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Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity

  • * Corresponding author: P. Poláčik

    * Corresponding author: P. Poláčik 

The first author was supported in part by the NSF Grant DMS-1565388. The second author was supported in part by CONICYT-Chile Becas Chile, Convocatoria 2010

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  • We consider the equation

    $\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

    where $ f $ is sufficiently regular, radially symmetric in $ x $, and $ f(\cdot,0)\equiv 0 $. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in $ y $ and decaying as $ |x|\to\infty $ 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,\cdot) $. In particular, our results apply to some quadratic nonlinearities.

    Mathematics Subject Classification: Primary: 35B08, 35B15, 35J61; Secondary: 37J40, 37K55.


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  • [1] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger Operators, volume 29 of Mathematical Notes, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982.
    [2] D. Bambusi, An introduction to Birkhoff normal form, Università di Milano, 2014.
    [3] H. W. BroerS. N. ChowY. Kim and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44 (1993), 389-432.  doi: 10.1007/BF00953660.
    [4] H. W. Broer and G. B. Huitema, A proof of the isoenergetic KAM-theorem from the "ordinary" one, Journal of Differential Equations, 90 (1991), 52-60.  doi: 10.1016/0022-0396(91)90160-B.
    [5] B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Séminaries and Congrès, 15 (2007), 1–46.
    [6] M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. doi: 10.1007/978-0-85729-112-7.
    [7] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994. doi: 10.1007/978-3-0348-8540-9.
    [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.
    [9] K. Kirchgässner, Wave solutions of reversible systems and applications, Journal of Differential Equations, 45 (1982), 113-127.  doi: 10.1016/0022-0396(82)90058-4.
    [10] A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489. Springer-Verlag, 1991. doi: 10.1007/BFb0097544.
    [11] P. Poláčik and D. A. Valdebenito, Existence of quasiperiodic solutions of elliptic equations on $\mathbb R^{N+1}$ via center manifold and KAM theorems, Journal of Differential Equations, 262 (2017), 6109-6164.  doi: 10.1016/j.jde.2017.02.027.
    [12] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.
    [13] M. Reed and  B. SimonMethods of Mathematical Physics, Volume Ⅳ, Academic Press, New York-London, 1979. 
    [14] J. Scheurle, Bifurcation of quasiperiodic solutions from equilibrium points of reversible dynamical systems, Arch. Rational Mech. Anal., 97 (1987), 103-139.  doi: 10.1007/BF00251911.
    [15] Y. Shi, J. Xu and X. Xu, On quasi-periodic solutions for generalized Boussinesq equation with quadratic nonlinearity, J. Math. Phys., 56 (2015), 022703, 15pp. doi: 10.1063/1.4906810.
    [16] J. M. Tuwankotta and F. Verhulst, Hamiltonian systems with widely separated frequencies, Nonlinearity, 16 (2003), 689-706.  doi: 10.1088/0951-7715/16/2/319.
    [17] C. Valls, Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain, Comentarii Mathematici Helvetici, 81 (2006), 783-800.  doi: 10.4171/CMH/73.
    [18] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported, Springer-Verlag, 1 (1992), 125–163.
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