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arXiv · 1511.08193

Eigenvalues for a nonlocal pseudo $p-$Laplacian

Leandro M. Del Pezzo·Julio D. Rossi
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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 $λ_n \to \infty$ 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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Leandro M. Del Pezzo, Julio D. Rossi. 2015-11-25. Eigenvalues for a nonlocal pseudo $p-$Laplacian. https://doi.org/10.3934/dcds.2016093

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