Critical points of the Moser-Trudinger functional on closed surfaces

Abstract

Given a closed Riemann surface $(\Sigma ,g)$ and any positive smooth weight, 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 $$J_{p,\beta }(u)=\frac{2-p}{2}{\left(\frac{p\Vert u{\Vert }_{H^1}^2}{2\beta }\right)}^{\frac{p}{2-p}}-\ln {\int }_{\Sigma }(e^{u_{+}^p}-1)fd{v}_g,$$ 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 $$F(u):={\int }_{\Sigma }(e^{u^2}-1)fd{v}_g$$ constrained to ${\mathcal{E}}_{\beta }:=\left\{v\text{ s.t. }\Vert v{\Vert }_{H^1}^2=\beta \right\}$ for any $\beta >0$.