Exact multiplicity for boundary blow-up solutions

The singularly perturbed boundary blow-up problem−ε2Δu=u(u−a)(1−u),u>0 in B,u=∞ on ∂B is studied in the unit ball B⊂RNB⊂RN (N⩾2N⩾2), a∈(1/2,1)a∈(1/2,1) is a constant. It is shown that for ε>0ε>0 sufficiently small, there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.