Singular Type Trudinger-Moser Inequalities with Logarithmic Weights and the Existence of Extremals

n this paper, we study the existence of extremals for thefollowing singular critical Trudinger-Moser inequality with logarithmicweights:supu is an element of W1,n0,r(B,omega beta),& Vert;u & Vert;omega beta <= 1 integral Bexp(alpha n,beta,sigma|u|n(n-1)(1-beta))|x|sigma dx<infinity, where B is the unit ball inRn,beta is an element of[0,1),sigma is an element of[0,n),alpha n,beta,sigma=(n-sigma)[omega 1n-1n-1(1-beta)]11-beta,W1,n0,r(B, omega beta) denotes the radial weighted Sobolevspace with the norm & Vert;u & Vert;omega beta=(integral B|del u|n omega beta(x)dx)1n,omega beta(x)=(loge|x|)beta(n-1).Moreover, form>0, we establish a singular supercritical Trudinger-Moser inequality with logarithmic weightssupu is an element of W1,n0,r(B,omega beta),& Vert;u & Vert;omega beta <= 1 integral Bexp((alpha n,beta,sigma+|x|m)|u|n(n-1)(1-beta))|x|sigma dx<infinity,and prove the existence of its extremal functions