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

摘要

In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation ??u = λ u γ + u 2 ??1 , x ∈ ?, u = 0, x ∈ ??, where ? 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 ?.