A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

TL;DR

This paper analyzes the bifurcation structure of radial solutions to semilinear elliptic equations with supercritical growth, deriving a limit equation and using transformations to classify solutions and their intersections.

Contribution

It introduces a limit equation approach for supercritical growth conditions and applies transformations to simplify and analyze bifurcation diagrams.

Findings

Constructed a radial singular solution.

Derived a limit equation at infinity.

Reduced limit equations to two canonical forms.

Abstract

We study radial solutions of the semilinear elliptic equation $\Delta u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\Delta u+u^p=0$ and $\Delta u+e^u=0$.