On the bifurcation for fractional Laplace equations

Abstract

In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-Δ)^{s} u = λu + f(λ,\,x,\,u)& \mbox{in }Ω, u = 0 &\mbox{in }\mathbb{R}^n\backslash Ω, \end{array} \end{eqnarray*} where $Ω\subset \mathbb{R}^n,\,n> 2s (0<s<1)$ is an open bounded subset with smooth boundary, $(-Δ)^{s}$ stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue $λ_1$ of the eigenvalue problem \begin{eqnarray*} \begin{gathered} (-Δ)^{s} v = λv\,\,\,\mbox{in}\,\,Ω, v = 0 \,\,\,\,\mbox{in}\,\,\,\,\mathbb{R}^n \backslashΩ, \end{gathered} \end{eqnarray*} and, conversely.