An improved singular Trudinger-Moser inequality in RN and its extremal functions

Let N≥2, W1,N(RN) be the standard Sobolev space. For any real numbers τ>0 and p≥N, we denoteλN,p,τ=infu∈W1,N(RN),u≢0⁡∫RN(|∇u|N+τ|u|N)dx(∫RN|u|pdx)N/p and define a norm in W1,N(RN) by‖u‖N,p,α,τ=(∫RN(|∇u|N+τ|u|N)dx−α(∫RN|u|pdx)N/p)1/N for any fixed α, 0≤α<λN,p,τ. If τ>0, 0<β<1, 0<α<λN,p,τ, p≥N≥2, and 0≤γ≤αN, then using blow-up analysis, we prove that the supremumsupu∈W1,N(RN),‖u‖N,p,α,τ≤1⁡∫RN1|x|Nβ(eγ(1−β)|u|NN−1−∑k=0N−2γk(1−β)k|u|kNN−1k!)dx can be attained by some function u∈W1,N(RN) with ‖u‖N,p,α,τ≤1, where αN=NωN−11/(N−1) and ωN−1 is the area of the unit sphere in RN. This improves an inequality of Adimurthi and Yang and the result I obtained recently with Yang.