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

TL;DR

This paper studies the bifurcation diagrams of radial solutions for a semilinear elliptic equation with exponential nonlinearity, revealing how perturbations influence the structure especially in the critical dimension N=10.

Contribution

It provides an optimal classification of bifurcation diagrams in the critical dimension and analyzes the effects of weight perturbations on the bifurcation structure.

Findings

Perturbations do not affect bifurcation structure for N≤9.

In the critical dimension N=10, perturbations alter the bifurcation diagram.

Identified specific singular solutions and analyzed their Morse index.

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.