Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension.

We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a positive radially symmetric weight in the unit ball. When the weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension 3 ≤ N ≤ 9 , and it has no turning points if N ≥ 10. In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when 3 ≤ N ≤ 9. Moreover, we find a one-parameter family of radial singular solutions for a parametrized weight and study the Morse index of the singular solution. As a result, we prove that the perturbation affects the bifurcation structure in the critical dimension N = 10. Moreover, we give a classification of the bifurcation diagrams in the critical dimension. [ABSTRACT FROM AUTHOR]