Critical points of the Moser-Trudinger functional on closed surfaces

Given a closed Riemann surface $(Σ,g)$ 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 $J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $F(u):=\int_Σ(e^{u^2}-1)f dv_g$ constrained to $\mathcal{E}_β:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=β\right\}$ for any $β>0$.

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