Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n.