Existence of quasiperiodic solutions of elliptic equations on RN+1 via center manifold and KAM theorems

We consider elliptic equations on RN+1 of the form(1)Δxu+uyy+g(x,u)=0,(x,y)∈RN×R, where g(x,u) is a sufficiently regular function with g(⋅,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. We discuss several classes of nonlinearities g to which our results apply.