Eigenvalue, Unilateral Global Bifurcation and Constant Sign Solution for a Fractional Laplace Problem

Abstract

In this paper, we apply the Ljusternik–Schnirelmann theory to deal with the eigenvalues and eigenfunctions of the fractional Laplace operator. It shows that there exists a simple and isolated principal eigenvalue [Formula: see text] such that under certain conditions on the perturbation function [Formula: see text], [Formula: see text] is a bifurcation point of the problem [Formula: see text] where [Formula: see text], [Formula: see text] is a smooth and bounded domain, [Formula: see text] [Formula: see text] with [Formula: see text] and [Formula: see text] satisfies the Carathéodory condition in the first two variables. There are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text], consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of the unilateral global bifurcation result, we extend our investigation to the existence of constant sign solutions for a class of related nonlinear fractional Laplace problems. Some known results on the fractional Laplace problem in the literature are generalized.