On the structure of positive solutions to an elliptic problem with a singular nonlinearity

We consider the following boundary value problemΔu=λ[u−p−u−q]in Ω,u=κ∈(0,∞)on ∂Ω, where p>q>0 and Ω is a bounded smooth domain in RN (N⩾2). It is shown that, for κ>1, there is at least one solution uλ of this problem satisfying uλ>κ in Ω for any λ>0. The profiles of uλ for λ→0 and λ→∞ are analyzed. For κ=1, it is clear that uλ≡1 is a constant solution to the problem. We also present the branch of solutions satisfying 0