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

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.
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. BroerS. N. ChowY. 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. BroerS. N. ChowY. 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]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[2]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[3]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[4]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[5]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[6]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[9]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[10]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[11]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[12]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[13]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[14]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403

[15]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[16]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[17]

Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078

[18]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[19]

Dingheng Pi. Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021080

[20]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (139)
  • HTML views (537)
  • Cited by (0)

Other articles
by authors

[Back to Top]