Global Bifurcation for Fractional $p$-Laplacian and an Application

Abstract

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation

bifurcating from the first eigenvalue. Here $(-\Delta {)}_p^s$ denotes the fractional $p$-Laplacian and $\Omega \subset {\mathbb{R}}^n$ is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray–Schauder degree by making an homotopy respect to $s$ (the order of the fractional $p$-Laplacian) and then to use results of local case (that is $s=1$) found in the paper of del Pino and Manásevich [J. Diff. Equ. 92 (1991)(2), 226–251]. Finally, we give some application to an existence result.