Existence and multiplicity of positive solutions for a nonlocal problem

In this work, we are interested in considering the following nonlocal problem ⎧⎨⎩−(a−b∫Ω|∇u|2dx)Δu=f(x)|u|p−2u,in Ω,u=0,on ∂Ω, where Ω⊂RN (N≥3)Ω⊂RN (N≥3) is a bounded domain with smooth boundary ∂Ω,a,b 0,1≤p 2∗∂Ω,a,b 0,1≤p 2∗, f∈L2∗/2∗−p(Ω)f∈L2∗2∗−p(Ω) is nonzero and nonnegative. By using the variational method, some existence and nonnegative. By using the variational method, some existence and multiplicity results are obtained.