  We investigate some fundamental features of a class of non-linear
relativistic lagrangian field theories with kinetic self-coupling. We focus our
attention upon theories admitting static, spherically symmetric solutions in
three space dimensions which are finite-energy and stable. We determine general
conditions for the existence and stability of these non-topological soliton
solutions. In particular, we perform a linear stability analysis that goes
beyond the usual Derrick-like criteria. On the basis of these considerations we
obtain a complete characterization of the soliton-supporting members of the
aforementioned class of non-linear field theories. We then classify the family
of soliton-supporting theories according to the central and asymptotic
behaviors of the soliton field, and provide illustrative explicit examples of
models belonging to each of the corresponding sub-families. In the present work
we restrict most of our considerations to one and many-components scalar
models. We show that in these cases the finite-energy static spherically
symmetric solutions are stable against charge-preserving perturbations,
provided that the vacuum energy of the model vanishes and the energy density is
positive definite. We also discuss briefly the extension of the present
approach to models involving other types of fields, but a detailed study of
this more general scenario will be addressed in a separate publication.
