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

TL;DR

This paper proves the existence of positive radial solutions with prescribed mass for a planar Choquard equation involving critical exponential growth, extending previous results and analyzing the variational structure of the problem.

Contribution

It establishes existence and ground state properties of solutions for a critical Choquard equation in the plane, extending prior work to this nonlocal setting.

Findings

Existence of positive radial solutions for all prescribed masses.

Solutions are ground states under monotonicity conditions.

Extension of previous results to the 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).