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

作者：Perera Kanishka^*; Squassina Marco; Yang Yang

来源：Advances in Nonlinear Analysis, 2015, 4(1): 13-23.

DOI：10.1515/anona-2014-0038

• Fractional p-Laplacian
• Dancer-Fucik spectrum
• critical groups

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.

• 出版日期2015-2
• 单位江南大学