LyX: Equation number overlaps equation

Instead of using eqnarray you should use align or some other environment provided by amsmath (the reasons can be found in Avoid eqnarray!); to reduce the length of your formula and increase readability, I would suggest you to use some abbreviation in the equation and then give its meaning; for example, since m_{x} + m_{y} appears several times, you could abbreviate it using \sigma_{xy} (or some other name), and similarly for \lvert m_{x} - m_{y}\rvert (for which I used \delta_{xy}); your equation now becomes:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{align}
  \tau_{0}^{-} & = \frac{\sigma_{xy} - \delta_{xy}\sqrt{1 - r_{H}}}{2e^{2}n_{s}\rho_{\text{xx}}} \\
  \tau_{vv}^{-} & = \frac{24\tau_{0}^{-}m_{x}m_{y}\sigma_{xy}\delta_{xy}\sqrt{1 - r_{H}}}{\left(10m_{x}m_{y} + 3m_{x}^{2} + 3m_{y}^{2}\right)(\delta_{xy}^{2} - \sigma_{xy}\delta_{xy}\sqrt{1 - r_{H}})},
\end{align}
where $\sigma_{xy} = m_{x} + m_{y}$ and $\delta_{xy} = \lvert m_{x} - m_{y}\rvert$.

\end{document}