An improved singular Trudinger-Moser inequality in unit ball

Let B subset of R (n >= 2) be the unit ball centered at the origin with radius 1. Let beta, 0 <= beta < n, be fixed. Define lambda(beta)(B)= inf(u is an element of 01,n) (B) , u not equivalent to 0 integral(B) vertical bar del vertical bar(n)dx/integral(B) vertical bar x vertical bar(-beta) vertical bar u vertical bar(n)dx Suppose that gamma satisfies gamma/alpha(n) + beta/n = 1, where alpha(n) = n omega(1/(n-1))(n-1) is the area of the unit sphere in R-n. Using rearrangement argument, we prove that for any alpha, 0 <= alpha <lambda(beta) (B), there holds u is an element of W-0(1,n) (B) integral(sup)(B) vertical bar del u vertical bar(n) dx <=integral vertical bar x vertical bar(-beta) e(gamma vertical bar u vertical bar) (n/n-1) ((1 + alpha) integral B vertical bar x vertical bar-beta vertical bar u vertical bar n) (dx) 1/n-1) dx < + infinity. Moreover, we prove that the above supremum is infinity for a >= lambda(beta) (B). This improves earlier results of Yang [15] and Adimurthi and Sandeep [2] in the unit ball. (C) 2015 Elsevier Inc. All rights reserved.