  Several results about positive solutions -in a Lipschitz domain- of a
nonlinear elliptic equation in a general form $ \Delta u(x)-g(x,u(x))=0$ are
proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$,
and a smooth domain. Our results include a characterization -in terms of a
natural capacity- of a (conditional) removability property, a characterization
of moderate solutions and of their boundary trace and a property relating
arbitrary positive solutions to moderate solutions. The proofs combine
techniques of non-linear p.d.e.\ with potential theoretic methods with respect
to linear Schr\"odinger equations. A general result describing the measures
that are diffuse with respect to certain capacities is also established and
used. The appendix by the first author provides classes of functions $g$ such
that the nonnegative solutions of $ \Delta u-g(.,u)=0$ has some "good"
properties which appear in the paper.
