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April  2020, 13(4): 1369-1393. doi: 10.3934/dcdss.2020077

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

Peter Poláčik 1,, and  Darío A. Valdebenito 2,

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
2. 

Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada

* Corresponding author: P. Poláčik

Received  September 2017 Revised  April 2018 Published  April 2019

Fund Project: 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

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.
Keywords: Elliptic equations on the entire space, quasiperiodic solutions, center manifold, Birkhoff normal form, KAM theorem.
Mathematics Subject Classification: Primary: 35B08, 35B15, 35J61; Secondary: 37J40, 37K55.
Citation: Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1369-1393. doi: 10.3934/dcdss.2020077
References:
[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.  Google Scholar

[2]

D. Bambusi, An introduction to Birkhoff normal form, Università di Milano, 2014. Google Scholar

[3]

H. W. Broer, S. N. Chow, Y. Kim and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44 (1993), 389-432.  doi: 10.1007/BF00953660.  Google Scholar

[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.  Google Scholar

[5]

B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Séminaries and Congrès, 15 (2007), 1–46.  Google Scholar

[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.  Google Scholar

[7]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[10]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489. Springer-Verlag, 1991. doi: 10.1007/BFb0097544.  Google Scholar

[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.  Google Scholar

[12]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[13] M. Reed and B. Simon, Methods of Mathematical Physics, Volume Ⅳ, Academic Press, New York-London, 1979.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported, Springer-Verlag, 1 (1992), 125–163.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

D. Bambusi, An introduction to Birkhoff normal form, Università di Milano, 2014. Google Scholar

[3]

H. W. Broer, S. N. Chow, Y. Kim and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44 (1993), 389-432.  doi: 10.1007/BF00953660.  Google Scholar

[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.  Google Scholar

[5]

B. Grébert, Birkhoff normal form and Hamiltonian PDEs, Séminaries and Congrès, 15 (2007), 1–46.  Google Scholar

[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.  Google Scholar

[7]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[8]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[10]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, Lecture Notes in Mathematics, 1489. Springer-Verlag, 1991. doi: 10.1007/BFb0097544.  Google Scholar

[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.  Google Scholar

[12]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[13] M. Reed and B. Simon, Methods of Mathematical Physics, Volume Ⅳ, Academic Press, New York-London, 1979.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported, Springer-Verlag, 1 (1992), 125–163.  Google Scholar

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