Remarks on a Brezis–Nirenberg's result

In this note, we revisit the equation with Sobolev critical exponent which is firstly studied by Brezis–Nirenberg in [4]:(P){−Δu+λu=μ|u|p−2u+β|u|2⁎−2u,in Ω,u∈H01(Ω), where Ω⊂RNΩ⊂RN is a bounded smooth domain, N≥3N≥3, 20β>0, λ,μ∈Rλ,μ∈R. We show that for any m∈Nm∈N, there exists βm>0βm>0 such that (P)(P) has m   solutions for any λ∈Rλ∈R, μ>0μ>0 and every β∈(0,βm)β∈(0,βm). When N≥4N≥4, λ<0λ<0, β>0β>0, we show that there is μ0>0μ0>0 such that problem (P)(P) has a nontrivial solution for every μ>−μ0μ>−μ0. Moreover, we get multiple sign changing solutions for problem (P)(P).