The successful achievement of demand flexibility relies heavily on effective building energy management (BEM) to manage users' diverse needs and priorities. Building on the day-Ahead demand response program, this paper proposes a many-objective BEM framework that supports user-centric decision-making by considering electricity cost, thermal comfort, inconvenience, and CO2 emissions objectives. The energy modelling of photovoltaic (PV), battery storage system (BSS), ground source heat pump (GSHP), and the flexible loads is first developed. A case study is conducted on a typical Swedish detached house, where the non-dominated sorting genetic algorithm (NSGA)-III is applied to solve this many objectives optimization problem and obtain the Pareto frontier set. When all objectives are weighed equally (0.25 each), results show the proposed BEM strategy helps save 43.8% of daily electricity cost and reduces 36.3% of CO2 emissions compared to the scenario with no PV or BSS used; 29.8% of daily electricity cost and 14.5% of CO2 emissions are lowered when compared with the scenario with PV and BSS applied but without management. These findings demonstrate the proposed BEM can effectively support users' different priorities for decision-making in practical applications to enhance demand flexibility.