  We are interested in the global bifurcation diagram of radial solutions for
the Gelfand problem with the exponential nonlinearity and a radially symmetric
weight $0<a(|x|)\in C^2(\overline{B_1})$ in the unit ball. When the weight is
constant, it is known that the bifurcation curve has infinitely many turning
points if the dimension $N\le 9$, and it has no turning points if $N\ge 10$.
  In this paper, we show that the perturbation of the weight does not affect
the bifurcation structure when $N\le 9$. Moreover, we find specific radial
singular solutions with specific weights and study the Morse index of the
solutions. As a consequence, we prove that the perturbation affects the
bifurcation structure in the critical dimension $N=10$. Moreover, we give an
optimal classification of the bifurcation diagrams in the critical dimension.
