Pool boiling serves as physical model problem for electronics cooling by means of phase-change heat transfer. Key for optimal and reliable cooling capacity is better understanding of the conditions that determine the critical heat flux (CHF). Exceeding CHF results in transition from efficient nucleate boiling to inefficient film boiling. This transition is intimately related to the formation and stability of multiple (steady) states on the fluid-heater interface. To this end the steady-state behaviour of a three-dimensional pool-boiling system has been studied in terms of a representative mathematical model problem. This model problem involves only the temperature field within the heater and models the heat exchange with the boiling medium via a nonlinearboundary condition imposed on the fluid-heater interface. The steady-state behaviour is investigated via a bifurcation analysis with a continuation algorithm based upon treatment of the model with the method of separation of variables and a Fourier-collocation method. This revealed that steady-state solutions with homogeneous interface temperatures may undergo bifurcations that result in multiple solutions with essentially heterogeneous interface temperatures. These heterogeneous states phenomenologically correspond with vapour patches (`dry spots') on the interface that characterise transitionconditions. The findings on the model problem are consistent with laboratory experiments.