Variational Problems with a p-Homogeneous Energy

We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1. Nonlinear spectral methods are used to find necessary and sufficient conditions for the existence of a minimizer. These methods are then combined with the pointwise order relation in a Sobolev space to obtain positivity and uniqueness of minimizers or critical points. A crucial restriction on the p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established.