Critical points of the Moser-Trudinger functional on closed surfaces

Given a closed Riemann surface $(\mathrm{\Sigma },g)$ 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 $$J_{p,\beta }(u)=\frac{2-p}{2}{\left(\frac{p\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2}{2\beta }\right)}^{\frac{p}{2-p}}-\ln {\int }_{\mathrm{\Sigma }}(e^{u_{+}^p}-1)fd{v}_g,$$ for every $p\in (1,2)$and$\beta>0$,orfor$p=1$and$\beta\in (0,\infty)\setminus 4\pi$\mathbb{N}$$\mathrm{.}Letting$p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):={\int }_{\mathrm{\Sigma }}(e^{u^2}-1)fd{v}_g$$ constrained to $$\mathcal{E}$\beta:=\left{v\text{ s.t. }|v|{H^1}^2=\beta\right}$forany$\beta>0$.