Existence and multiplicity results for resonant fractional boundary value problems

TL;DR

This paper investigates a fractional boundary value problem with resonance, proving the existence of multiple solutions using eigenvalue properties and Morse theory, including conditions for three distinct solutions.

Contribution

It introduces new existence and multiplicity results for resonant fractional boundary value problems using spectral analysis and Morse theory techniques.

Findings

Existence of a non-trivial solution at resonance.

Presence of at least three solutions under certain conditions.

Application of weighted fractional eigenvalues and Morse theory.

Abstract

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.