Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity

We consider the equation ∆u + uyy + f(x, u) = 0, (x, y) ∈ R^N × R (1) where f is sufficiently regular, radially symmetric in x, and f(·, 0) ≡ 0. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in y and decaying as |x| → ∞ uniformly in y. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of fu(x, 0) and fuu(x, 0), and is independent of higher-order terms in the Taylor expansion of f(x, ·). In particular, our results apply to some quadratic nonlinearities.