A macroscopic first-principles mathematical model of a solid oxide fuel cell (SOFC) with a BaCe1-xSmxO3-a type electrolyte is developed. This class of electrolytes exhibits both proton and oxygen-anion conductivity. The existence of steady-state multiplicity with respect to bifurcation parameters such as cell outlet voltage, cell current, ohmic external load, and cell power density is investigated. The analyses with respect to the first two bifurcation parameters represent potentiostatic and galvanostatic modes of operation, respectively. The cell can have up to three steady states with respect to the external load resistance and cell outlet voltage, and a unique steady state with respect to the cell current. The cell can have four steady states with respect to cell power density (when power density is the bifurcation parameter). The same qualitative steady-state behavior had been observed in oxygen-ion conducting and proton conducting SOFCs. Furthermore, thermal and concentration multiplicities coexist in this class of SOFCs; ignition in the solid temperature is accompanied by (a) extinction in the fuel and oxygen concentrations, and (b) ignition and extinction in concentrations of water in the anode and cathode sides, respectively. The cell exhibits steady-state multiplicity with respect to the fuel and air flow rates under fixed ohmic load, potentiostatic, and fixed cell power density modes of operation, but does not show this behavior under a galvanostatic mode of operation. The dynamic response of the cell also shows that the cell is highly nonlinear and some of the cell variables can show inverse response.