Principal eigenvalues for problems with indefinite weight function on ${\bf R}\sp n$

We investigate the existence of positive principal eigenvalues of the problem $- \Delta u(x) = \lambda g(x)u$ for $x \in {R^n};u(x) \to 0$ as $x \to \infty$ where the weight function $g$ changes sign on ${R^n}$. It is proved that such eigenvalues exist if $g$ is negative and bounded away from 0 at $\infty$ or if $n \geq 3$ and $\vert g(x)\vert$ is sufficiently small at $\infty$ but do not exist if $n = 1\,{\text{or}}\,2$ and $\int_{{R^n}} {g(x)dx > 0}$.