# Critical points of the Moser-Trudinger functional on closed surfaces

created by malchiodi on 07 Oct 2020

[BibTeX]

Preprint

Inserted: 7 oct 2020
Last Updated: 7 oct 2020

Pages: 58
Year: 2020

Abstract:

Given a closed Riemann surface $(\Sigma,g)$, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $J_{p,\beta}(u)=\frac{2-p}{2}\left(\frac{p | u |_{H^1}^2}{2\beta} \right)^{\frac{p}{2-p}}-\ln \int_\Sigma (e^{u_+^p}-1) dv_g\,,$ for every $p\in (1,2)$ and $\beta>0$, or for $p=1$ and $\beta\in (0,\infty)\setminus 4\pi\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $F(u):=\int_\Sigma (e^{u^2}-1)dv_g$ constrained to $\mathcal{E}_\beta:=\left\{v\text{ s.t. } | v | _{H^1}^2=\beta\right\}$ for any $\beta>0$.