A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions

We consider the following semilinear elliptic equation: where B 1 is the unit ball in ℝ d , d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λ p,∞ such that ( * ) has a solution ( λ p,∞ , W p ) satisfying lim |x|→ 0 W p ( 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 λ p,∞ when 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in the d ≥ 11 case.