  We establish a positivity property for a class of semilinear elliptic
problems involving indefinite sublinear nonlinearities. Namely, we show that
any nontrivial nonnegative solution is positive for a class of problems the
strong maximum principle does not apply to. Our approach is based on a
continuity argument combined with variational techniques, the sub and
supersolutions method and some a priori bounds. Both Dirichlet and Neumann
homogeneous boundary conditions are considered. As a byproduct, we deduce some
existence and uniqueness results. Finally, as an application, we derive some
positivity results for indefinite concave-convex type problems.
