Taiwanese Journal of Mathematics


Dongdong Qin and Xianhua Tang

Full-text: Open access


This paper is concerned with the following Schrödinger equation:\[\begin{cases}-\triangle u + V(x)u = f(x, u), &\textrm{for $x \in \mathbb{R}^{N}$}, \\u(x) \to 0, &\textrm{as $|x| \to \infty$},\end{cases}\] where the potential $V$ and $f$ are periodic with respect to $x$ and $0$ is a boundary point of the spectrum $\sigma(-\triangle+V)$. By a new technique for showing the boundedness of Cerami sequences, we are able to obtain the existence of nontrivial solutions with mild assumptions on $f$.

Article information

Taiwanese J. Math., Volume 19, Number 4 (2015), 977-993.

First available in Project Euclid: 4 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Schrödinger equation strongly indefinite functional spectrum point zero superlinear


Qin, Dongdong; Tang, Xianhua. NEW CONDITIONS ON SOLUTIONS FOR PERIODIC SCHRÖDINGER EQUATIONS WITH SPECTRUM ZERO. Taiwanese J. Math. 19 (2015), no. 4, 977--993. doi:10.11650/tjm.19.2015.4227. https://projecteuclid.org/euclid.twjm/1499133685

Export citation


  • S. Alama and Y. Y. Li, On multibump bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41 (1992), 983-1026.
  • T. Bartsch and Y. H. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313 (1999), 15-37.
  • T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3&4 (2011), 549-569.
  • Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163.
  • Y. H. Ding, Variational methods for strongly indefinite problems, Interdiscip. Math. Sci., Vol. 7, World Scientific Publishing Co., Hackensack, NJ, 2007.
  • L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on ${\R}^N$, Peoc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
  • W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equation, 3 (1998), 441-472.
  • P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincarée Anal. Non Linéeaire, 1 (1984), 223-283.
  • S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. PDE., 45 (2012), 1-9.
  • G. Li and A. Szulkin, An asymptotically periodic equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
  • Y. Q. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(6) (2006), 829-837.
  • Z. L. Liu and Z. Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nolinear Studies, 4 (2004), 561-572.
  • J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, arXiv: 1308.4320v1 [math.AP] 20 Aug 2013.
  • A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.
  • D. D. Qin, X. H. Tang and Z. Jian, Multiple solutions for semilinear elliptic equations with sign-changing potential and nonlinearity, Electron. J. Diff. Equ., 2013(207) (2013), 1-9.
  • D. D. Qin, F. F. Liao and Y. Chen, Multipel solutions for periodic Schrödinger equations with spectrum point zero, Taiwanese J. Math., 18(4) (2014), 1185-1202.
  • M. Reed and B. Simon, Methods of Mordern Mathematical Physics, Vol. IV, Analysis of Operators, Academic Press, New York, 1978.
  • P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
  • M. Schechter, Superlinear Schrödinger operators, J. Func. Anal., 262 (2012), 2677-2694.
  • M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Soblev exponent, ESAIM Contral Optim. Calc. Var., 9 (2003), 601-619 (electronic).
  • M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonion Systems, Springer-Verlag, Berlin, 2000.
  • A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257(12) (2009), 3802-3822.
  • X. H. Tang, Infinitely many solutins for semilinear Schrödinger equation with sign-changing potential and nonlinearity, J. Math. Anal. Appl., 401 (2013), 407-415.
  • X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nolinear Studies, 14 (2014), 349-361.
  • X. H. Tang, New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl., 413(1) (2014), 392-410.
  • X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Australian Math. Socie., (2014), 1-13, DOI: 10.1017/S144678871400041X.
  • X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math., DOI: 10.11650/tjm.18.2014.3541.
  • M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
  • M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana. Univ. Math. J., 52 (2003), 109-132.
  • M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Noli. Anal., 72 (2010), 2620-2627.
  • M. Yang, W. Chen and Y. Ding, Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, J. Math. Anal. Appl., 364(2) (2010), 404-413.
  • Coti-Zelati, P. Rabinowitz, Homoclinic type solutions for a smilinear elliptic PDE on ${\R}^N$, Comm. Pure Appl. Math., 46 (1992), 1217-1269.