Milan Journal of Mathematics

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

Self-focusing Multibump Standing Waves in Expanding Waveguides



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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© 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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