An exact estimate result for a semilinear equation with critical exponent and prescribed singularity

We consider the singular boundary value problem{−Δu−λV(x)u=h(x)u−γ+μu2⁎−1inΩ,u(x)>0inΩ,u(x)=0on∂Ω, where 0<λ<λ¯=(N−22)2,0<γ<1,2⁎=2NN−2, h(x)h(x) is a given function and V(x)V(x) has prescribed finitely many singular points. Our goal in this paper is to establish some existence and multiplicity results for above problem when μ∈(0,μ⁎)μ∈(0,μ⁎) for some μ⁎>0μ⁎>0 and obtain exact estimate for extremal value μ⁎=μ⁎(Ω,γ,2⁎,h(x))>0μ⁎=μ⁎(Ω,γ,2⁎,h(x))>0 for above problem.