Existence of partially localized quasiperiodic solutions of homogeneous elliptic equations on RN+1

We consider the equation Delta u + u(yy) + f(u) = 0, (x, y) is an element of R-N x R, (1) where 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 vertical bar x vertical bar -> infinity 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 Delta u + f (u) = 0 on R-N.