Critical points of the Moser-Trudinger functional on a disk

On the unit disk B-1 subset of R-2 we study the Moser-Trudinger functional
E(u) = integral(B1) (e(u2) - 1)dx, is an element of H-0(1) (B-1)
and its restrictions E vertical bar M-Lambda, where M-Lambda := {u is an element of H-0(1) (B-1): parallel to u parallel to(2)(H01) = Lambda} for Lambda > 0. We prove that if a sequence u(k) of positive critical points of E vertical bar(M Lambda k) ( for some Lambda(k) > 0) blows up as k -> infinity, then Lambda(k) -> 4 pi, and u(k) -> 0 weakly in H-0(1) (B-1) and strongly in C-loc(1) ((B) over bar (1) \ {0}).
Using this fact we also prove that when Lambda is large enough, then E vertical bar M-Lambda has no positive critical point, complementing previous existence results by Carleson-Chang, Struwe and Lamm-Robert-Struwe.