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In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation −∆u = λ uγ + u2 −1, x ∈ Ω, u = 0, x ∈ ∂Ω, where Ω is a smooth bounded domain in RN (N ≥ 3), 2∗ = 2N N−2 , γ ∈ (0, 1) and λ > 0 is a real parameter. We show by the variational methods and perturbation functional that the problem has at least two positive solutions w0(x) and w1(x) with w0(x) < w1(x) in Ω.

New multiple positive solutions for elliptic equations with singularity and critical growth