A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators

Kanishka Perera; Marco Squassina; Yang Yang

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arXiv · 1410.0402

A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators

Kanishka Perera·Marco Squassina·Yang Yang

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Abstract

We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We construct an unbounded sequence of decreasing curves in the spectrum using a suitable minimax scheme. For p=2, we present a very accurate local analysis. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sufficient condition for the region between them to be nonempty, and show that it is free of the spectrum in the case of a simple eigenvalue. Finally we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions, and prove a shifting theorem that gives a finite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and infinity.

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Kanishka Perera, Marco Squassina, Yang Yang. 2014-10-19. A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators. https://arxiv.org/abs/1410.0402

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