Critical points of the Moser-Trudinger functional on closed surfaces. (English) Zbl 1506.35050
Authors’ 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
\[
I_{p,\beta}(u)=\frac{2-p}{2}\left(\frac{p \|u\|^2_{H^1}}{2\beta} \right)^\frac{p}{2-p} - \ln \int_{\Sigma}\left(e^{u^p_+} -1 \right) f d v_{g_0},
\]
for every \(p\in (1,2)\) and \(\beta>0\), or for \(p=1\) and \(\beta \in (0,\infty)\backslash 4\pi\mathbb{N}\). Letting \(p \uparrow 2\) we obtain positive critical points of the Moser-Trudinger functional
\[
F(u):= \int_\Sigma \left(e^{u^2} -1 \right) f d v_{g_0}
\]
constrained to \(\mathcal{E}_\beta := \left\{v\ \ s.t. \ \|v\|^2_{H^1}=\beta \right\}\) for any \(\beta >0\).
Reviewer: David Kapanadze (Tbilisi)
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 49J35 | Existence of solutions for minimax problems |
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