Critical points of the Moser-Trudinger functional on closed surfaces

TL;DR

This paper establishes the existence of positive critical points for a class of Moser-Trudinger functionals on closed surfaces using minmax schemes, compactness, and energy estimates, extending results to the limiting case as p approaches 2.

Contribution

It introduces a novel minmax approach combined with compactness and quantization techniques to find critical points of the Moser-Trudinger functional on closed surfaces for a range of parameters.

Findings

Existence of positive critical points for the functional $J_{p,eta}$ for $p o 2$.

Extension of critical point existence to the Moser-Trudinger functional constrained on energy spheres.

Application of sharp energy estimates and quantization results to the problem.

Abstract

Given a closed Riemann surface $(\Sigma,g)$ and any positive smooth weight, 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 $$J_{p,\beta}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2\beta} \right)^{\frac{p}{2-p}}-\ln \int_\Sigma (e^{u_+^p}-1) f dv_g,$$ 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 $$F(u):=\int_\Sigma (e^{u^2}-1)f dv_g$$ constrained to $\mathcal{E}_\beta:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=\beta\right\}$ for any $\beta>0$.