JournalszaaVol. 35, No. 4pp. 411–447

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

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Abstract

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

(Δ)psu=λup2u+f(x,u,λ)in Ω,u=0in RnΩ,(-\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 (Δ)ps(-\Delta)^s_p denotes the fractional pp-Laplacian and ΩRn\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 ss (the order of the fractional pp-Laplacian) and then to use results of local case (that is s=1s=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 pp-Laplacian and an Application. Z. Anal. Anwend. 35 (2016), no. 4, pp. 411–447

DOI 10.4171/ZAA/1572