Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions

TL;DR

This paper studies semilinear elliptic equations with exponential nonlinearities in two dimensions, constructing singular solutions with precise asymptotics and analyzing complex bifurcation structures without requiring analyticity.

Contribution

It introduces a generalized Emden-type transformation to analyze bifurcation diagrams and relaxes conditions for singular solution existence in two-dimensional cases.

Findings

Constructed singular solutions with explicit asymptotics.

Proved bifurcation curves oscillate infinitely many times.

Extended bifurcation analysis without assuming nonlinearities are analytic.

Abstract

In this paper, we investigate semilinear elliptic equations with general exponential-type nonlinearities in two dimensions. For such nonlinearities, we establish two main results. The first is the construction of a singular solution. Recently, Fujishima, Ioku, Ruf, and Terraneo [10] proved the existence of singular solutions under certain assumptions for nonlinearities. We succeed in relaxing these conditions by providing the precise asymptotic form of a singular solution. Our second result concerns the bifurcation diagram of regular solutions. While the bifurcation structure has been extensively studied in three or higher dimensions, comparatively little was known in two dimensions until recently. In [18], the second author proved that the bifurcation curve possesses infinitely many turning points for supercritical analytic nonlinearities. In the present work, we refine this analysis by showing the bifurcation curve oscillates infinitely many times around some point, without assuming analyticity of the nonlinearities. The novelty of our approach lies in the introduction of a generalized Emden-type transformation.