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