In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem (− △u + u = f(x, u) + μh(x), x ∈ RN, u ∈ H1(RN),(∗)μ where h ∈ H−1(RN), N ≥ 3, |f(x, u)| ≤ C1up−1 + C2u with C1 > 0, C2 ∈ [0, 1) being some constants and 2 < p < +∞. Under some assumptions on f and h, they prove that there exists a positive constant μ < +∞ such that problem (∗)μ has at least one positive solution uμ if μ ∈ (0, μ), there are no solutions for (∗)μ if μ > μ and uμ is increasing with respect to μ ∈ (0, μ); furthermore, problem (∗)μ has at least two positive solution for μ ∈ (0, μ) and a unique positive solution for μ = μ if p ≤ 2N N−2 .
