Positive solutions with prescribed mass for a planar Choquard equation with critical growth
Ling Huang, Giulio Romani
TL;DR
This paper studies normalised solutions of the planar Choquard equation $- abla u+\{u}$ in $\mathbb{R}^2$ with a prescribed mass $\|u\|_2^2=a^2$, where the nonlinearity $f$ has exponential critical growth and the nonlocal term involves the Riesz kernel $I_\alpha(x)=\frac{c_{2,\alpha}}{|x|^{2-\alpha}}$. Using a variational framework on the $L^2$-sphere $S_a$ and a mountain-pass geometry, the authors construct a Palais–Smale–Pohozaev sequence that yields a nontrivial, positive radial solution $(u_a,\lambda_a)$ for every $a>0$, with $\lambda_a>0$ and $u_a>0$. A detailed analysis with a two-parameter scaling and Pohozaev functional provides an upper bound for $\lambda_a$ in terms of the problem’s data and shows that, under a monotonicity assumption on $\widetilde F(t)=f(t)t-(2+\alpha)F(t)/2$, $u_a$ is a normalised ground state in $H^1(\mathbb{R}^2)$. The results extend and sharpen prior work on planar Choquard equations by removing a lower-bound constraint and proving existence for all masses, with a precise energy and frequency control, and a rearrangement-based argument confirming global minimality of the radial solution.
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).