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