Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions
Hiroaki Kikuchi, Kenta Kumagai
TL;DR
The paper studies radial singular solutions and bifurcation structures for the 2D semilinear elliptic equation $-\,\Delta u=f(u)$ with general exponential-type nonlinearities. It develops a generalized Emden-type transformation and a Lyapunov-energy framework to obtain sharp singular-solution asymptotics and to prove that the regular-solution bifurcation diagram exhibits infinitely many turning points near a critical parameter $\lambda_\infty$, without requiring analyticity of $f$. These results extend prior higher-dimensional and specific-nonlinearity findings to the two-dimensional, Trudinger–Moser-type setting, and provide a robust method for understanding multiplicity and interaction between singular and regular solutions. The approach yields precise first- and second-order expansions for the singular solution $U_\infty$ and establishes intersection properties that underlie infinite multiplicity at the bifurcation point, with potential implications for uniqueness questions in 2D. Overall, the work advances the theory of two-dimensional exponential-type nonlinearities and offers a versatile analytical toolkit for singular- and bifurcation-analysis in radially symmetric settings.
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.