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

We consider the following boundary value problem Δ u = λ [ u − p − u − q ] in Ω , u = κ ∈ ( 0 , ∞ ) on ∂ Ω , where p > q > 0 and Ω is a bounded smooth domain in R N ( N ⩾ 2 ). It is shown that, for κ > 1 , there is at least one solution u λ of this problem satisfying u λ > κ in Ω for any λ > 0 . The profiles of u λ for λ → 0 and λ → ∞ are analyzed. For κ = 1 , it is clear that u λ ≡ 1 is a constant solution to the problem. We also present the branch of solutions satisfying 0 < u λ < 1 in Ω for the equation with κ = 1 . We obtain more information on the structure of positive radial solutions of the problem when Ω is an annulus.