In this talk we present a new center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders. Inspired by a recent work of Faye-Scheel (TAMS '18), this reduction is “without a phase space” in the sense that we never explicitly reformulate our PDE as an evolution equation. Under suitable hypotheses, the resulting center manifold is able to characterize all sufficiently small bounded solutions. Compared with classical methods, we find our reduction theorem to be more directly related to the original physical problems and  particularly convenient for calculations. Moreover our analysis is casted in Holder spaces, which is often desirable for elliptic problems. We then apply this machinery to construct small bounded solutions to two examples, one from nonlinear elasticity and the other from hydrodynamics. This is a joint work with Samuel Walsh and Miles Wheeler.