Pool-boiling serves as the physical model problem for electronics cooling by means of phase-change heat-transfer. The key for optimal and reliable cooling capacity is better understanding of the conditions that determine the critical heat-flux (CHF). Exceeding CHF results in the 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 behavior 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 nonlinear boundary condition imposed on the fluid-heater interface. The steady-state behavior is investigated via a bifurcation analysis with a continuation algorithm based on the 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 vapor patches (“dry spots”) on the interface that characterize transition conditions. The findings on the model problem are consistent with laboratory experiments.