We study, on the entire space ℝN(N [ges ] 1), the diffusive logistic equation
and its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important
role in understanding various population models and some other problems in applied
mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and
KPP equation, due to the pioneering works of Fisher  and Kolmogoroff, Petrovsky
and Piscounoff .