Critical points of the Moser-Trudinger functional on a disk

Source abstract

On the unit disk B_1\subset \mathbb R{2} we study the Moser-Trudinger functional E(u)=\int_{B_1}\Big(e^{u^2}-1\Big)dx,\quad u\in H^1_0(B_1) and its restrictions E|_{M_\Lambda} , where M_{\Lambda}:=\{u\in H^1_0(B_1):\|u\|^2_{H^1_0}=\Lambda\} for \Lambda>0 . We prove that if a sequence u_k of positive critical points of E|_{M_{\Lambda_k}} (for some \Lambda_k>0 ) blows up as k\to\infty , then \Lambda_k\to 4\pi , and u_k\to 0 weakly in H^1_0(B_1) and strongly in C^1_{\mathrm {loc}}(\overline B_1\setminus\{0\}) . Using this fact we also prove that when \Lambda is large enough, then E|_{M_\Lambda} has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.