A Neumann Problem with Asymmetric Nonlinearity and a Related Minimizing Problem

Variational methods are used to prove the existence of solution of some classes of nonlinear ordinary differential equations with zero Neumann boundary conditions. Among other very mild conditions, the nonlinearities we consider have a behavior at −∞ at least like a linearity of slope less thanπ2/4. A variational characterization of this valuevis used to prove both geometrical and compactness properties of the associated Euler functional. A partial differential equation version of these problems is also considered.