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.
Categories: math.AP