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Fractional eigenvalues

  • Erik Lindgren
  • Peter Lindqvist
Article

Abstract

We study the non-local eigenvalue problem
$$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$
for large values of \(p\) and derive the limit equation as \(p\rightarrow \infty \). Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.

Mathematics Subject Classification (2000)

35J60 35P30 35R11 

Notes

Acknowledgments

We thank Evgenia Malinnikova for helping us to verify an inequality. We thank the referees for a careful reading of the manuscript and for drawing our attention to the article [9].

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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