We consider the following semilinear elliptic equation:

where B1 is the unit ball in ℝd, d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λp,∞ such that (*) has a solution (λp,∞,Wp) satisfying lim|x|→0Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (*). It follows from the result of Dancer that (*) has an unbounded bifurcation branch of regular solutions that emanates from (λ, u) = (0, 0). Here, using the singular solution, we show that the bifurcation branch has infinitely many turning points around λp,∞ when 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in the d ≥ 11 case.