Summary: Let \(\Delta _{p }\) denote the p-Laplacian operator and \(\Omega \) be a bounded domain in \({\mathbb{R}^N}\). We consider the eigenvalue problem \[ -\Delta_p u +V(x) |u|^{p-2}u=\lambda m(x) |u|^{p-2} u, \, \quad u \in W_0^{1,p} (\Omega) \] for a potential \(V\) and a weight function \(m\) that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below. The main feature here is the introduction of a value \(\alpha \)(V, m) that guarantees the boundedness of the energy over the weighted sphere \({M=\{u \in W_0^{1,p}(\Omega); \int_{\Omega}m|u|^p\, dx= 1\}}\). We show that the above equation has a principal eigenvalue if and only if either \(m \geq 0\) and \(\alpha (V, m) > 0\) or \(m\) changes sign and \(\alpha (V, m) \geq 0\). The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence with respect to \(V\) and \(m\).