Archive for Rational Mechanics and Analysis

, Volume 207, Issue 3, pp 1075–1089 | Cite as

A Concentration Phenomenon for Semilinear Elliptic Equations

Article

Abstract

For a domain \({\Omega \subset \mathbb{R}^{N}}\) we consider the equation
$$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$
with zero Dirichlet boundary conditions and \({p\in(2, 2^*)}\). Here \({V \geqq 0}\) and Qn are bounded functions that are positive in a region contained in \({\Omega}\) and negative outside, and such that the sets {Qn > 0} shrink to a point \({x_0 \in \Omega}\) as \({n \to \infty}\). We show that if un is a nontrivial solution corresponding to Qn, then the sequence (un) concentrates at x0 with respect to the H1 and certain Lq-norms. We also show that if the sets {Qn > 0} shrink to two points and un are ground state solutions, then they concentrate at one of these points.

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References

  1. 1.
    Ambrosetti, A., Arcoya, D., Gámez, J.L.: Asymmetric bound states of differential equations in nonlinear optics. Rend. Sem. Math. Univ. Padova 100, 231–247 (1998). http://www.numdam.org/item?id=RSMUP_1998__100__231_0
  2. 2.
    Bandle, C., Marcus, M.: “Large” solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour. J. Anal. Math. 58, 9–24 (1992). doi:10.1007/BF02790355 (Festschrift on the occasion of the 70th birthday of Shmuel Agmon)
  3. 3.
    Berestycki H., Capuzzo-Dolcetta I., Nirenberg L.: Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differ. Equ. Appl. 2(4), 553–572 (1995)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bonheure D., Gomes J.M., Habets P.: Multiple positive solutions of superlinear elliptic problems with sign-changing weight. J. Differ. Equ. 214(1), 36–64 (2005). doi:10.1016/j.jde.2004.08.009 MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brézis H., Véron L.: Removable singularities for some nonlinear elliptic equations. Arch. Ration. Mech. Anal. 75(1), 1–6 (1980). doi:10.1007/BF00284616 MATHGoogle Scholar
  6. 6.
    Buryak A.V., Trapani P.D., Skryabin D.V., Trillo S.: Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370(2), 63–235 (2002). doi:10.1016/S0370-1573(02)00196-5 MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Costa D.G., Tehrani H.: Existence of positive solutions for a class of indefinite elliptic problems in \({\mathbb{R}^{N}}\). Calc. Var. Partial Differ. Equ. 13(2), 159–189 (2001)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dror, N., Malomed, B.A.: Solitons supported by localized nonlinearities in periodic media. Phys. Rev. A 83, 033,828 (2011). doi:10.1103/PhysRevA.83.033828 Google Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, Vol. 224, 2nd edn. Springer, Berlin, 1983Google Scholar
  10. 10.
    Girão P.M., Gomes J.M.: Multibump nodal solutions for an indefinite superlinear elliptic problem. J. Differ. Equ. 247(4), 1001–1012 (2009). doi:10.1016/j.jde.2009.04.018 MATHCrossRefGoogle Scholar
  11. 11.
    Kartashov Y.V., Malomed B.A., Torner L.: Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247–305 (2011). doi:10.1103/RevModPhys.83.247 ADSCrossRefGoogle Scholar
  12. 12.
    López-Gómez J.: Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems. Trans. Am. Math. Soc. 352(4), 1825–1858 (2000). doi:10.1090/S0002-9947-99-02352-1 MATHCrossRefGoogle Scholar
  13. 13.
    Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006). doi:10.1126/science.1125907 MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Shalaev V.M.: Optical negative-index metamaterials. Nat. Photon. 1(1), 41–48 (2007). doi:10.1038/nphoton.2006.49 MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Smith D.R., Pendry J.B., Wiltshire M.C.K.: Metamaterials and negative refractive index. Science 305(5685), 788–792 (2004). doi:10.1126/science.1096796 ADSCrossRefGoogle Scholar
  16. 16.
    Strauss, W.A.: The nonlinear Schrödinger equation. Contemporary developments in continuum mechanics and partial differential equations. Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. North-Holland Math. Stud., Vol. 30. North-Holland, Amsterdam, 452–465, 1978Google Scholar
  17. 17.
    Stuart C.A.: Bifurcation in L p(R N) for a semilinear elliptic equation. Proc. London Math. Soc. (3) 57(3), 511–541 (1988). doi:10.1112/plms/s3-57.3.511 MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Stuart C.A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Ration. Mech. Anal. 113(1), 65–96 (1991). doi:10.1007/BF00380816 MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Stuart C.A.: Guidance properties of nonlinear planar waveguides. Arch. Ration. Mech. Anal. 125(2), 145–200 (1993). doi:10.1007/BF00376812 MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Stuart C.A.: Existence and stability of TE modes in a stratified non-linear dielectric. IMA J. Appl. Math. 72(5), 659–679 (2007). doi:10.1093/imamat/hxm033 MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of ε and μ. Physics-Uspekhi 10(4), 509–514 (1968). doi:10.1070/PU1968v010n04ABEH003699. http://ufn.ru/en/articles/1968/4/e/

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México, Circuito Exterior, C.U.MexicoMexico
  2. 2.Department of MathematicsStockholm UniversityStockholmSweden

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