Eigenvalues for a nonlocal pseudo $p-$Laplacian

Leandro M. Del  Pezzo; Julio D.  Rossi

doi:10.3934/dcds.2016093

Article Contents

2016, Volume 36, Issue 12: 6737-6765. Doi: 10.3934/dcds.2016093

This issue Previous Article Normal forms of planar switching systems Next Article Classification of positive solutions to a Lane-Emden type integral system with negative exponents

Eigenvalues for a nonlocal pseudo $p-$Laplacian

Leandro M. Del Pezzo^1, and
Julio D. Rossi^1,

1.
CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria, Buenos Aires, 1428

Received: October 31, 2015

Revised: July 31, 2016

Published: May 2017

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Abstract

In this paper we study the eigenvalue problems for a nonlocal operator of order $s$ that is analogous to the local pseudo $p-$Laplacian. We show that there is a sequence of eigenvalues $\lambda_n \to \infty$ and that the first one is positive, simple, isolated and has a positive and bounded associated eigenfunction. For the first eigenvalue we also analyze the limits as $p\to \infty$ (obtaining a limit nonlocal eigenvalue problem analogous to the pseudo infinity Laplacian) and as $s\to 1^-$ (obtaining the first eigenvalue for a local operator of $p-$Laplacian type). To perform this study we have to introduce anisotropic fractional Sobolev spaces and prove some of their properties.

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Mathematics Subject Classification: Primary: 35P30, 35J92, 35R11.

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