Using spatial dynamics and results from the KAM theory, we develop a framework to find solutions of semilinear elliptic equations on the entire space which are quasiperiodic in one variable, decaying in the other variables. These results apply to a wide class of nonhomogeneous (and some homogeneous) problems. A careful application of Birkhoff normal form allows us to obtain a nondegeneracy condition for KAM that works even for some purely quadratic nonlinearities.