Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions

TL;DR

This paper investigates the bifurcation structure of semilinear elliptic equations with supercritical nonlinearities in two dimensions, revealing the existence of singular solutions and infinitely many turning points in the bifurcation diagram.

Contribution

It extends known bifurcation results from higher dimensions to two dimensions, showing how supercritical nonlinearities influence solution structure and bifurcation behavior.

Findings

No unstable solutions for small positive parameters

Bifurcation curves have infinitely many turning points

Existence of radial singular solutions

Abstract

We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $\lambda$. The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $\lambda=0$. As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in $N$ dimensions with $3\le N \le 9$ and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $2\le N\le 9$.