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

TL;DR

This paper investigates the Dancer-Fucik spectrum of the fractional p-Laplacian, constructing spectral curves, analyzing local behavior, and applying critical group theory to find solutions of perturbed nonlinear problems.

Contribution

It introduces a minimax scheme to construct spectral curves and provides a detailed local analysis for p=2, advancing understanding of the spectrum's structure and solutions.

Findings

Constructed an unbounded sequence of spectral curves.

Identified conditions for the spectrum region between minimal and maximal curves.

Computed critical groups and established a shifting theorem for solution existence.

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.