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

We investigate the minimization of the positive principal eigenvalue of the problem −Δpu=λm|u|p−2u in Ω, ∂u/∂ν=0 on ∂Ω, over a class of sign-changing weights m with ∫Ωm<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.