Milan Journal of Mathematics

, Volume 79, Issue 1, pp 221–232 | Cite as

Self-focusing Multibump Standing Waves in Expanding Waveguides

Article

Abstract

Let M be a smooth k-dimensional closed submanifold of \({\mathbb{R}^N, N \geq 2}\), and let ΩR be the open tubular neighborhood of radius 1 of the expanded manifold \({M_R := \{R_x : x \in M\}}\). For R sufficiently large we show the existence of positive multibump solutions to the problem
$$ -\Delta u + \lambda u = f(u)\,{\rm in}\,\Omega_R,\quad u= 0\,{\rm on}\,\partial\Omega_R. $$
The function f is superlinear and subcritical, and λ >  −λ1, where λ1 is the first Dirichlet eigenvalue of −Δ in the unit ball in \({\mathbb{R}^{N-k}}\).

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References

  1. 1.
    N. Ackermann, M. Clapp, and F. Pacella, Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains, 40 pages, preprint, 2011.Google Scholar
  2. 2.
    T. Bartsch, M. Clapp, M. Grossi, and F. Pacella, Asymptotically radial solutions in expanding annular domains, Preprint.Google Scholar
  3. 3.
    H. Berestycki, L.A. Caffarelli, and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. I, Duke Math. J. 81 (1996), no. 2, 467-494, A celebration of John F. Nash, Jr.Google Scholar
  4. 4.
    Byeon J.: Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli. J. Differential Equations 136(1), 136–165 (1997)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Catrina F., Wang Z.Q.: Nonlinear elliptic equations on expanding symmetric domains. J. Differential Equations 156(1), 153–181 (1999)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Coffman C.V.: A nonlinear boundary value problem with many positive solutions. J. Differential Equations 54(3), 429–437 (1984)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dancer E.N.: Real analyticity and non-degeneracy. Math. Ann. 325(2), 369–392 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dancer E.N., Yan S.: Multibump solutions for an elliptic problem in expanding domains. Comm. Partial Differential Equations 27(1-2), 23–55 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fromm S.J.: Potential space estimates for Green potentials in convex domains. Proc. Amer. Math. Soc. 119(1), 225–233 (1993)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Lee M.G., Lin S.S.: Multiplicity of positive solutions for nonlinear elliptic equations on annulus. Chinese J. Math. 19(3), 257–276 (1991)MathSciNetMATHGoogle Scholar
  11. 11.
    Li Y.Y.: Existence of many positive solutions of semilinear elliptic equations on annulus. J. Differential Equations 83(2), 348–367 (1990)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lin S.S.: Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus. J. Differential Equations 103(2), 338–349 (1993)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    C. Sulem and P.L. Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999, Self-focusing and wave collapse.Google Scholar
  14. 14.
    Suzuki T.: Positive solutions for semilinear elliptic equations on expanding annuli: mountain pass approach. Funkcial. Ekvac. 39(1), 143–164 (1996)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Nils Ackermann
    • 1
  • Mónica Clapp
    • 1
  • Filomena Pacella
    • 2
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico D.F.Mexico
  2. 2.Dipartimento di MatematicaUniversità “La Sapienza” di RomaRomaItaly

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