A TRUDINGER-MOSER TYPE INEQUALITY AND ITS EXTREMAL FUNCTIONS IN DIMENSION TWO

Let Omega be a smooth bounded domain in R-2, W-0(1,2) (Omega) be the usual Sobolev space and. (Omega) be the first eigenvalue of the Laplace-Beltrami operator, say lambda(Omega) = inf u epsilon W-0(1,2)(Omega),integral(u2)(Omega)dx=1 integral(Omega)vertical bar del u vertical bar(2)dx. Using blow-up analysis, we prove that for real numbers alpha < lambda (Omega) and beta < 4 pi, the supremum sup (u epsilon W01,2 (Omega), integral Omega vertical bar del u vertical bar 2dx-alpha alpha integral Omega u2dx <= 1) integral(Omega)(e(4 pi u2) - beta u(2)) dx can be attained by some function u epsilon W-0(1,2) (Omega) with integral(Omega) vertical bar del u vertical bar(2)dx-alpha alpha integral(Omega) u(2)dx = 1. In the case beta = 0, this is reduced to a result of Yang [24].