Critical points of the Moser-Trudinger functional on closed surfaces

2020/10/14 by Francesca De Marchis, De Marchis, Francesca, Andrea Malchiodi +5 · 1 citation
Mathematics · #35J20 #35J60 #49J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2010.07397

openalex publication_date 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Jp,β(u)=(2-p)/(2)(\fracp‖u‖H122β )(p)/(2-p)-ln ∫Σ(eu+p-1) f dvg, for every p∈ (1,2) and β>0, or for p=1 and β∈ (0,∞)∖ 4πℕ. Letting p\uparrow 2 we obtain positive critical points of the Moser-Trudinger functional F(u):=∫Σ(eu2-1)f dvg constrained to Eβ:=\v s.t. ‖v‖H12=β\ for any β>0.

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