Classification of bifurcation structure for semilinear elliptic equations in a ball

TL;DR

This paper investigates the bifurcation structure of solutions to a semilinear elliptic equation in a ball, revealing that the structure depends on more than just the growth rate of the nonlinearity, and providing a new classification criterion.

Contribution

It introduces a new class of nonlinearities with high growth rates that still exhibit infinitely many bifurcation turning points, challenging previous assumptions.

Findings

Bifurcation structure depends on more than growth rate.

Existence of nonlinearities with high growth and multiple bifurcation points.

A general criterion for classifying bifurcation structures.

Abstract

We consider the Gelfand problem with Sobolev supercritical nonlinearities $f$ in the unit ball. In the case where $f$ is a power type nonlinearity or the exponential nonlinearity, it is well-known that the bifurcation curve has infinitely many turning points when the growth rate of $f$ is smaller than that of the specific nonlinearity (called the Joseph-Lundgren critical nonlinearity), while the bifurcation curve has no turning point when the growth rate of $f$ is greater than or equal to that of the Joseph-Lundgren critical nonlinearity. In this paper, we give a new type of nonlinearity $f$ such that the growth rate is greater than or equal to that of the Joseph-Lundgren critical nonlinearity, while the bifurcation curve has infinitely many turning points. This result shows that the bifurcation structure is not determined solely by the comparison between the growth rate of $f$ and that of the Joseph-Lundgren critical nonlinearity. In fact, we find a general criterion which determines the bifurcation structure; and give a classification of the bifurcation structure.