Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension
Kenta Kumagai · Journal of Differential Equations · 2024
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.