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This article investigates nonlocal, fully nonlinear generalizations of theclassical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonicoperators appear naturally in the variational characterization of the optimalfractional Poincar\'e constants in Bessel potential spaces. We study thefollowing basic questions for anisotropic fractional $p$-biharmonic systems:existence and uniqueness of weak solutions to the associated interior sourceand exterior value problems, unique continuation properties (UCP), monotonicityrelations, and inverse problems for the exterior Dirichlet-to-Neumann maps.Furthermore, we show the UCP for the fractional Laplacian in all Besselpotential 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 toobtain a UCP for the fractional $p$-biharmonic systems and plays a central rolein the analysis of the associated inverse problems. Our proofs use variationalmethods and the Caffarelli-Silvestre extension.

The fractional $p\,$-biharmonic systems: optimal Poincar\'e constants, unique continuation and inverse problems