We investigate the minimization of the positive principal eigenvalue of the problem − p u = λm|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.

We prove existence and multiplicity results for sign‐changing solutions of a prescribed mean curvature equation with Dirichlet boundary conditions. Our arguments involve a perturbation of the degenerate part \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1/\sqrt{1+s}$\end{document}, which allows us to use classical variational techniques and to localize small regular solutions.

Abstract. A quasilinear parabolic problem is investigated. It models the evolution of a single population species with a nonlinear diffusion and a logistic reaction function. We present a new treatment combining standard theory of monotone operators in L 2 (Ω) with some orderpreserving properties of the evolutionary equation. The advantage of our approach is that we are able to obtain the existence and long-time asymptotic behavior of a weak solution almost simultaneously. We do not employ any uniqueness results; we rely on the uniqueness of the minimal and maximal solutions instead. At last, we answer the question of (long-time) survival of the population in terms of a critical value of a spectral parameter.Mathematics subject classification (2010): 35B40, 35K59, 35K92, 92D25.

We prove the existence of one positive, one negative, and one sign-changing solution of a p-Laplacian equation on R N , with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of R N have only been scarcely investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.