Given its highly radioactive, highly chemically reactive nature, and rarity, Francium is not likely to have many practical patentable applications. However patent databases do contain patent documents which actually contain references to francium.
These patents are predominantly catch-all disclosures which include, or exclude from, the claimed invention, all or some of the group 1, alkali metals. In these cases francium is mentioned for the sake of speculative completeness. For instance US2100130272 (titanate photocatalyst (2011)) relates to a compound of formula HmAx-mTiyOzDn wherein: A is a cation selected [included] from the group consisting of lithium, sodium, potassium, rubidium, caesium and francium. On the other hand GB2538658 relating to drilling fluids (Weighted well fluids (2016)) discloses that the fluids are substantially free of [exclude] lithium (Li), sodium (Na), potassium (K), rubidium (Rb), cesium (Cs), and francium (Fr) salts and/or ions.