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

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space RN(N≥2). Precisely, if 0≤β<1 and 0<γ≤1−β, then there holds for any τ>0,supu∈W1,N(RN),∫RN(|∇u|N+τ|u|N)dx≤1⁡∫RN1|x|Nβ(eαNγ|u|NN−1−∑k=0N−2αNkγk|u|kNN−1k!)dx<∞, where αN=NωN−11/(N−1) and ωN−1 is the area of the unit sphere in RN. The above inequality is sharp in the sense that if γ>1−β, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β=0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case 0<β<1 and prove that for all τ>0, 0<β<1 and 0<γ≤1−β, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.