Preprint

Inserted: 7 oct 2020
Last Updated: 22 jul 2022

Pages: 60
Year: 2020
Notes:

Invent. Math., to appear.

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 ${I_{p,\beta}}(u)=\frac{2-p}{2}\left(\frac{p|u|_{H^1}^2}{2\beta} \right)^{\frac{p}{2-p}}-\ln \int_\Sigma (e^{u_+^p}-1) {f}\, dv_{{g_0}}\,,$ 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 $F(u):=\int_\Sigma (e^{u^2}-1){f}\,dv_{{g_0}}$ constrained to $\mathcal{E}_\beta:=\left\{v\text{ s.t. }|v|_{H^1}^2=\beta\right\}$ for any $\beta>0$.

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