EXISTENCE OF QUASIPERIODIC SOLUTIONS OF ELLIPTIC EQUATIONS ON THE ENTIRE SPACE WITH A QUADRATIC NONLINEARITY

摘要

We consider the equation Δu + u_(yy) + f(x, u) = 0, (x, y)∈ R~N × R (1) where f is suciently 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 f_u(x,0) and f_(uu)(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.