American Institute of Mathematical Sciences

Advanced
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

[1]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
[2]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65
[3]

Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure & Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719
[4]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[5]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[6]

Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607
[7]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[8]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[9]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252
[10]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[11]

Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335
[12]

Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115
[13]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[14]

Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715
[15]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020364
[16]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[17]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
[18]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
[19]

Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051
[20]

Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (131)
  • HTML views (534)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences