Theories combining nuclear density functional approach (DFT) and effects beyond the independent particle/quasi-particle limit have attracted much attention recently. In particular, such theories, generically referred as “beyond mean-field” (BMF) seem unavoidable to account for both single-particle effects and complex quantum internal phenomena in nuclear finite many-body nuclear systems. It has been realized recently that BMF theories might lead to specific difficulties when applied within the nuclear DFT context. An example is the appearance of divergences in configuration mixing approaches. A short summary of the difficulties is given here. One source of problem is the use of energy functional of non-integer powers of the density. We show that such dependence can be mimicked by a suitable choice of a three-body interaction. Application on infinite nuclear matter in various spin-isospin channels will be given.