Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [20]), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space R-N (N >= 2). Precisely, if 0 <= beta < 1 and 0 < gamma <= 1 - beta, then there holds for any tau > 0, u is an element of W-1,W-N(R-N), integral(sup)(RN)(vertical bar del u vertical bar(N)+tau vertical bar u vertical bar(N))dx <= 1 integral(RN) 1/vertical bar x vertical bar(N beta) (e(alpha N gamma vertical bar u vertical bar N/N-1) - Sigma(N-2)(k=0) N-alpha k(gamma k)vertical bar u vertical bar(kN/N-1)/k!) dx < infinity, where alpha(N) = N-omega N-1(1/(N-1)) and omega(N-1) is the area of the unit sphere in R-N. The above inequality is sharp in the sense that if gamma > 1 - beta, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case beta = 0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case 0 < beta < 1 and prove that for all tau > 0, 0 < beta < 1 and 0 < y <= 1 - beta, extremal functions for the above inequalities exist. The proof is based on blow-up analysis. (C) 2017 Elsevier Inc. All rights reserved.