Critical points of the Moser–Trudinger functional on closed surfaces

Given a closed Riemann surface ( Σ , g 0 ) and any positive weight f ∈ C ∞ ( Σ ) , 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 , β ( u ) = 2 - p 2 p ‖ u ‖ H 1 2 2 β p 2 - p - ln ∫ Σ e u + p - 1 f d v g 0 , for every p ∈ ( 1 , 2 ) and β > 0 , or for p = 1 and β ∈ ( 0 , ∞ ) \ 4 π N . Letting p ↑ 2 we obtain positive critical points of the Moser-Trudinger functional F ( u ) : = ∫ Σ e u 2 - 1 f d v g 0 constrained to E β : = v s.t. ‖ v ‖ H 1 2 = β for any β > 0 .