The authors deal with the nonlinear eigenvalue problem: \[ -\Delta_ p u+ f(x, u)- \lambda m(x)| u|^{p-2} u= \mu| u|^{p-2}u \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega, \tag{P} \] where \(\Delta_ p u= \text{div}( | \nabla u|^{p-2} \nabla u)\) is the \(p\)-Laplacian with \(p\in (1,\infty)\) and \(\Omega\subset \mathbb{R}^ N\) is a bounded domain, \(m\in L^ \infty (\Omega)\). By using the variational method, they prove the existence of a family of eigenvalues \(\{mu_ k (\lambda, \alpha) \}^ \infty_{k=1}\) and eigenfunctions \(\{u_ k (\lambda, \alpha) \}^ \infty_{k=1}\) with their properties in \(\lambda\) and \(\alpha\). They also consider the bifurcation problem of (P) from the trivial solution \(u=0\) and obtain the relation to the unperturbed problem \[ -\Delta_ pu -\lambda m(x)| u|^{p-2} u=\mu | u|^{p-2} u \quad \text{in } \Omega, \qquad u=0 \quad \text{on } \partial \Omega. \]