Global bifurcation for fractional $p$-Laplacian and application

TL;DR

This paper establishes the existence of an unbounded branch of solutions for a fractional p-Laplacian equation using bifurcation theory, extending classical results to non-local operators and providing applications to existence problems.

Contribution

It introduces a bifurcation framework for the fractional p-Laplacian, connecting local and non-local cases via Leray--Schauder degree homotopy, and applies it to prove solution existence.

Findings

Existence of an unbounded bifurcation branch from the first eigenvalue.

Method of homotopy in fractional order s to compute degree.

Application to existence results for fractional p-Laplacian equations.

Abstract

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