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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ in Ω, ⎨ ⎩ $|∇u|^{p-2} ∂u/∂ν = λϱ(x)|u|^{p-2}u + μ|u|^{p-2}u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.