# American Institute of Mathematical Sciences

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. 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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