The $p$-bending energy is defined by the $L^p$-norm of the curvature, and critical points of the $p$-bending energy under the fixed length constraint are called $p$-elasticae. The aim of this talk is to give a complete classification of planar $p$-elasticae. In addition, by introducing a new type of generalization of the elliptic functions, we also obtain explicit formulae of planar $p$-elasticae and their optimal regularity.