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 $eta>0$.

Critical points of the Moser-Trudinger functional on closed surfaces

Andrea Malchiodi;
2020

Abstract

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 $eta>0$.
File in questo prodotto:
File Dimensione Formato  
DMMT.pdf

accesso aperto

Tipologia: Submitted version (pre-print)
Licenza: Dominio pubblico
Dimensione 721.44 kB
Formato Adobe PDF
721.44 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/94397
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact