The author considers the linear eigenvalue problem \[ -\Delta u(x) = \lambda g(x) u(x) \text{ in } \mathbb{R}^ N,\;u(x) \to 0 \text{ as } | x | \to \infty, \tag{1} \] where \(N \geq 3\), \(\Delta\) denotes the Laplacian and \(g\) is a real-valued function which changes sign. The aim of this paper is to determine positive eigenvalues \(\lambda\) of (1) so that there is a corresponding eigenfunction \(u\) which does not change sign.