Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 4, pp 1399–1430 | Cite as

Positive solutions for asymptotically linear problems in exterior domains

  • Liliane A. Maia
  • Benedetta Pellacci


The existence of a positive solution for a class of asymptotically linear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.


Asymptotically linear problems Exterior domains Schrödinger equation 

Mathematics Subject Classification

35J20 35J25 35J61 35Q55 



Part of this work has been done while the second author was visiting the University of Brasilia. She wishes to thank all the departamento de Matemática for the warm hospitality. A special thank goes to the Liliane’s family for the hospitality and the friendly atmosphere. The authors thank the anonymous referee whose important comments helped them to improve their work.


  1. 1.
    Ackermann, N., Clapp, M., Pacella, F.: Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains. Commun. Partial Differ. Equ. 38(5), 751–779 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82(13), 2661–2664 (1999)CrossRefGoogle Scholar
  3. 3.
    Akhmediev, N., Królinowski, W., Snyder, A.: Partially coherent solitons of variable shape. Phys. Rev. Lett. 81(21), 4632–4635 (1998)CrossRefGoogle Scholar
  4. 4.
    Ambrosetti, A., Cerami, G., Ruiz, D.: Solitons of linearly coupled systems of semilinear non-autonomous equations on \(R^{n}\). J. Funct. Anal. 254(11), 2816–2845 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ambrosetti, A., Malchiodi, A.: Nonlinear analysis and semilinear elliptic problems. In: Cambridge Studies in Advanced Mathematics, vol. 104. Cambridge University Press, Cambridge (2006)Google Scholar
  6. 6.
    Bahri, A., Li, Y.Y.: On a min–max procedure for the existence of a positive solution for certain scalar field equations in \(\mathbb{R}^N\). Rev. Math. Iberoamericana 6, 1–15 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bahri, A., Lions, P.L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(3), 365–413 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bartsch, T., Weth, T.: Three nodal solutions of singularly elliptic equations on domains without topology. Ann. I. H. Poincaré Anal. Non Linéaire 22(3), 259–281 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benci, V., Cerami, G.: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Ration. Mech. Anal. 99(4), 283–300 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I and II. Arch. Rational Mech. Anal. 82(4), 313–345 (1983) and 347–375Google Scholar
  11. 11.
    Berestycki, H., Gallouet, T., Kavian, O., Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris série I 297, 307–310 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Brezis, H., Lieb, E.: A relation between pointwise convergence and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cerami, G.: Un criterio di esistenza per i punti critici su varietà illimitate. Rend. Accad. Sci. Lett. Inst. Lombardo 112, 332–336 (1978)zbMATHGoogle Scholar
  14. 14.
    Cerami, G.: Some nonlinear elliptic problems in unbounded domains. Milan J. Math. 74, 47–77 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cerami, G., Clapp, M.: Sign-changing solutions of semilinear elliptic problems in exterior domains. Calc. Var. Partial Differ. Equ. 30(3), 353–367 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cerami, G., Molle, R., Passaseo, D.: Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary. Ann. I. H. Poincaré Anal. Non Linéaire 24(1), 41–60 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cerami, G., Passaseo, D.: Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with “rich” topology. Nonlinear Anal. TMA 18(2), 109–119 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cerami, G., Passaseo, D.: Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains. Nonlinear Anal. TMA 24(11), 1533–1547 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Citti, G.: On the exterior Dirichlet problem for \(\Delta u-u+f(x, u)=0\). Rendiconti del seminario matematico dell’università di Padova 88, 83–110 (1992)zbMATHGoogle Scholar
  20. 20.
    Clapp, M., Salazar, D.: Multiple sign changing solutions of nonlinear elliptic problems in exterior domains. Adv. Nonlinear Stud. 12(3), 427–443 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Costa, D., Magalhaes, C.A.: Variational elliptic problems which are non quadratic at infinity. Nonlinear Anal. TMA 23(11), 1401–1412 (1994)CrossRefzbMATHGoogle Scholar
  22. 22.
    Esteban, M., Lions, P.L.: Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinb. Sect. A 93(1–2), 1–14 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Evéquoz, G., Weth, T.: Entire solutions to nonlinear scalar field equations with indefinite linear part. Adv. Nonlinear Stud. 12(2), 281–314 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, G., Zheng, G.: The existence of positive solution to some asymptotically linear elliptic equations in exterior domains. Rev. Mat. Iberoamericana 22(2), 559–590 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. I. H. Poincaré. A. N. 1(1–2), 109–145 (1984) and 223–283Google Scholar
  26. 26.
    Litchinitser, N.M., Królikowski, W., Akhmediev, N.N., Agrawal, G.P.: Asymmetric partially coherent solitons in saturable nonlinear media. Phys. Rev. E 60, 2377–2380 (1999)CrossRefGoogle Scholar
  27. 27.
    Maia, L.A., Miyagaki, O.H., Soares, S.H.M.: Sign-changing solution for an asymptotically linear Schrödinger equation. Proc. Edin. Math. Soc. 58(3), 697–716 (2015)CrossRefzbMATHGoogle Scholar
  28. 28.
    Maia, L., Montefusco, E., Pellacci, B.: Weakly coupled nonlinear Schrödinger systems: the saturation effect. Calc. Var. Partial Differ. Equ. 46(1–2), 325–351 (2013)CrossRefzbMATHGoogle Scholar
  29. 29.
    Nehari, Z.: Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105, 141–175 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nehari, Z.: A nonlinear oscillation theorem. Duke Math. J. 42, 183–189 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Noris, B., Verzini, G.: A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems. J. Differ. Equ. 254(3), 1529–1547 (2013)CrossRefzbMATHGoogle Scholar
  32. 32.
    Ostrovskaya, E.A., Kivshar, Y.S.: Multi-hump optical solitons in a saturable medium. J. Opt. B: Quantum Semiclassical Opt. 1, 77–83 (1999)CrossRefGoogle Scholar
  33. 33.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Serrin, J., Tang, M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49, 897–923 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Stegeman, G.I., Christodoulides, D.N., Segev, M.: Optical spatial solitons: historical Perspectives. IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000)CrossRefGoogle Scholar
  36. 36.
    Stuart, C.A., Zhou, H.S.: Applying the mountain pass theorem to an asymptotically linear elliptic equation on \({\mathbb{R}}^{N}\). Commun. Partial Differ. Equ. 24(9–10), 1731–1758 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Struwe, M.: Variational Methods, vol. 34. Springer, Berlin (2008)zbMATHGoogle Scholar
  38. 38.
    Szulkin, A., Weth, T.: The method of Nehari manifold. In: Handbook of Nonconvex Analysis and Applications, pp. 597–632. Int Press, Somerville (2010)Google Scholar
  39. 39.
    Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser, Boston (1996)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Dipartimento di Scienze e TecnologieUniversità di Napoli “Parthenope”NaplesItaly

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