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Non-Nehari manifold method for asymptotically periodic Schrödinger equations

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Abstract

We consider the semilinear Schrödinger equation

$$\left\{ {\begin{array}{*{20}c} { - \Delta u + V(x)u = f(x,u), \mathbb{R}^N ,} \\ {u \in H^1 (\mathbb{R}^N ),} \\ \end{array} } \right.$$

where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V 0(x) + V 1(x), V 0 ∈ C(ℝN), V 0(x) is 1-periodic in each of x 1, x 2, …, x N and sup[σ(−Δ + V 0) ∩ (−∞, 0)] < 0 < inf[σ(−Δ + V 0) ∩ (0, ∞)], V 1 ∈ C(ℝN) and lim|x|→∞ V 1(x) = 0. Inspired by previous work of Li et al. (2006), Pankov (2005) and Szulkin and Weth (2009), we develop a more direct approach to generalize the main result of Szulkin and Weth (2009) by removing the “strictly increasing” condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N 0 by using the diagonal method.

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References

  1. Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14: 349–381

    Article  MATH  MathSciNet  Google Scholar 

  2. Bartsch T, Wang Z-Q. Existence and multiplicity results for some superlinear elliptic problems on ℝN. Comm Partial Differential Equations, 1995, 20: 1725–1741

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen G W, Ma S W. Asymptotically or super linear cooperative elliptic systems in the whole space. Sci China Math, 2013, 56: 1181–1194

    Article  MATH  MathSciNet  Google Scholar 

  4. Coti Zelati V, Rabinowitz P H. Homoclinic type solutions for a semilinear elliptic PDE on ℝN. Comm Pure Appl Math, 1992, 14: 1217–1269

    MathSciNet  Google Scholar 

  5. Ding Y. Variational Methods for Strongly Indefinite Problems. Singapore: World Scientific, 2007

    MATH  Google Scholar 

  6. Ding Y, Lee C. Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J Differential Equations 2006, 222: 137–163

    Article  MATH  MathSciNet  Google Scholar 

  7. Ding Y, Szulkin A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc Var Partial Differential Equations. 2007, 29: 397–419

    Article  MATH  MathSciNet  Google Scholar 

  8. Edmunds D E, Evans W D. Spectral Theory and Differential Operators. Oxford: Clarendon Press, 1987

    MATH  Google Scholar 

  9. Egorov Y, Kondratiev V. On Spectral Theory of Elliptic Operators. Basel: Birkhäuser, 1996

    Book  MATH  Google Scholar 

  10. He Y, Li G B. The existence and concentration of weak solutions to a class of p-Laplacian type problems in unbounded domains. Sci China Math, 2014, 57: 1927–1952

    Article  MATH  MathSciNet  Google Scholar 

  11. Jiang Y S, Zhou H S. Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential. Sci China Math, 2014, 57: 1163–1174

    Article  MATH  MathSciNet  Google Scholar 

  12. Kryszewski W, Szulkin A. Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv Differ Equ, 1998, 3: 441–472

    MATH  MathSciNet  Google Scholar 

  13. Li G B, Szulkin A. An asymptotically periodic Schrödinger equation with indefinite linear part. Commun Contemp Math, 2002, 4: 763–776

    Article  MATH  MathSciNet  Google Scholar 

  14. Li Y Q, Wang Z-Q, Zeng J. Ground states of nonlinear Schrödinger equations with potentials. Ann Inst H Poincaré Anal Non Linéaire, 2006, 23: 829–837

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin X, Tang X H. Semiclassical solutions of perturbed p-Laplacian equations with critical nonlinearity. J Math Anal Appl, 2014, 413: 438–449

    Article  MathSciNet  Google Scholar 

  16. Lin X, Tang X H. Nehari-type ground state solutions for superlinear asymptotically periodic Schrödinger equation. Abstr Appl Anal, 2014, 2014: ID 607078, 7pp

    Google Scholar 

  17. Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case, part 2. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 223–283

    MATH  Google Scholar 

  18. Liu S. On superlinear Schrödinger equations with periodic potential. Calc Var Partial Differential Equations, 2012, 45: 1–9

    Article  MathSciNet  Google Scholar 

  19. Liu Z L, Wang Z-Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4: 561–572

    Google Scholar 

  20. Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J Math, 2005, 73: 259–287

    Article  MATH  MathSciNet  Google Scholar 

  21. Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43: 270–291

    Article  MATH  MathSciNet  Google Scholar 

  22. Schechter M. Superlinear Schrödinger operators. J Funct Anal, 2012, 262: 2677–2694

    Article  MATH  MathSciNet  Google Scholar 

  23. Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal, 2009, 257: 3802–3822

    Article  MATH  MathSciNet  Google Scholar 

  24. Tang X H. Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity. J Math Anal Appl, 2013, 401: 407–415

    Article  MATH  MathSciNet  Google Scholar 

  25. Tang X H. New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation. Adv Nonlinear Studies, 2014, 14: 361–373

    MATH  Google Scholar 

  26. Tang X H. Non-Nehari manifold method for superlinear Schrödinger equation. Taiwan J Math, 2014, 18: 1957–1979

    Google Scholar 

  27. Tang X H. New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum. J Math Anal Appl, 2014, 413: 392–410

    Article  MathSciNet  Google Scholar 

  28. Tang X H. Non-Nehari manifold method for asymptotically linear Schrödinger equation. J Aust Math Soc, doi:10.1017/S144678871400041X, 2014

    Google Scholar 

  29. Troestler C, Willem M. Nontrivial solution of a semilinear Schrödinger equation. Commun Partial Differ Equ, 1996, 21: 1431–1449

    Article  MATH  MathSciNet  Google Scholar 

  30. Willem M. Minimax Theorems. Boston: Birkhäuser, 1996

    Book  MATH  Google Scholar 

  31. Yang M. Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities. Nonlinear Anal, 2010, 72: 2620–2627

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang J, Zou W M. The critical case for a Berestycki-Lions theorem. Sci China Math, 2014, 57: 541–554

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhong X, Zou W M. Ground state and multiple solutions via generalized Nehari manifold. Nonlinear Anal, 2014, 102: 251–263

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    XianHua Tang

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Tang, X. Non-Nehari manifold method for asymptotically periodic Schrödinger equations. Sci. China Math. 58, 715–728 (2015). https://doi.org/10.1007/s11425-014-4957-1

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  • DOI: https://doi.org/10.1007/s11425-014-4957-1

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