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

Abstract

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.