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} (-\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.