An improved singular Trudinger–Moser inequality in unit ball

Abstract

Let B⊂Rn(n≥2) be the unit ball centered at the origin with radius 1. Let β, 0≤β<n, be fixed. Defineλβ(B)=infu∈W01,n(B),u≢0⁡∫B|∇u|ndx∫B|x|−β|u|ndx. Suppose that γ satisfies γαn+βn=1, where αn=nωn−11/(n−1), ωn−1 is the area of the unit sphere in Rn. Using rearrangement argument, we prove that for any α, 0≤α<λβ(B), there holdssupu∈W01,n(B),∫B|∇u|ndx≤1⁡∫B|x|−βeγ|u|nn−1(1+α∫B|x|−β|u|ndx)1n−1dx<+∞. Moreover, we prove that the above supremum is infinity for α≥λβ(B). This improves earlier results of Yang [15] and Adimurthi and Sandeep [2] in the unit ball.