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Acta Mathematica Sinica, English Series

, Volume 30, Issue 12, pp 2014–2026 | Cite as

A concentration behavior for semilinear elliptic systems with indefinite weight

  • Xue Xiu Zhong
  • Wen Ming Zou
Article

Abstract

Consider the Schrödinger system
$$\left\{ \begin{gathered} - \Delta u + V_{1,n} u = \alpha Q_n (x)|u|^{\alpha - 2} u|v|^\beta , \hfill \\ - \Delta v + V_{2,n} v = \beta Q_n (x)|u|^\alpha |v|^{\beta - 2} v, \hfill \\ u,v \in H_0^1 (\Omega ), \hfill \\ \end{gathered} \right.$$
where Ω ⊂ ℝ N , α, β > 1, α + β < 2* and the spectrum σ(−Δ + V i,n ) ⊂ (0, + ∞), i = 1, 2; Q n is a bounded function and is positive in a region contained in Ω and negative outside. Moreover, the sets {Q n > 0} shrink to a point x 0 ∈ Ω as n → +∞. We obtain the concentration phenomenon. Precisely, we first show that the system has a nontrivial solution (u n , v n ) corresponding to Q n , then we prove that the sequences (u n ) and (v n ) concentrate at x 0 with respect to the H 1-norm. Moreover, if the sets {Q n > 0} shrink to finite points and (u n , v n ) is a ground state solution, then we must have that both u n and v n concentrate at exactly one of these points. Surprisingly, the concentration of u n and v n occurs at the same point. Hence, we generalize the results due to Ackermann and Szulkin [Arch. Rational Mech. Anal., 207, 1075–1089 (2013)].

Keywords

Schrödinger systems concentration ground state solution 

MR(2010) Subject Classification

35D30 35J20 35J47 35J50 35J60 46E30 46E35 

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References

  1. [1]
    Ackermann, N., Szulkin, A.: A concentration phenomenon for semilinear elliptic equations. Arch. Ration. Mech. Anal., 207, 1075–1089 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett., 82, 2661–2664 (1999)CrossRefGoogle Scholar
  3. [3]
    Ambrosetti, A., Arcoya, D., Gámez, J. L.: Asymmetric bound states of differential equations in nonlinear optics. Rend. Sem. Mat. Univ. Padova, 100, 231–247 (1998)zbMATHMathSciNetGoogle Scholar
  4. [4]
    Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differential Equations Appl., 2, 553–572 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    Bonheure, D., Gomes, J. M., Habets, P.: Multiple positive solutions of superlinear elliptic problems with sign-changing weight. J. Differential Equations, 214(1), 36–64 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Brown, K. J., Wu, T. F.: A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function. J. Math. Anal. Appl., 337, 1326–1336 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    Brown, K. J., Zhang, Y.: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differential Equations, 193, 481–499 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Buryak, A. V., Trapani, P. D., Skryabin, D. V., et al.: Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep., 370(2), 63–235 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    Chabrowski, J.: Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd, Singapore, 1999CrossRefzbMATHGoogle Scholar
  10. [10]
    Dror, N., Malomed, B. A.: Solitons supported by localized nonlinearities in periodic media. Phys. Rev. A (3), 83, 033828 (2011)CrossRefGoogle Scholar
  11. [11]
    Evangelides, S. G., Mollenauer, Jr. L. F., Gordon, J. P., et al.: Polarization multiplexing with solitons. J. Lightwave Technol., 10, 28–35 (1992)CrossRefGoogle Scholar
  12. [12]
    Girão, P. M., Gomes, M. J.: Multibump nodal solutions for an indefinite superlinear elliptic problem. J. Differential Equations, 247, 1001–1012 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    Kaminow, I. P.: Polarization in optical fibers. IEEE J. Quantum Electron, 17, 15–22 (1981)CrossRefGoogle Scholar
  14. [14]
    Kartashov, Y. V., Malomed, B. A., Torner, L.: Solitons in nonlinear lattices. Rev. Modern. Phys., 83, 247–305 (2011)CrossRefGoogle Scholar
  15. [15]
    López-Gómez, J.: Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems. Trans. Amer. Math. Soc., 352, 1825–1858 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Menyuk, C. R.: Nonlinear pulse propagation in birefringence optical fiber. IEEE J. Quantum Electron, 23, 174–176 (1987)CrossRefGoogle Scholar
  17. [17]
    Menyuk, C. R.: Pulse propagation in an elliptically birefringent Kerr medium. IEEE J. Quantum Electron, 25, 2674–2682 (1989)CrossRefGoogle Scholar
  18. [18]
    Pendry, J. B., Schurig, D., Smith, D. R.: Controlling electromagnetic fields. Science, 312(5781), 1780–1782 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    Ramos, M., Terracini, S., Troestler, C.: Superlinear indefinite elliptic problems and Pohozaev type identities. J. Funct. Anal., 159, 596–628 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    Shalaev, V. M.: Optical negative-indexmetamaterials. Nat. Photon., 1(1), 41–48 (2007)CrossRefMathSciNetGoogle Scholar
  21. [21]
    Smith, D. R., Pendry, J. B., Wiltshire, M. C. K.: Metamaterials and negative refractive index. Science, 305(5685), 788–792 (2004)CrossRefGoogle Scholar
  22. [22]
    Strauss, W. A.: The nonlinear Schröinger equation. Contemporary developments in continuum mechanics and partial differential equations. Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. North-Holland Math. Stud., Vol. 30, North-Holland, Amsterdam, 452–465 (1978)Google Scholar
  23. [23]
    Stuart, C. A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Ration. Mech. Anal., 113(1), 65–96 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    Stuart, C. A.: Guidance properties of nonlinear planarwaveguides. Arch. Ration. Mech. Anal., 125(2), 145–200 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Stuart, C. A.: Existence and stability of TE modes in a stratified non-linear dielectric. IMA J. Appl. Math., 72(5), 659–679 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    Szulkin, A., Willem, M.: Eigenvalue problems with indefinite weight. Studia Math., 135, 191–201 (1999)zbMATHMathSciNetGoogle Scholar
  27. [27]
    Veselago, V. G.: The electrodynamics of substances with simultaneously negative values of ɛ and µ., Physics- Uspekhi, 10(4), 509–514 (1968)CrossRefGoogle Scholar
  28. [28]
    Wai, P. K. A., Menyuk., C. R., Chen, H. H.: Stability of solitons in randomly varying birefringent fibers. Optim. Lett., 7, 40–42 (1982)CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingP. R. China

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