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

Leandro M. Del Pezzo; Alexander Quaas

doi:10.4171/zaa/1572

JournalszaaVol. 35, No. 4pp. 411–447

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

Leandro M. Del Pezzo
Universidad Torcuato di Tella, C. A. de Buenos Aires, Argentina
Alexander Quaas
Universidad Técnica Federico Santa María, Valparaíso, Chile

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Abstract

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

(- Δ)_{p}^{s} u = λ ∣ u ∣^{p - 2} u + f (x, u, λ) in Ω, u = 0 in R^{n} ∖ Ω,

bifurcating from the first eigenvalue. Here $(- Δ)_{p}^{s}$ denotes the fractional $p$ -Laplacian and $Ω \subset R^{n}$ 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.

Cite this article

Leandro M. Del Pezzo, Alexander Quaas, Global Bifurcation for Fractional $p$ -Laplacian and an Application. Z. Anal. Anwend. 35 (2016), no. 4, pp. 411–447

DOI 10.4171/ZAA/1572