  In this paper, we will establish the best constants for certain classes of
weighted Moser-Trudinger inequalities on the entire Euclidean spaces
$\mathbb{R}^N$. We will also prove the existence of maximizers of these sharp
weighted inequalities. The class of functions considered here are not
necessarily spherically symmetric. Our inequality in Theorem 1.1 improves the
earlier one where such type of inequality was only considered for spherically
symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in \cite{INW}
(except in the case $s\not=0$). Since
$\int_{\mathbb{R}^N}\Phi_N(\alpha|u|^{N/(N-1)})\frac{dx}{|x|^t}\le
\int_{\mathbb{R}^N}e^{\alpha|u|^{N/(N-1)}}|u|^N\frac{dx}{|x|^t}$, our
inequality in Theorem 1.2 is stronger than the inequality in Theorem 1.1.
  We note that it suffices for us to prove the above inequalities for all
functions not necessarily radially symmetric when $s=t$ by the well-known
Caffareli-Kokn-Nirenberg inequalities \cite{CKN}.
