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 $$ (-Δ)^s_p u=λ|u|^{p-2}u + f(x,u,λ) \quad\text{in}\quad Ω,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminusΩ, $$ bifurcating from the first eigenvalue. Here $(-Δ)^s_p$ denotes the fractional $p$-Laplacian and $Ω\subset\mathbb{R}^n$ 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