Minimization of eigenvalues for a quasilinear elliptic Neumann problem with indefinite weight

We investigate the minimization of the positive principal eigenvalue of the problem − Δ p u = λ m | u | p − 2 u in Ω, ∂ u / ∂ ν = 0 on ∂Ω, over a class of sign-changing weights m with ∫ Ω m < 0 . It is proved that minimizers exist and satisfy a bang–bang type property. In dimension one, we obtain a complete description of the minimizers. This problem is motivated by applications from population dynamics.