Spikes of the two-component elliptic system in \({\mathbb {R}}^4\) with the critical Sobolev exponent

  • Yuanze WuEmail author
  • Wenming Zou


Consider the following elliptic system:
$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u_1+\lambda _1u_1=\mu _1u_1^3+\alpha _1u_1^{p-1}+\beta u_2^2u_1 \quad \text {in }\quad \Omega ,\\&-\varepsilon ^2\Delta u_2+\lambda _2u_2=\mu _2u_2^3+\alpha _2u_2^{p-1}+\beta u_1^2u_2 \quad \text {in }\quad \Omega ,\\&u_1,u_2>0\quad \text {in }\quad \Omega ,\quad u_1=u_2=0\quad \text {on }\quad \partial \Omega , \end{aligned}\right. \end{aligned}$$
where \(\Omega \subset {\mathbb {R}}^4\) is a bounded domain, \(\lambda _i,\mu _i,\alpha _i>0\) \((i=1,2)\) and \(\beta \not =0\) are constants, \(\varepsilon >0\) is a small parameter and \(2<p<2^*=4\). By using variational methods, we study the existence of ground state solutions to this system for sufficiently small \(\varepsilon >0\). The concentration behaviors of least-energy solutions as \(\varepsilon \rightarrow 0^+\) are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as \(\varepsilon \rightarrow 0^+\).

Mathematics Subject Classification

35B09 35B33 35J50 



Y. Wu is supported by NSFC (11701554, 11771319), the Fundamental Research Funds for the Central Universities (2017XKQY091) and Jiangsu overseas visiting scholar program for university prominent young and middle-aged teachers and presidents. W. Zou is supported by NSFC (11771234). The authors also would like to thank the anonymous referee for very carefully reading the manuscript and wonderfully valuable comments that greatly improve this paper.


  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.
    Bartsch, T., Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. PDEs 37, 345–361 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Byeon, J., Jeanjean, L., Maris, M.: Symmetric and monotonicity of least energy solutions. Calc. Var. PDEs 36, 481–492 (2009)CrossRefGoogle Scholar
  5. 5.
    Brezís, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–77 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Byeon, J.: Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity. Trans. Am. Math. Soc. 362, 1981–2001 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Byeon, J.: Semi-classical standing waves for nonlinear Schrödinger systems. Calc. Var. PDEs 54, 2287–2340 (2015)CrossRefGoogle Scholar
  8. 8.
    Byeon, J., Zhang, J., Zou, W.: Singularly perturbed nonlinear Dirichlet problems involving critical growth. Calc. Var. PDEs 47, 65–85 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Conti, M., Terracini, S., Verzini, G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195, 524–560 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205, 515–551 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Z., Zou, W.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case. Calc. Var. PDEs 52, 423–467 (2015)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    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
  14. 14.
    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
  15. 15.
    Chen, Z., Lin, C.-S.: Asymptotic behavior of least energy solutions for a critical elliptic system. Int. Math. Res. Not. 21, 11045–11082 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dancer, E., Wei, J.: Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction. Trans. Am. Math. Soc. 361, 1189–1208 (2009)CrossRefGoogle Scholar
  17. 17.
    Dancer, E., Santra, S., Wei, J.: Least energy nodal solution of a singular perturbed problem with jumping nonlinearity. Annali della Scuola Normale Superiore di Pisa 10, 19–36 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Esry, B., Greene, C., Burke, J., Bohn, J.: Hartree–Fock theory for double condesates. Phys. Rev. Lett. 78, 3594–3597 (1997)CrossRefGoogle Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1998)zbMATHGoogle Scholar
  20. 20.
    Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^n\). In: Mathematical Analysis and Applications, Part A, in: Advanced Mathematical Supplemented Studies, vol. 7A. Academic Press, New York, pp. 369–402 (1981)Google Scholar
  21. 21.
    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
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Li, Y., Ni, W.-M.: Radial symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^n\). Commun. PDEs 18, 1043–1054 (1993)CrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    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
  27. 27.
    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
  28. 28.
    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
  29. 29.
    Liu, Z., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282, 721–731 (2008)CrossRefGoogle Scholar
  30. 30.
    Liu, Z., Wang, Z.-Q.: Ground states and bound states of a nonlinear Schrödinger system. Adv. Nonlinear Stud. 10, 175–193 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Long, W., Peng, S.: Segregated vector solutions for a class of Bose–Einstein systems. J. Differ. Equ. 257, 207–230 (2014)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Luo, S., Zou, W.: Existence, nonexistence, symmetry and uniqueness of ground state for critical Schrödinger system involving Hardy term. arXiv:1608.01123v1 [math.AP]
  33. 33.
    Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schröodinger systems. J. Eur. Math. Soc. 10, 47–71 (2006)zbMATHGoogle Scholar
  34. 34.
    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
  35. 35.
    Noris, B., Tavares, H., Terracini, S., Verzini, G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63, 267–302 (2010)CrossRefGoogle Scholar
  36. 36.
    Pistoia, A., Tavares, H.: Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions. J. Fixed Point Theory Appl. 19, 407–446 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pistoia, A., Soave, N.: On Coron’s problem for weakly coupled elliptic systems. Proc. Lond. Math. Soc. arXiv:1610.07762
  38. 38.
    Royo-Letelier, J.: Segregation and symmetry breaking of strongly coupled two-component Bose–Einstein condensates in a harmonic trap. Calc. Var. PDEs 49, 103–124 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}}^N\). Commun. Math. Phys. 271, 199–221 (2007)CrossRefGoogle Scholar
  40. 40.
    Terracini, S., Verzini, G.: Multipulse phases in k-mixtures of Bose–Einstein condensates. Arch. Ration. Mech. Anal. 194, 717–741 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Tavares, H., Terracini, S.: Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 279–300 (2012)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153, 229–244 (1993)CrossRefGoogle Scholar
  43. 43.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)CrossRefGoogle Scholar
  44. 44.
    Wei, J., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190, 83–106 (2008)MathSciNetCrossRefGoogle Scholar
  45. 45.
    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
  46. 46.
    Wu, Y., Wu, T.-F., Zou, W.: On a two-component Bose–Einstein condensate with steep potential wells. Ann. Mat. 196, 1695–1737 (2017)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Wu, Y.: On a \(K\)-component elliptic system with the Sobolev critical exponent in high dimensions: the repulsive case. Calc. Var. PDEs 56, 51 (2017). (article 151)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Wu, Y.: On the semiclassical solutions of a two-component elliptic system in R4 with trapping potentials and Sobolev critical exponent: the repulsive case. Z. Angew. Math. Phys. 69, 17 (2018). (article 111)CrossRefGoogle Scholar
  49. 49.
    Wu, Y.: Least energy sign-changing solutions of the singularly perturbed Brezis–Nirenberg problem. Nonlinear Anal. 171, 85–101 (2018)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhang, J., Zou, W.: A Berestycki–Lion theorem revisited. Commun. Contemp. Math. 14, 1250033 (14 pages) (2012)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zhang, J., do Marcos, O.J.: Spiked vector solutions of coupled Schrödinger systems with critical exponent and solutions concentrating on spheres. Preprint (2015)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

Personalised recommendations