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On the semiclassical solutions of a two-component elliptic system in \(\mathbb {R}^4\) with trapping potentials and Sobolev critical exponent: the repulsive case

  • Yuanze Wu
Article
  • 47 Downloads

Abstract

Consider the following elliptic system:
$$\begin{aligned} \left\{ \begin{array}{ll} -\varepsilon ^2\Delta u_1+V_1(x)u_1=\mu _1u_1^3+\alpha _1u_1^{p-1}+\beta u_2^2u_1&{}\quad \text {in }\mathbb {R}^4,\\ -\varepsilon ^2\Delta u_2+V_2(x)u_2=\mu _2u_2^3+\alpha _2u_2^{p-1}+\beta u_1^2u_2&{}\quad \text {in }\mathbb {R}^4,\\ u_1,u_2>0\quad \text {in }\mathbb {R}^4,\quad u_1,u_2\in H^1(\mathbb {R}^4), &{}\\ \end{array}\right. \end{aligned}$$
where \(V_i(x)\) are trapping potentials, \(\mu _i,\alpha _i>0(i=1,2)\) and \(\beta <0\) are constants, \(\varepsilon >0\) is a small parameter, and \(2<p<2^*=4\). By using the variational method, we obtain a solution to this system for \(\varepsilon >0\) small enough. The concentration behaviors of this solution as \(\varepsilon \rightarrow 0^+\), involving the location of the spikes, are also studied by combining the uniformly elliptic estimates and local energy estimates. To the best of our knowledge, this is the first result devoted to the spikes in the Bose–Einstein condensate with trapping potentials in \(\mathbb {R}^4\) for the repulsive case.

Keywords

Elliptic system Sobolev critical exponent Spike Trapping potential Variational method 

Mathematics Subject Classification

35B09 35B33 35J50 

Notes

Acknowledgments

The author was supported by the Fundamental Research Funds for the Central Universities (2017XKQY091). The author also would like to thank the anonymous referee for very carefully reading the manuscript and wonderful valuable comments that greatly improve this paper.

References

  1. 1.
    Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999)CrossRefGoogle Scholar
  2. 2.
    Abdellaoui, B., Felli, V., Peral, I.: Some remarks on systems of elliptic equations doubly critical the whole \(\mathbb{R}^N\). Calc. Var. PDEs 34, 97–137 (2009)CrossRefGoogle Scholar
  3. 3.
    Byeon, J.: Singularly rerturbed nonlinear Dirichlet problems with a general nonlinearity. Trans. Am. Math. Soc. 362, 1981–2001 (2010)CrossRefGoogle Scholar
  4. 4.
    Byeon, J.: Semi-classical standing waves for nonlinear Schrödinger systems. Calc. Var. PDEs 54, 2287–2340 (2015)CrossRefGoogle Scholar
  5. 5.
    Byeon, J., Zhang, J., Zou, W.: Singularly perturbed nonlinear Dirichlet problems involving critical growth. Calc. Var. PDEs 47, 65–85 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Rational Mech. Anal. 205, 515–551 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, Z., Lin, C.-S., Zou, W.: Sign-changing solutions and phase separation for an elliptic system with critical exponent. Commun. PDEs 39, 1827–1859 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Z., Zou, W.: Existence and symmetry of positive ground states for a doubly critical Schrödinger system. Trans. Am. Math. Soc. 367, 3599–3646 (2015)CrossRefGoogle Scholar
  9. 9.
    Huang, Y., Wu, T.-F., Wu, Y.: Multiple positive solutions for a class of concave-convex elliptic problems in \(\mathbb{R}^N\) involving sign-changing weight (II). Commun. Contemp. Math., 17, 1450045 (35 pages) (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, Z., Lin, C.-S.: Removable singularity of positive solutions for a critical elliptic system with isolated singularity. Math. Ann. 363, 501–523 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Esry, B., Greene, C., Burke, J., Bohn, J.: Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997)CrossRefGoogle Scholar
  12. 12.
    Figueiredo, G., Furtado, M.: Multiple positive solutions for a quasilinear system of Schrödinger equations. NoDEA 15, 309–333 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hall, D., Matthews, M., Ensher, J., Wieman, C., Cornell, E.: Dynamics of component separation in a binary mixture of Bose–Einstein condensates. Phys. Rev. Lett. 81, 1539–1542 (1998)CrossRefGoogle Scholar
  14. 14.
    Ikoma, N., Tanaka, K.: A local mountain pass type result for a system of nonlinear Schrödinger equations. Calc. Var. PDEs 40, 449–480 (2011)CrossRefGoogle Scholar
  15. 15.
    Lin, T.-C., Wei, J.: Ground state of \(N\) coupled nonlinear Schrödinger equations in \(\mathbb{R}^n\), \(n\le 3\). Commun. Math. Phys. 255, 629–653 (2005)CrossRefGoogle Scholar
  16. 16.
    Lin, T.-C., Wei, J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 403–439 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lin, T.-C., Wei, J.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229, 538–569 (2006)CrossRefGoogle Scholar
  18. 18.
    Lin, T.-C., Wu, T.-F.: Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 33, 2911–2938 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Long, W., Peng, S.: Segregated vector solutions for a class of Bose-Einstein systems. J. Differ. Equ. 257, 207–230 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schröodinger systems. J. Eur. Math. Soc. 10, 47–71 (2006)zbMATHGoogle Scholar
  21. 21.
    Ni, W.-M., Wei, J.: On the location and profile of spike-Layer solutions to singularly perturbed semilinear Dirichlet problems. Commun. Pure Appl. Math. 48, 731–768 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Terracini, S., Verzini, G.: Multipulse phases in k-mixtures of Bose–Einstein condensates. Arch. Rational Mech. Anal. 194, 717–741 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153, 229–244 (1993)CrossRefGoogle Scholar
  24. 24.
    Wang, J., Shi, J.: Standing waves of a weakly coupled Schrödinger system with distinct potential functions. J. Differ. Equ. 260, 1830–1864 (2016)CrossRefGoogle Scholar
  25. 25.
    Wu, Y., Wu, T.-F., Zou, W.: On a two-component Bose–Einstein condensate with steep potential wells. Annali di Matematica 196, 1695–1737 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wu, Y.: On a \(K\)-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case. Calc. Var. PDEs, 56, article 151, 51pp (2017)Google Scholar
  27. 27.
    Wu, Y., Zou, W.: Spikes of the two-component elliptic system in \(\mathbb{R}^4\) with Sobolev critical exponent. arXiv:1804.00400v1 [math.AP]
  28. 28.
    Wu, Y.: Least energy sign-changing solutions of the singularly perturbed Brezis–Nirenberg problem. Nonlinear Anal. 171, 85–101 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wu, Y.: Sign-changing semi-classical solutions of the Brezis–Nirenberg problems with jump nonlinearities in high dimensions. J. Math. Anal. Appl. 461, 7–23 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang, J., Zou, W.: A Berestycki-Lion theorem revisited. Commun. Contemp. Math., 14, 1250033 (14 pages), (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhang, J., Chen, Z., Zou, W.: Standing wave for nonlinear Schröding equations involving critical growth. J. Lond. Math. Soc. 90, 827–844 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhang, J., Zou, W.: Solutions concentrating around the saddle points of the potential for critical Schrd̈inger equations. Calc. Var. PDEs 54, 4119–4142 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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