An improved singular Trudinger-Moser inequality in dimension two

Let Omega subset of R-2 be a smooth bounded domain and W-0(1,2)(Omega) be the usual Sobolev space. Let beta, 0 <= beta < 2, be fixed. Define for any real number p > 1, lambda(p,beta)(Omega) = inf(u is an element of W01,2 (Omega), u not equivalent to 0) parallel to del u parallel to(2)(2)/parallel to u parallel to(2)(p,beta), where parallel to.parallel to(2) denotes the standard L-2-norm in Omega and parallel to u parallel to(p,beta) = (integral Omega vertical bar x vertical bar(-beta)vertical bar u vertical bar(p)dx)(1/p) Suppose that gamma satisfies gamma/4 pi + beta/2 = 1. Using a rearrangement argument, the author proves that sup(u is an element of W01,2 (Omega), parallel to del u parallel to 2 <= 1)integral(Omega) vertical bar x vertical bar(-beta)e(gamma u2(1+alpha parallel to u parallel to p,beta 2)) dx is finite for any alpha, 0 <= alpha < lambda(p,beta)(B-R), where B-R stands for the disc centered at the origin with radius R verifying that pi R-2 is equal to the area of Omega. Moreover, when Omega = B-R, the above supremum is infinity if alpha >= lambda(p,beta)(B-R). This extends earlier results of Adimurthi and Druet, Y. Yang, Adimurthi and Sandeep, Adimurthi and Yang, Lu and Yang, and J. Zhu in dimension two.