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In this paper we prove Bourgain-Brezis-Mironescu's type results (cf.\cite{BBM2001}) (BBM for short) for an energy functional which is stronglyrelated to the fractional anisotropic p-Laplacian. We also provide with theanalogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for theseenergies and finally we apply these results to analyze the stability ofsolutions to anisotropic fractional $p-$laplacian equations.

A Bourgain-Brezis-Mironescu formula for anisotropic fractional Sobolev spaces and applications to anisotropic fractional differential equations