  This article investigates nonlocal, fully nonlinear generalizations of the
classical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonic
operators appear naturally in the variational characterization of the optimal
fractional Poincar\'e constants in Bessel potential spaces. We study the
following basic questions for anisotropic fractional $p$-biharmonic systems:
existence and uniqueness of weak solutions to the associated interior source
and exterior value problems, unique continuation properties (UCP), monotonicity
relations, and inverse problems for the exterior Dirichlet-to-Neumann maps.
Furthermore, we show the UCP for the fractional Laplacian in all Bessel
potential spaces $H^{t,p}$ for any $t\in \mathbb{R}$, $1 \leq p < \infty$ and
$s \in \mathbb{R}_+ \setminus \mathbb{N}$: If $u\in H^{t,p}(\mathbb{R}^n)$
satisfies $(-\Delta)^su=u=0$ in a nonempty open set $V$, then $u\equiv 0$ in
$\mathbb{R}^n$. This property of the fractional Laplacian is then used to
obtain a UCP for the fractional $p$-biharmonic systems and plays a central role
in the analysis of the associated inverse problems. Our proofs use variational
methods and the Caffarelli-Silvestre extension.
