Skip to main content
SpringerLink
Log in

A Concentration Phenomenon for Semilinear Elliptic Equations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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 Q n are bounded functions that are positive in a region contained in \({\Omega}\) and negative outside, and such that the sets {Q n  > 0} shrink to a point \({x_0 \in \Omega}\) as \({n \to \infty}\). We show that if u n is a nontrivial solution corresponding to Q n , then the sequence (u n ) concentrates at x 0 with respect to the H 1 and certain L q-norms. We also show that if the sets {Q n  > 0} shrink to two points and u n are ground state solutions, then they concentrate at one of these points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    MATH  Google Scholar 

  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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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, 1983

  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

    Article  MATH  Google Scholar 

  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

    Article  ADS  Google Scholar 

  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

    Article  MATH  Google Scholar 

  13. Pendry J.B., Schurig D., Smith D.R.: Controlling electromagnetic fields. Science 312(5781), 1780–1782 (2006). doi:10.1126/science.1125907

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Shalaev V.M.: Optical negative-index metamaterials. Nat. Photon. 1(1), 41–48 (2007). doi:10.1038/nphoton.2006.49

    Article  MathSciNet  ADS  Google Scholar 

  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

    Article  ADS  Google Scholar 

  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, 1978

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  19. Stuart C.A.: Guidance properties of nonlinear planar waveguides. Arch. Ration. Mech. Anal. 125(2), 145–200 (1993). doi:10.1007/BF00376812

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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/

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Mexico, DF, Mexico

    Nils Ackermann

  2. Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden

    Andrzej Szulkin

Authors
  1. Nils Ackermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrzej Szulkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nils Ackermann.

Additional information

Communicated by P. Rabinowitz

Nils Aickermann was supported by CONACYT grant 129847 and PAPIIT-DGAPA-UNAM grants IN101209 and IN106612 (Mexico). Aindrzej Sizulkin was supported in part by the Swedish Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ackermann, N., Szulkin, A. A Concentration Phenomenon for Semilinear Elliptic Equations. Arch Rational Mech Anal 207, 1075–1089 (2013). https://doi.org/10.1007/s00205-012-0589-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-012-0589-1

Keywords

Search

Navigation