Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient f ( z ) of the critical nonlinearity. For sufficiently small λ , μ > 0 , there are at least k positive solutions of the semilinear elliptic systems { − Δ u = λ g ( z ) | u | p − 2 u + α α + β f ( z ) | u | α − 2 u | v | β in Ω ; − Δ v = μ h ( z ) | v | p − 2 v + β α + β f ( z ) | u | α | v | β − 2 v in Ω ; u = v = 0 on ∂ Ω , where 0 ∈ Ω ⊂ R N is a bounded domain, α > 1 , β > 1 and 2 < p < α + β = 2 ⁎ for N > 4 .