Bifurcation diagrams for semilinear elliptic equations with singular weights in two dimensions

TL;DR

This paper investigates the bifurcation diagrams of radial solutions to the Gelfand problem with singular weights in two dimensions, demonstrating the realization of all bifurcation types by appropriate weight selection.

Contribution

It is the first to show that in two dimensions, the bifurcation curve can exhibit all three types, including a case with no turning points, by choosing weights carefully.

Findings

Bifurcation curves can exhibit all three types in two dimensions.

Existence of bifurcation curves with no turning points in two dimensions.

Weight manipulation enables diverse bifurcation behaviors.

Abstract

We consider the bifurcation diagram of radial solutions for the Gelfand problem with a positive radially symmetric weight in the unit ball. We deal with the exponential nonlinearity and a power-type nonlinearity. When the weight is constant, it is well-known that the bifurcation curve exhibits three different types depending on the dimension and the exponent of power for higher dimensions, while the curve exhibits only one type in two dimensions. In this paper, we succeed in realizing in two dimensions a phenomenon such that the bifurcation curve exhibits all of the three types, by choosing the weight appropriately. In particular, to the best of the author's knowledge, it is the first result to establish in two dimensions the bifurcation curve having no turning points.