Critical points of the Moser-Trudinger functional on closed surfaces

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 $$J_{p,eta}(u)=rac{2-p}{2}left(rac{p|u|_{H^1}^2}{2eta} ight)^{rac{p}{2-p}}-ln int_Sigma (e^{u_+^p}-1) dv_g,,$$ for every $pin (1,2)$ and $eta>0$, {or} for $p=1$ and $etain (0,infty)setminus 4pimathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=int_Sigma (e^{u^2}-1)dv_g$$ constrained to $mathcal{E}_eta:=left{v ext{ s.t. }|v|_{H^1}^2=eta ight}$ for any $eta>0$