Positive solutions for semilinear elliptic equations with singular forcing terms

We consider the existence of solutions to the semilinear elliptic problemequation(∗){−Δu+u=up+κ∑i=1mciδaiin D′(RN),u>0a.e. inRNandu(x)→0as|x|→∞,with prescribed given finite points {ai}i=1m in RNRN and positive numbers {ci}i=1m, where N⩾3N⩾3, 10κ∗>0 such that (∗)κ has at least two solutions if κ∈(0,κ∗)κ∈(0,κ∗), a unique solution if κ=κ∗κ=κ∗, and no solution if κ>κ∗κ>κ∗.