EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (6): 1966-1980.

• 论文 • 上一篇

EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS

王文波^1,2, 李全清³

1 Wuhan Institute of Physics and Mathematics, CAS, Wuhan 430071, China;
2 University of Chinese Academy of Sciences, Beijing 100049, China;
3 Department of Mathematics, Honghe University, Mengzi 661100, China

收稿日期:2017-03-06 修回日期:2018-05-18 出版日期:2018-12-25 发布日期:2018-12-28
作者简介:Wenbo WANG,E-mail:wenbowangmath@163.com;Quanqing LI,E-mail:shili06171987@126.com
基金资助:
This work was partially supported by the Honghe University Doctoral Research Program (XJ17B11) and Yunnan Province Applied Basic Research for Youths and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013).

EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS

Wenbo WANG^1,2, Quanqing LI³

1 Wuhan Institute of Physics and Mathematics, CAS, Wuhan 430071, China;
2 University of Chinese Academy of Sciences, Beijing 100049, China;
3 Department of Mathematics, Honghe University, Mengzi 661100, China

Received:2017-03-06 Revised:2018-05-18 Online:2018-12-25 Published:2018-12-28
Supported by:
This work was partially supported by the Honghe University Doctoral Research Program (XJ17B11) and Yunnan Province Applied Basic Research for Youths and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013).

摘要/Abstract

摘要： In this paper, we investigate nonlinear Hamiltonian elliptic system

where N ≥ 3, τ > 0 is a positive parameter and V, K are nonnegative continuous functions, f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.

关键词: Hamiltonian elliptic system, strongly indefinite functional, generalized Nehari manifold

Abstract: In this paper, we investigate nonlinear Hamiltonian elliptic system

where N ≥ 3, τ > 0 is a positive parameter and V, K are nonnegative continuous functions, f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.

Key words: Hamiltonian elliptic system, strongly indefinite functional, generalized Nehari manifold

王文波, 李全清. EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS[J]. 数学物理学报(英文版), 2018, 38(6): 1966-1980.

Wenbo WANG, Quanqing LI. EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS[J]. Acta mathematica scientia,Series B, 2018, 38(6): 1966-1980.

参考文献

[1] Ambrosetti A, Felli V, Malchiodi A. Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7(1):117-144
[2] Alves C O, Souto M A S. Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J Differential Equations, 2013, 254(4):1977-1991
[3] Bartsch T, Ding Y H. Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr, 2006, 279(12):1-22
[4] Bartsch T, Figueiredo D G. Infinitely many solutions of n onlinear elliptic systems. Progr Nonlinear Differential Equations Appl, 1999, 35:51-67
[5] Chang K C, Guo M Z. Teaching Materials of Functional Analysis (Chinese). Beijing:Peking University Press, 1990
[6] Ding Y H. Variational Methods for Strong Infinity Problems. Wrold Scientific Press, 2008
[7] Dautray R, Lions J L. Mathematical Analysis and Numerical Methods for Science and Technology. Vol 3, Spectral Theory and Applications. Berlin:Springer-Verlag, 1990
[8] Edmunds D E, Evans W D. Spectral Theory and Differential Operators. New York:Oxford University Press, 1987
[9] Figueiredo D G. Semilinear elliptic problems//Proc 2nd School on Nonlin Funct Anal and Appl to Diff Equ. ICTP Trieste, 1997:122-152
[10] Figueiredo D G. Semilinear elliptic systems:existence, multiplicity, symmetry of solutions//Handbook of Differential Equations, Stationary Partial Differential Equations, Volume 5. Elsevier, 2008
[11] Figueiredo D G, Ding Y H. Strongly indefinite functionals and multiple solutions of elliptic systems. Trans Amer Math Soc, 2003, 355(7):2973-2989
[12] Figueiredo D G and Felmer P L. On superquadratic elliptic systems. Trans Amer Math Soc, 1994, 343(1):99-116
[13] Figueiredo G. M, Pimenta M T O. Existence of ground state solutions to Dirac equations with vanishing potentials at infinity. J Differential Equations, 2017, 262(1):486-505
[14] Figueiredo D G, Yang J. Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal, 1998, 331(3):211-234
[15] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equtions of Second Order. Berlin, New York:Springer-verlag, 2001
[16] Itô S. Diffusion Equations. Transl Math Monogr, Vol 114. Providence, RI:Amer Math Soc, 1992
[17] Kato T. Perturbation Thory for Linear Operator. 2nd ed. New York, Heidelberg, Berlin, Tokyo:SpringerVerlag, 1984
[18] Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, 1971
[19] Xia L R, Zhang J, Zhao F K. Ground state solutions for superlinear elliptic systems on R^N. J Math Anal Appl, 2013, 401(2):518-525
[20] Nagasawa M. Schrödinger Equations and Diffusion Theory. Birkhäuser, 1993
[21] do O J M, Miyagaki O H, Squassina M. Critical and subcritical frá ctional problems with vanishing potentials. Commun Contemp Math, 2016, 18(6):1550063
[22] Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J Math, 2005, 73(1):259-287
[23] Rabinowitz P H. Minimax Methods in Critical Point Theory with Application to Differential Equations. CBMS Reg Conf Ser Math, Vol 65. Providence, RI:American Mathematical Society, 1986
[24] Reed M, Simon B. Methods of Modern Mathematical Physics, I, Fourier Analysis, Self-Adjointness. New York:Academic Press, 1978
[25] Reed M, Simon B. Methods of Modern Mathematical Physics, Ⅱ, Fourier Analysis, Self-Adjointness. New York:Academic Press, 1978
[26] Sirakov B. On the existence of solutions of Hamiltonian elliptic systems in R^N. Adv Differential Equations, 2000, 5:1445-1464
[27] Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal, 2009, 257(12):3802-3822
[28] Szulkin A, Weth T. The method of Nehari manifold//Handbook of Nonconvex Analysis and Applications. Internation Press, 2010:597-632
[29] Willem M, Minimax theorems//Progress in Nonlinear Differential Equations and their Applications. Vol 24. Boston, MA:Birkhäuser Boston, Inc, 1996
[30] Zhao F K, Ding Y H. On hamiltonian elliptic systems with periodic or non-periodic potentials. J Differential Equations, 2010, 249(12):2964-2985
[31] Zhang J, Qin W P, Zhao F K. Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system. J Math Anal Appl, 2013, 399(2):433-441
[32] Zhao F K, Zhao L G, Ding Y H. Multiple solutions for a superlinear and periodic eliiptic system on RN. Z Angew Math Phys, 2011, 62(3):495-511

EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS

EXISTENCE OF GROUND STATE SOLUTIONS TO HAMILTONIAN ELLIPTIC SYSTEM WITH POTENTIALS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

Metrics

本文评价

推荐阅读 10