Uniqueness of point rupture solutions to semilinear elliptic equations

We prove the uniqueness of positive solutions for{u″+αru′+f(u)=0,r>0,limr→0+⁡u=0, in the case where α≥1, f<0 andlimt→0+⁡−tf′(t)f(t)=p∈(−1,∞).