A note on the Dancer–Fučík spectra of the fractional <i>p</i> -Laplacian and Laplacian operators

Abstract

Abstract We study the Dancer–Fučík 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.