Critical points of the Moser–Trudinger functional on closed surfaces

Abstract

Given a closed Riemann surface $(\Sigma ,{g_0})$ and any positive weight $f\in C^\infty (\Sigma )$, 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

$$\begin{aligned} {I_{p,\beta }}(u)=\frac{2-p}{2}\left( \frac{p\Vert u\Vert _{H^1}^2}{2\beta } \right) ^{\frac{p}{2-p}}-\ln \int _\Sigma \left( e^{u_+^p}-1\right) {f}\, dv_{{g_0}}, \end{aligned}$$

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

$$\begin{aligned} F(u):=\int _\Sigma \left( e^{u^2}-1\right) {f}\,dv_{{g_0}} \end{aligned}$$

constrained to ${\mathcal {E}}_\beta :=\left\{ v\text { s.t. }\Vert v\Vert _{H^1}^2=\beta \right\}$ for any $\beta >0$.