Bifurcations of Some Elliptic Problems with a Singular Nonlinearity via Morse Index

We study the boundary value problem ∆u = λ|x|f(u) in Ω, u = 1 on ∂Ω (1) where λ > 0, α ≥ 0, Ω is a bounded smooth domain in RN (N ≥ 2) containing 0 and f is a C1 function satisfying lims→0+ s pf(s) = 1. We show that for each α ≥ 0, there is a critical power pc(α) > 0, which is decreasing in α, such that the branch of positive solutions possesses infinitely many bifurcation points provided p > pc(α) or p > pc(0), and this relies on the shape of the domain Ω. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that p > pc(α) and the Morse index of any radial solution (regular or singular) in this branch is finite provided that 0 < p ≤ pc(α). This implies that the structure of the radial solution branch of (1) changes for 0 < p ≤ pc(α) and p > pc(α).