JournalsjemsVol. 16, No. 5pp. 893–908

Critical points of the Moser-Trudinger functional on a disk

  • Andrea Malchiodi

    Scuola Normale Superiore, Pisa, Italy
  • Luca Martinazzi

    Rutgers University, Piscataway, Switzerland
Critical points of the Moser-Trudinger functional on a disk cover
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Abstract

On the unit disk B1R2B_1\subset \mathbb R{2} we study the Moser-Trudinger functional

E(u)=B1(eu21)dx,uH01(B1)E(u)=\int_{B_1}\Big(e^{u^2}-1\Big)dx,\quad u\in H^1_0(B_1)

and its restrictions EMΛE|_{M_\Lambda}, where MΛ:={uH01(B1):uH012=Λ}M_{\Lambda}:=\{u\in H^1_0(B_1):\|u\|^2_{H^1_0}=\Lambda\} for Λ>0\Lambda>0. We prove that if a sequence uku_k of positive critical points of EMΛkE|_{M_{\Lambda_k}} (for some Λk>0\Lambda_k>0) blows up as kk\to\infty, then Λk4π\Lambda_k\to 4\pi, and uk0u_k\to 0 weakly in H01(B1)H^1_0(B_1) and strongly in Cloc1(B1{0})C^1_{\mathrm {loc}}(\overline B_1\setminus\{0\}). Using this fact we also prove that when Λ\Lambda is large enough, then EMΛE|_{M_\Lambda} has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

Cite this article

Andrea Malchiodi, Luca Martinazzi, Critical points of the Moser-Trudinger functional on a disk. J. Eur. Math. Soc. 16 (2014), no. 5, pp. 893–908

DOI 10.4171/JEMS/450