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Zeitschrift für Analysis und ihre Anwendungen

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Published online: 2016-10-05

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

Leandro M. Del Pezzo^[1] and Alexander Quaas^[2]

(1) Universidad Torcuato di Tella, C. A. de Buenos Aires, Argentina
(2) Universidad Técnica Federico Santa María, Valparaíso, Chile

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$(-\Delta)^s_p u=\lambda |u|^{p-2}u + f(x,u,\lambda) \quad\text{in } \Omega,\quad u=0 \quad\text{in } \mathbb R^n\setminus\Omega ,$$ bifurcating from the first eigenvalue. Here $(-\Delta)^s_p$ 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.

Keywords: Bifurcation, fractional $p$-Laplacian, existence results

Del Pezzo Leandro, Quaas Alexander: Global Bifurcation for Fractional $p$-Laplacian and an Application. Z. Anal. Anwend. 35 (2016), 411-447. doi: 10.4171/ZAA/1572