We report a combined experimental and theoretical study of micellization of block copolymer with hydrophilic nonionic corona-forming blocks and weak polyelectrolyte (wPE) core-forming blocks with pH-triggered solubility in aqueous solutions. We demonstrate that in addition to micelles with neutral cores, there exist two other types of micelles with PE- or ionomer-like cores, in which monovalent counterions are released or condensed on core wPE block, respectively. The transition between the two types of micelles occurred upon changes in ionization of the PE core block and resulted in nonmonotonous changes of aggregation number as a function of pH. Such micelles with stimulus responsive cores represent promising nanocarriers for controlled delivery applications.