Global bifurcation for fractional $p$-Laplacian and application

Abstract

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$(-\Delta {)}_p^{s}u=\lambda |u{|}^{p-2}u+f(x,u,\lambda )\text{in}\Omega ,u=0\text{in}{\mathbb{R}}^n\setminus \Omega ,$$ 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 [17]. Finally, we give some application to an existence result.

38 pages

References in corpus (3)

• Local behavior of fractional $p$-minimizers
• Fractional p-eigenvalues
• Bifurcation and multiplicity results for critical fractional p-Laplacian problems

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