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

International audienceGiven 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$$J_{p,\beta}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2\beta} \right)^{\frac{p}{2-p}}-\ln \int_\Sigma \left(e^{u_+^p}-1\right) dv_g\,,$$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$$F(u):=\int_\Sigma \left(e^{u^2}-1\right)dv_g$$constrained to $\mathcal{E}_\beta:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=\beta\right\}$ for any $\beta>0$