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

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. We find a singular solution explicitly for a suitable weight. Moreover, we find a new Emden-Fowler type transformation for the same weight. These enable us to study the stability of the singular solution and to apply a phase plane analysis. As a result, we classify the bifurcation diagrams completely.