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Pré-Publication, Document De Travail Année : 2020

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

Résumé

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 $$J_{p,\beta}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2\beta} \right)^{\frac{p}{2-p}}-\ln \int_\Sigma \left(e^{u_+^p}-1\right) dv_g\,,$$ 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 $\beta>0$.
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Dates et versions

hal-02960649 , version 1 (07-10-2020)
hal-02960649 , version 2 (01-07-2022)

Identifiants

  • HAL Id : hal-02960649 , version 1

Citer

Francesca de Marchis, Andrea Malchiodi, Luca Martinazzi, Pierre-Damien Thizy. Critical points of the Moser-Trudinger functional on closed surfaces.. 2020. ⟨hal-02960649v1⟩

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