Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on R^{N+1}

Abstract

We consider the equationwhere N ≥ 2 and f is a smooth function satisfying f (0) = 0 and f (0) < 0. We show that for suitable nonlinearities f of this form equation (1) possesses uncountably many positive solutions which are quasiperiodic in y, radially symmetric in x, and decaying as |x| → ∞ uniformly in y. Our method is based on center manifold and KAM-type results and involves analysis of solutions of (1) in a vicinity of a y-independent solution u * (x)-a ground state of the equation ∆u + f (u) = 0 on R N .