Abstract: Let f(x, t):R²×R→ R be a C²-function with respect to t∈R, f(x,0)=0, f(x, t)~e^bt2 as t→+∞ for somc b>0. Under suitable conditions on f(x, t), author shows that for g∈L² (R²), g(x) ≥ (≠) 0, the following semilinear clliptic problem:
-△u+u=f(x,u)+λg(x),x∈R², u∈H¹(R²),λ>0,
has at least two distinct positive solutions for any λ∈(0, λ_*), at least one positive solution for any λ∈[λ_*, λ^*] and has no positive solntion for all λ>λ^*. It is also proved that λ_* ≤ λ^*< +∞.

Key words: Positive solution, elliptic equations, critical growth