We consider the equation Δxu+uyy+f(u)=0,x=(x1,⋯,xN)∈RN,y∈R,where N≥ 2 and f is a sufficiently smooth function satisfying f(0) = 0 , f′(0) < 0 , and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in x′= (x1, … , xN-1) and decaying as | x′| → ∞, periodic in xN, and quasiperiodic in y. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results.
Bibliographical noteFunding Information:
Supported in part by the NSF Grant DMS–1856491.
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
- Center manifold
- Elliptic equations
- Entire solutions
- KAM theorems
- Partially localized solutions
- Quasiperiodic solutions