The singularly perturbed boundary blow-up problem − ε 2 Δ u = u ( u − a ) ( 1 − u ) , u > 0 in B , u = ∞ on ∂ B is studied in the unit ball B ⊂ R N ( N ⩾ 2 ), a ∈ ( 1 / 2 , 1 ) is a constant. It is shown that for ε > 0 sufficiently small, there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.