  In this paper, we present a novel center manifold reduction theorem for
quasilinear elliptic equations posed on infinite cylinders. This is done
without a phase space in the sense that we avoid explicitly reformulating the
PDE as an evolution problem. Under suitable hypotheses, the resulting center
manifold is finite dimensional and captures all sufficiently small bounded
solutions. Compared with classical methods, the reduced ODE on the manifold is
more directly related to the original physical problem and also easier to
compute. The analysis is conducted directly in H\"older spaces, which is often
desirable for elliptic equations.
  We then use this machinery to construct small bounded solutions to a variety
of systems. These include heteroclinic and homoclinic solutions of the
anti-plane shear problem from nonlinear elasticity; exact slow moving invasion
fronts in a two-dimensional Fisher--KPP equation; and hydrodynamic bores with
vorticity in a channel. The last example is particularly interesting in that we
find solutions with critical layers and distinctive "half cat's eye" streamline
patterns.
