Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

  • Angela Pistoia
  • Hugo TavaresEmail author


In this paper we deal with the nonlinear Schrödinger system
$$\begin{aligned} -\Delta u_i =\mu _i u_i^3 + \beta u_i \sum _{j\ne i} u_j^2 + \lambda _i u_i, \qquad u_1,\ldots , u_m\in H^1_0(\Omega ) \end{aligned}$$
in dimension 4, a problem with critical Sobolev exponent. In the competitive case (\(\beta <0\) fixed or \(\beta \rightarrow -\infty \)) or in the weakly cooperative case (\(\beta \ge 0\) small), we construct, under suitable assumptions on the Robin function associated to the domain \(\Omega \), families of positive solutions which blowup and concentrate at different points as \(\lambda _1,\ldots , \lambda _m\rightarrow 0\). This problem can be seen as a generalization for systems of a Brezis–Nirenberg type problem.


Blowup and concentrating solutions Brezis–Nirenberg type problems Competitive and weakly cooperative systems Critical Sobolev Exponent Cubic Schrödinger systems Lyapunov–Schmidt reduction 

Mathematics Subject Classification

35A15 35J20 35J47 



Angela Pistoia was supported by GNAMPA and Sapienza Fondi di Ricerca. Hugo Tavares was partially supported by Fundação para a Ciência e Tecnologia through the program Investigador FCT and the project PEst-OE/EEI/-LA0009/2013, as well as by the ERC Advanced Grant 2013 N.339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”.


  1. 1.
    Akhmediev, N.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82(13), 2661–2664 (1999)CrossRefGoogle Scholar
  2. 2.
    Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ayed, M.B., Mehdi, K.E., Pacella, F.: Blow-up and symmetry of sign-changing solutions to some critical elliptic equations. J. Differ. Equ. 230(2), 771–795 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartsch, T., Micheletti, A.M., Pistoia, A.: On the existence and the profile of nodal solutions of elliptic equations involving critical growth. Calc. Var. Partial Differ. Equ. 26(3), 265–282 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100(1), 18–24 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Castro, A., Clapp, M.: The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain. Nonlinearity 16(2), 579 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z., Lin, C.-S.: Asymptotic behavior of least energy solutions for a critical elliptic system. Int. Math. Res. Not. 21, 11045–11082 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Z., Lin, C.-S., Zou, W.: Sign-changing solutions and phase separation for an elliptic system with critical exponent. Commun. Partial Differ. Equ. 39(10), 1827–1859 (2014)MathSciNetCrossRefzbMATHGoogle 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(2), 515–551 (2012)MathSciNetCrossRefzbMATHGoogle 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. Partial Differ. Equ. 52(1), 423–467 (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    Correia, S., Oliveira, F., Tavares, H.: Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with \(d\ge 3\) equations. J. Funct. Anal. 271, 2247–2273 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Han, Z.-C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(2), 159–174 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iacopetti, A., Vaira, G.: Sign-changing blowing-up solutions for the Brezis–Nirenberg problem in dimensions four and five (2015). arXiv:1504.05010 (Preprint)
  15. 15.
    Li, Y.: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 2(6), 955–980 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lin, T.-C., Wei, J.: Erratum: “Ground state of \(N\) coupled nonlinear Schrödinger equations in \({\bf R}^n\), \(n\le 3\)”. Comm. Math. Phys. 277(2):573–576 [Comm. Math. Phys. 255(3), 629–653 (2005); MR2135447]Google Scholar
  17. 17.
    Lin, T.-C., Wei, J.: Ground state of \(N\) coupled nonlinear Schrödinger equations in \({\mathbf{R}}^n\), \(n\le 3\). Commun. Math. Phys. 255(3), 629–653 (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mandel, R.: Minimal energy solutions for cooperative nonlinear Schrödinger systems. NoDEA Nonlinear Differ. Equ. Appl. 22(2), 239–262 (2015)CrossRefzbMATHGoogle Scholar
  20. 20.
    Micheletti, A.M., Pistoia, A.: On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth. Nonlinearity 17(3), 851–866 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Micheletti, A.M., Pistoia, A.: Non degeneracy of critical points of the Robin function with respect to deformations of the domain. Potential Anal. 40(2), 103–116 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Musso, M., Pistoia, A.: Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent. Indiana Univ. Math. J. 51(3), 541–579 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Noris, B., Terracini, S., Tavares, H., Verzini, G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267–302 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Oliveira, F., Tavares, H.: Ground states for a nonlinear Schrödinger system with sublinear coupling terms. Adv. Nonlinear Stud. 16(2), 381–387 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pistoia, A.: The Ljapunov-Schmidt reduction for some critical problems. In: Concentration analysis and applications to PDE–ICTS Workshop. Bangalore, January 2012, Trends in Mathematics, pp. 69–83. Springer, Basel (2013)Google Scholar
  26. 26.
    Rey, O.: Proof of two conjectures of H. Brézis and L. A. Peletier. Manuscr. Math. 65(1), 19–37 (1989)CrossRefzbMATHGoogle Scholar
  27. 27.
    Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89(1), 1–52 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Soave, N.: On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition. Calc. Var. Partial Differ. Equ. 53(3), 689–718 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Soave, N., Tavares, H.: New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms. J. Differ. Equ. 261(1), 505–537 (2016)CrossRefzbMATHGoogle Scholar
  30. 30.
    Soave, N., Tavares, H., Terracini, S., Zilio, A.: Hölder bounds and regularity of emerging free boundaries for strongly competing schrödinger equations with nontrivial grouping. Nonlinear Anal. Theory Methods Appl. 138, 388–427 (2016). (Nonlinear Partial Differential Equations, in honor of Juan Luis Vázquez for his 70th birthday)CrossRefzbMATHGoogle Scholar
  31. 31.
    Soave, N., Zilio, A.: Uniform bounds for strongly competing systems: the optimal Lipschitz case. Arch. Ratio. Mech. Anal. 218(2), 647–697 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Timmermans, E.: Phase separation of Bose–Einstein condensates. Phys. Rev. Lett. 81(26), 5718–5721 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Dipartimento di Metodi e Modelli MatematiciUniversità di Roma “La Sapienza”RomaItaly
  2. 2.CAMGSD, Instituto Superior Técnico, Pavilhão de MatemáticaLisboaPortugal

Personalised recommendations