On the bifurcation for fractional Laplace equations

TL;DR

This paper investigates bifurcation phenomena for fractional Laplace equations, demonstrating the emergence of solution continua from the principal eigenvalue in bounded domains with smooth boundaries.

Contribution

It establishes the existence of bifurcation from the principal eigenvalue for fractional Laplace equations, extending bifurcation theory to nonlocal operators.

Findings

A continuum of solutions bifurcates from the principal eigenvalue.

Bifurcation occurs both ways from the principal eigenvalue.

Results apply to bounded domains with smooth boundaries.

Abstract

In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-\Delta)^{s} u = \lambda u + f(\lambda,\,x,\,u)& \mbox{in }\Omega, u = 0 &\mbox{in }\mathbb{R}^n\backslash \Omega, \end{array} \end{eqnarray*} where $\Omega\subset \mathbb{R}^n,\,n> 2s (0<s<1)$ is an open bounded subset with smooth boundary, $(-\Delta)^{s}$ stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue $\lambda_1$ of the eigenvalue problem \begin{eqnarray*} \begin{gathered} (-\Delta)^{s} v = \lambda v\,\,\,\mbox{in}\,\,\Omega, v = 0 \,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^n \backslash\Omega, \end{gathered} \end{eqnarray*} and, conversely.