results by Dou, Huang, and Zhong (J Geom Anal 34(10):317, 2024) to the Choquard setting, improving in several directions those by Deng and Yu in (Z Angew Math Phys 74(3):103, 2023).