  We consider a semilinear elliptic equation with Dirichlet boundary conditions
in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce
a condition on the size of the domain that implies the existence of a positive
solution satisfying a uniform pointwise estimate. Here, uniform means that the
estimate is independent of the domain. The main advantage of our approach is
that it allows us to remove a restrictive monotonicity assumption that was
imposed in the recent paper. In addition, we can remove a non-degeneracy
condition that was assumed in the latter reference. Furthermore, we can
generalize an old result, concerning semilinear elliptic nonlinear eigenvalue
problems. Moreover, we study the boundary layer of global minimizers of the
corresponding singular perturbation problem. For the above applications, our
approach is based on a refinement of a result, concerning the behavior of
global minimizers of the associated energy over large balls, subject to
Dirichlet conditions. Combining this refinement with global bifurcation theory
and the sliding method, we can prove uniform estimates for solutions away from
their nodal set. Combining our approach with a-priori estimates that we obtain
by blow-up, a doubling lemma, and known Liouville type theorems, we can give a
new proof of a known Liouville type theorem without using boundary blow-up
solutions. We can also provide an alternative proof, and a useful extension, of
a Liouville theorem, involving the presence of an obstacle. Making use of the
latter extension, we consider the singular perturbation problem with mixed
boundary conditions. Moreover, we prove some new one-dimensional symmetry and
rigidity properties of certain entire solutions to Allen-Cahn type equations,
as well as in half spaces, convex cylindrical domains. In particular, we
provide a new proof of Gibbons' conjecture in two dimensions.
