Positive ground states for a system of Schrödinger equations with critically growing nonlinearities

Article

Abstract

We study the following problem
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + u^{2^*-2} v &{}\quad \hbox {in }\quad \Omega ,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} &{}\quad \hbox {in }\quad \Omega ,\\ u> 0,v> 0 &{}\quad \hbox {in }\quad \Omega ,\\ u=v=0 &{}\quad \hbox {on }\quad \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a bounded domain of \(\mathbb {R}^N\), \(N\ge 4\), \(2^*=2N/(N-2)\), \(\lambda \in \mathbb {R}\) and \(\mu \ge 0\) and we obtain existence and nonexistence results, depending on the value of the parameters \(\lambda \) and \(\mu \).

Mathematics Subject Classification

35J57 35A01 35B33 35J50 

Notes

Acknowledgments

We would like to thank Giusi Vaira for her valuable comments concerning the problem in case \(N\ge 6\).

References

  1. 1.
    Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2665 (1999)CrossRefGoogle Scholar
  2. 2.
    Ambrosetti, A., Cerami, G., Ruiz, D.: Solitons of linearly coupled systems of semilinear non-autonomous equations on \(\mathbb{R}^n.\) J. Funct. Anal. 254, 2816–2845 (2008)Google Scholar
  3. 3.
    Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75, 67–82 (2007)Google Scholar
  4. 4.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)MATHMathSciNetGoogle Scholar
  5. 5.
    Azzollini, A., d’Avenia, P.: On a system involving a critically growing nonlinearity. J. Math. Anal. Appl. 387, 433–438 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bahri, A., Coron, J.-M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253–294 (1988)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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. Partial Differ. Equ. 37, 345–361 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bartsch, T., Wang, Z.-Q., Wei, J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2, 353–367 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Brezis, H., Lieb, E.H.: Minimum action solutions of some vector field equations. Commun. Math. Phys. 96, 97–113 (1984)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Brezis, H., Niremberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)CrossRefMATHGoogle Scholar
  11. 11.
    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)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, Z., Zou, W.: Ground states for a system of Schrödinger equations with critical exponent. J. Funct. Anal. 262, 3091–3107 (2012)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, Z., Zou, W.: An optimal constant for the existence of least energy solutions of a coupled Schrödinger system. Calc. Var. Partial Differ. Equ. 48, 695–711 (2013)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Crandall, M.G., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207–218 (1975)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Dancer, E.N., Wei, J., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 953–969 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Esry, B.D., Greene, C.H., Burke, J.P., Bohn, J.L.: Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997)CrossRefGoogle Scholar
  17. 17.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, Berlin (2001)MATHGoogle Scholar
  18. 18.
    Ikoma, N., Tanaka, K.: A local mountain pass type result for a system of nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 40, 449–480 (2011)Google Scholar
  19. 19.
    Kavian, O.: Introduction à la Théorie des Points Critiques et applications aux problémes elliptiques. Mathématiques et Applications, vol. 13. Springer, Paris (1993)Google Scholar
  20. 20.
    Lin, T.-C., Wei, J.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229, 538–569 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743–767 (2006)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Merle, F.: Sur la non-existence de solutions positives d’équations elliptiques surlinéaires. C. R. Acad. Sci. Paris Sér. I Math. 306, 313–316 (1988)Google Scholar
  23. 23.
    Montefusco, E., Pellacci, B., Squassina, M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. J. Eur. Math. Soc. 10, 47–71 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    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)MATHMathSciNetGoogle Scholar
  25. 25.
    Pomponio, A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 227, 258–281 (2006)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Pomponio, A., Secchi, S.: A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities. Commun. Pure Appl. Anal. 9, 741–750 (2010)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Sirakov, B.: Least-energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}}^n.\) Commun. Math. Phys. 271, 199–221 (2007)Google Scholar
  28. 28.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)Google Scholar
  29. 29.
    Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 281–304 (1992)MATHMathSciNetGoogle Scholar
  30. 30.
    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)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)Google Scholar
  32. 32.
    Zheng, X.M.: Un résultat de non-existence de solution positive pour une équation elliptique. Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 91–96 (1990)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly
  2. 2.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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