Trudinger type inequalities in RN and their best exponents

We study Trudinger type inequalities in R-N and their best exponents alpha(N). We show for alpha is an element of (0, alpha(N)), alpha(N) = N omega(N-1)((N-1)) (omega(N-1) is the surface area of the unit sphere in R-N), there exists a constant C-alpha>0 such that [GRAPHICS] for all u is an element of W-1,W-N (R-N) \ {0}. Here Phi(N) (xi) is defined by [GRAPHICS] is also shown that (*) with alpha greater than or equal to alpha(N) is false, which is different from the usual Trudinger's inequalities in bounded domains.