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Elliptic equations with jumping nonlinearities involving high eigenvalues

  • Riccardo MolleEmail author
  • Donato Passaseo
Article

Abstract

We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer \(k\) there exists a positive integer \(n(k)\) such that, if the number of jumped eigenvalues is greater than \(n(k),\) then the problem has at least a solution which presents \(k\) peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc\(\ldots \)

Mathematics Subject Classification (2000)

35J20 35J60 35J65 

Notes

Acknowledgments

Work supported by the Italian national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Dipartimento di Matematica “E. De Giorgi”Università di LecceLecceItaly

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