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

2014/10/01 by Perera, Kanishka, Squassina, Marco, Yang, Yang
#35P30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35R11 #Secondary 35A15

paper · doi:10.48550/arxiv.1410.0402

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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