A Bahri–Lions theorem revisited

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:−Δu=|u|p−2u+f(x,u),u∈H01(Ω), where Ω   is a bounded smooth domain of RNRN, 2