Bifurcation diagram of solutions to elliptic equation with exponential nonlinearity in higher dimensions

We consider the following semilinear elliptic equation:    −∆u = λeup in B1, u = 0 on ∂B1, (0.1) where B1 is the unit ball in R, d ≥ 3, λ > 0 and p > 0. First, following Merle and Peletier [13], we show that there exists a unique eigenvalue λp,∞ such that (0.1) has a solution (λp,∞,Wp) satisfying lim|x|→0 Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (0.1). It follows from the result of Dancer [4] that (0.1) has an unbounded bifurcation branch of regular solutions which emanates from (λ, u) = (0, 0). Here, using the singular solution, we show that the bifurcation branch has infinitely many turning points around λp,∞ in case of 3 ≤ d ≤ 9. We also investigate the Morse index of the singular solution in case of d ≥ 11.